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Our objective in this series of two articles, of which the present article is the ﬁrst, is to give a Perrin-Riou-style construction of p-adic L-functions (of Bellaïche and Stevens) over the eigencurve. As the ﬁrst ingredient, we interpolate the Beilinson–Kato elements over the eigencurve (including the neighborhoods of θ-critical points). Along the way, we prove étale variants of Bellaïche’s results describing the local properties of the eigencurve. We also develop the local framework to construct and establish the interpolative properties of these p-adic L-functions away from θ-critical points. Résumé Ce texte est le premier d’une série de deux articles consacrés à la construction des fonctions Lp-adiques de Bellaïche et Stevens sur la courbe de Hecke par la méthode de Perrin-Riou. Nous commençons par interpoler les éléments de Beilinson–Kato le long de la courbe de Hecke, y compris aux voisinages des points θ-critiques. Chemin faisant, nous prouvons une version étale des résultats de Bellaïche sur la structure locale de la courbe. Nous donnons un cadre pour construire les fonctions Lp-adiques et établir leurs propriétés d’interpolation en dehors des points θ-critiques. Keywords Eigencurve · θ-criticality · Triangulations · Beilinson–Kato elements · p-adic L-functions Mathematics Subject Classiﬁcation 11F11 · 11F67 (primary ) · 11R23 (secondary ) To Bernadette Perrin-Riou on the occasion of her 65th birthday, with admiration. Kâzım Büyükboduk kazim.buyukboduk@ucd.ie Denis Benois denis.benois@math.u-bordeaux.fr Institut de Mathématiques, Université de Bordeaux, 351, Cours de la Libération, 33405 Talence, France UCD School of Mathematics and Statistics, University College, Dublin, Ireland 123 232 D. Benois, K. Büyükboduk Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 1.1 Set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 1.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 1.2.1 θ-critical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 1.2.2 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 2 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 2.1 Tate algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 2.2 Generalized eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 2.3 Specializations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 3 Exponential maps and triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 3.1 Cohomology of (ϕ, )-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 3.2 The Perrin-Riou exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 3.3 Triangulations in families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4 Two variable p-adic L-functions: the abstract deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . 246 4.1 Duality (bis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 4.2 Perrin-Riou-style two variable p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . 247 5 Local description of the eigencurve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.1 Modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.2 Local description of the Coleman–Mazur–Buzzard eigencurve . . . . . . . . . . . . . . . . . 252 5.3 A variant of Bellaïche’s construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6 Interpolation of Beilinson–Kato elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.1 Modular curves and Iwasawa sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.2 Big Beilinson–Kato elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 6.3 Overconvergent étale sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 6.4 Big Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 6.5 Beilinson–Kato elements over the eigencurve . . . . . . . . . . . . . . . . . . . . . . . . . . 270 6.6 Normalized Beilinson–Kato elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 7 “Étale” construction of p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Appendix A: Integrality of normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 A.1 The normalization factors revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 A.2 Dependence on c and d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 A.3 Integrality of partial normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 1 Introduction Let p ≥ 5 be a prime number and let us ﬁx forever an embedding ι : Q → C as well as an isomorphism ι : C − → C . Let us also put ι := ι◦ι .Let f = a ( f )q ∈ S ( ( N)∩ p p ∞ n k+2 1 ( p), ε) be a cuspidal eigenform (for all Hecke operators {T } and {U , } )of 0 N |Np weight k + 2 ≥ 2 with p N . When ord ι (a ( f )) < k + 1, Amice–Vélu in [2]and Višik p p p in [34] have given a construction of a p-adic L-function L ( f , s), which is characterized with the property that it interpolates the critical values of Hecke L-functions attached to (twists) of f . The analogous result in the extreme case when ord ι (a ( f )) = k + 1 (in which case p p p we say that f has critical slope) was established by Pollack–Stevens in [32] and Bellaïche [10]. Note that if ord ι (a ( f )) = k + 1then f is necessarily p-old unless k = 0. It p p p is worthwhile to note that the p-adic L-functions of Pollack–Stevens and Bellaïche are not characterized in terms of their interpolation property, but rather via the properties of the f -isotypic Hecke eigensubspace of the space of modular symbols. The work of Pollack– k+1 Stevens assumes in addition that f is not in the image of the p-adic θ-operator θ := 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 233 k+1 (q(d/dq)) on the space of overconvergent modular forms of weight −k (i.e., f is not θ-critical); Bellaïche’s work removes this restriction. Both constructions in [10,32] take place in Betti cohomology. Our objective in this series of two articles, of which the present article is the ﬁrst, is to recover these results in the context of p-adic (étale) cohomology. More precisely, we shall recast the results of Bellaïche and Pollack–Stevens in terms of the Beilinson–Kato elements and the triangulation over the Coleman–Mazur–Buzzard eigencurve. Along the way, we interpolate the Beilinson–Kato elements over a neighborhood on the eigencurve (including the neighborhoods of θ-critical points). 1.1 Set up We put = ( N) ∩ ( p) and let f ∈ S ( ) be a p-stabilized cuspidal eigenform, p 1 0 k +2 p 0 0 α α α 0 0 0 where U f = α f and N is coprime to p.Welet f be the newform associated to f p 0 0 0 0 0 α α 0 0 (note that it may happen that f = f , in which case f is not of critical slope). We ﬁx 0 0 a real number ν ≥ v (α ),where v (·) is the p-adic valuation on Q normalized so that p 0 p v ( p) = 1. We shall call Hom (Z ,G ) the weight space, which we think of as a rigid analytic cts m space over Q .Let W be a nice afﬁnoid neighborhood (in the sense of [10], Deﬁnition 3.5) about k of the weight space, which is adapted to slope ν (in the sense of [10], §3.2.4). We will adjust our choice of W on shrinking it as necessary for our arguments. Let C be the Coleman–Mazur–Buzzard eigencurve and let x ∈ C be the point that cor- responds to f .Welet C ⊂ C denote an open afﬁnoid subspace of the eigencurve that W,ν lies over W and U acts by slope at most ν. By shrinking W as necessary, there is a unique connected component X ⊂ C that contains x , which is an afﬁnoid neighborhood of x . W,ν 0 0 For any E-valued point x ∈ X ( E) (where E is a sufﬁciently large extension ofQ ,which contains the image of the Hecke ﬁeld of f under the ﬁxed isomorphism ι)ofclassical weight w(x) ∈ Z in the irreducible component X ⊂ C,welet f ∈ S () denote the ≥0 x w(x)+2 corresponding p-stabilized eigenform. We let V denote Deligne’s representation attached to f ; see §6.4.1 for its precise description. There is a natural free O -module V of rank x X 2, which is equipped with an O -linear continuous G -action such that V ⊗ E − → V X Q x X f (c.f. (51)and (55)below). As in [22, §5], let ξ denote either the symbol a( B) with a, B ∈ Z and B ≥ 1 or an element of SL (Z). For each positive integer m coprime to p,weset S := primes(mBp), if ξ = a( B) , where primes( M) stands for the set of prime divisors of primes(mN p), if ξ ∈ SL (Z) M.Let (c, d) be a pair of positive integers satisfying the following conditions: (cd, 6) = 1 = (d, N) and prime(cd) ∩ S =∅. For each integer r and natural number n,welet n n BK ( f , j, r,ξ) ∈ H (Q(ζ ), V (2 − r)) c,d N,mp x mp denote the Beilinson–Kato element associated to the eigenform f , given as in Theo- rem 6.8(iv) (see also Sect. 6.2 for details). For any abelian group G, let us denote by X(G) its character group. We also put (G) := Z [[G]] for its completed group ring. We shall denote by (G) for the free (G)-module −1 × of rank one on which G acts via the character g → g ∈ (G) . For each positive integer m, let us put := Gal(Q(ζ ∞)/Q(ζ )) and set := . Q(ζ ) mp m Q We denote by χ the p-adic cyclotomic character. We put H () forthe imageoftempered 123 234 D. Benois, K. Büyükboduk distributions of order ν under the Amice transform and deﬁne H () := lim H ().Wealso − → set H () := H () ⊗ (?),where (?) = E, O , O . ? W X Fix a generator ε of Z (1). 1.2 Main results We will brieﬂy overview of the results in this article. In a nutshell, our work has two threads: The ﬁrst concerns the interpolation of Beilinson–Kato elements along the eigencurve C,the second concerns p-local aspects, such as the (properties of the) triangulation over C and the formalism of Perrin-Riou exponential maps. The ﬁrst part of Theorem A below belongs to the ﬁrst thread and corresponds to Theorem 6.8 below. The second part corresponds to Theorem 7.3 in the main body of our article and dwells on the second thread, granted the ﬁrst. Theorem A (i) For each m ≥ 1 and coprime to p, there exists a cohomology class [X ] 1 ι BK ( j,ξ) ∈ H (Q(ζ ), V ⊗ ( ) (1)) c,d m Q(ζ ) N,m X m cl which interpolates the Beilinson–Kato classes BK ( f , j, r,ξ) as x ∈ X ( E) c,d N,mp x and integers r, nvary. [X ] 1 ι ii) Suppose that X is étale over W.Welet BK ( j,ξ) ∈ H Z[1/S], V ⊗ () (1) N X denote the partially normalized Beilinson–Kato element, given as in Deﬁnition 6.14 and Proposition 6.15 below. Let L (X ; j,ξ) ∈ H () denote the images of the ±-parts of p X [X ] the class BK ( j,ξ) under the Perrin-Riou dual exponential map (see Deﬁnition 7.1). On shrinking the neighborhood X of x as neccesary, the following hold true. + − (a) There exist ( j ,ξ ) and ( j ,ξ ) such that L (X ; j ,ξ , x ) and L (X ; j ,ξ , x ) + + − − + + 0 − − 0 p p are nonzero elements of H (). cl (b) Assume that ( j ,ξ ) satisfy the conditions in (a). Then for each x ∈ X ( E) such ± ± that v (α(x)) < w(x) + 1, the p-adic L -functions L (X ; j ,ξ , x) agree with the p ± ± Manin–Višik p-adic L -functions attached to f , up to multiplication by D E (x) x N ± × where D ∈ E and E (x) is the product of bad Euler factors, given as in (64). (c) Assume that v (α ) = k + 1, but f is not θ -critical. Then L (X ; j ,ξ , x ) agree p 0 0 ± ± 0 with the one-variable p-adic L -functions of Pollack–Stevens [32] up to multiplication ± ± × by D E (x),where D ∈ E . (d) Let L (X, ) denote the two-variable p-adic L -functions of Bellaïche and Stevens associated to modular symbols ∈ Symb (X)(c.f. [10], Theorem 3). Then there ± × exist functions u (x) ∈ O such that ± ± ± L (X ; j ,ξ ) = u (x)E (x)L (X, ). ± ± N p [X ] See Deﬁnition 6.7(iii) where the class BK ( j,ξ) is introduced and see Deﬁnition 6.14 c,d N,m where we introduce its partial normalization (as well as Appendix A where we establish the regularity properties of this normalization). We call the p-adic L-function L (X ; j,ξ) ∈ H () given as in Deﬁnition 7.1(i) (which is denoted by L (f ; j,ξ) in the main body X p,η of our text) the arithmetic p-adic L-function. Remark 1.1 Results similar to Theorem A are proved in the recent paper of Wang [35]via different methods. Some parts of this result has been also announced in the independent 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 235 preprints of Hansen [17] and Ochiai [29]. Our approach to establish Theorem A(i) (which applies even when X is not étale over W; this is crucial for our results in the θ-critical case) is a synthesis of the techniques of [17,26], relying on the overconvergent étale cohomology of Andreatta–Iovita–Stevens [3] and Kings’ theory of -adic sheaves developed in [23]. Along the way, we verify (relying on the results of Ash–Stevens and Pollack–Stevens) that the construction of the local pieces (including neighborhoods of θ-critical points, where X is no longer étale over W) of the cuspidal eigencurve using different types of distribution spaces over the weight space, or compactly vs non-compactly supported cohomology produce the same end result. See Sect. 5.3, §5.1.5 and Sect. 6.3 for further details (most particularly, Lemma 5.2, Proposition 5.8(ii) and Proposition 6.5) concerning this technical point, which we believe is of independent interest. Remark 1.2 We note that in the particular case when f = f is a newform of level (Np) 0 0 k /2 and weight k + 2 with α = a ( f ) = p , the conclusions of Theorem 7.3 play a crucial 0 0 p 0 role in [6]. 1.2.1 -critical case We conclude Sect. 1.2 with a brief summary of our results in our companion article [5], where we focus on the θ-critical scenario (i.e. in the situation when X is not étale over W) and give an étale construction of Bellaïche’s p-adic L-function. The ﬁrst key ingredient in [5] is Theorem A(i), namely the construction of a big Beilinson– Kato class about a θ-critical point on the eigencurve. The local aspects turn out to be signiﬁcantly more challenging when X fails to be étale over W. To prove that the Perrin- Riou style p-adic L-functions L (X ; j,ξ), deﬁned in an analogous way, has the required properties, we introduce in op. cit. a new local argument (called the “eigenspace transition via differentiation”) and prove the following results (which we state in vague form to avoid digression and refer readers to [5] for details): Theorem B Suppose that the ramiﬁcation index of X over W at x equals to 2.Asbefore, [X ] we let L (X ; j,ξ) ∈ H () denote the images of the ±-parts of the class BK ( j,ξ) p X under the Perrin-Riou dual exponential map. There exists two pairs ( j ,ξ ) and ( j ,ξ ) + + − − with the following properties: ± r (i) We have L (x ; j ,ξ ,ρχ ) = 0 for all integers 1 ≤ r ≤ k + 1 and characters 0 ± ± 0 ρ ∈ X() of ﬁnite order. (ii) We deﬁne the improved arithmetic p-adic L -functions at the critical point x on setting [1],± ± L (x ; j ,ξ ) := L ( X ; j ,ξ ) ∈ H () 0 ± ± ± ± E p p ∂ X X =0 Here, as in [10, §4.4], we denote by X a ﬁxed choice of a uniformizer of X about x and consider the p-adic L -functions L (X ; j,ξ) in the neighborhood X of x as the functions L ( X ; j,ξ) with X in a neighborhood of 0. Then the improved arithmetic p- adic L -functions verify the same interpolation property that Bellaïche’s improved p-adic L -functions do, up to the bad Euler factors E (x ) and constants that depend only on N 0 the choices of Shimura periods. In an unpublished note (see [9] Proposition 1), Bellaïche explains that a conjecture of Jannsen combined with Greenberg’s conjecture (that “locally split implies CM”) and the important of result of Breuil-Emerton (“θ-critical implies locally split”) would yield e = 2. We are grateful to R. Pollack for bringing Bellaïche’s note to our attention. 123 236 D. Benois, K. Büyükboduk This theorem supplies us with a new construction of Bellaïche’s p-adic L-functions (with Euler factors at primes dividing the tame conductor removed). One of the consequences of the étale construction of the p-adic L-functions is the leading term formulae for these p-adic L-functions. These will be explored in the sequels to the present article. 1.2.2 Layout We close our introduction reviewing the layout of our article. After a very general preparation in Sect. 2 (where we axiomatise various constructions in [10], §4.3), we give a general overview of triangulations in Sect. 3. In Sect. 3.2,wealsodeﬁne the Perrin-Riou exponential map, which is one of the crucial inputs deﬁning the “arithmetic” p-adic L-functions. We then introduce Perrin-Riou-style (abstract) two-variable p-adic L-functions in Sect. 4 and study their interpolative properties. These results are later applied in Sect. 7 in the context of the Coleman–Mazur–Buzzard eigencurve and with the Beilinson–Kato element [X ] BK ( j,ξ) of Theorem A. c,d N,m In Sect. 5, we review Bellaïche’s results in [10] on the local description of the eigencurve and prove variants (in Sect. 5.3) of these results involving slightly different local systems as coefﬁcients and non-compactly supported cohomology (at a level of generality that cov- ers also the neighborhoods of θ-critical point). We utilize these in Sect. 6 (together with the work of Andreatta–Iovita–Stevens [3]) to deduce the required properties for the Galois representation V , where the interpolated Beilinson–Kato elements take coefﬁcients in. The variants of Bellaïche’s construction that we discuss in Sect. 5.3 allow us (through Proposi- tion 6.5) to establish the properties of V as an O -module, where the images of Perrin-Riou X X exponential maps take coefﬁcients in. In Sect. 6.5, we introduce the interpolated Beilinson–Kato elements (which are denoted [X ] by BK ( j,ξ) in Theorem A(i)) as part of Deﬁnition 6.7(iii) (see also Sect. 6.6 where c,d N,m we introduce their normalized versions). Our construction builds primarily on the ideas in [23,24,26]. As we have noted in Remark 1.1, our approach in this portions is, in some sense, a synthesis of the techniques of [26]and [17] (which we crucially enhance to apply also about θ-critical points). In Sect. 7, we apply the general results in Sect. 4 to deﬁne the “arithmetic” p-adic L-functions and study their interpolation properties, proving our Theorem A(ii). 2 Linear algebra In Sect. 2, we ﬁx some notation and conventions from linear algebra, which will be used in the remainder of the paper (as well as in the companion paper [5]). We also axiomatize various constructions in [10, §4.3]. 2.1 Tate algebras Let E be a ﬁnite extension of Q . Fix an integer e ≥ 1 and denote by R the Tate algebra re e R = E Y / p , where r ≥ 0 is some ﬁxed integer. Let A = R[ X ]/( X − Y ). Then A = E X/ p Set W = Spm( R) and X = Spm( A). We will consider W as a weight space in the re−1 re−1 following sense. Fix an integer k ≥ 2 and denote by D(k , p ) = k + p Z the 0 0 0 p re−1 re−1 closed disk with center k and radius 1/ p . We identify each y ∈ D(k , p ) with the 0 0 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 237 point of W that corresponds to the maximal ideal y−k m = (1 + Y ) − (1 + p) . We have a canonical morphism w : X → W, (1) which we will call the weight map. If x ∈ X , we say that y = w(x) is the weight of x. Let x ∈ X denote the unique point such that w(x ) = k . Set 0 0 0 cl X ={x ∈ X | w(x) ∈ Z,w(x) ≥ 2}. 2.2 Generalized eigenspaces 2.2.1. Let M be an A-module. For any x ∈ X ( E) and y ∈ W( E) deﬁne M = M ⊗ A/m , x A x M = M ⊗ R/m . y R y We denote by π : M −→ M x w(x) x the canonical projection. Consider X as a function on X and denote by X(x) the value of X at x ∈ X ( E). For any endomorphism f of M, set M [ f ]= ker( f ). Deﬁnition 2.1 We denote by M [x]= M [ X − X(x)], the submodule of M annihilated w(x) w(x) by X − X(x) ∈ A. We also put M [[x ]] := ∪ M [( X − X(x)) ] and call it the generalized w(x) n≥1 eigenspace associated to x. ∼ ∼ 2.2.2. Suppose that M A is a free A-module of rank d,sothatwealsohave M = = e e X − X(x) e d ( R[ X ]/( X − Y )) . Let us put P ( X) := ∈ E [ X ]. Then, X − X(x) e e M = E [ X ]/( X − X(x) ) E [ X ] , w(x) e e M [x]= P ( X) E [ X ]/( X − X(x) ) E [ X ] , M = ( E [ X ]/( X − X(x))) , and the multiplication by P ( X) gives an isomorphism M − −−−→ M [x ]. × P ( X) We consider separately the following cases: e e (a) When x = x (so w(x) = k ), then the polynomial X − X(x) is separable. In this 0 0 scenario, the A-module M is semi-simple and M [x]= P ( X) M ⊂ M is an w(x) x w(x) w(x) A-direct summand. The restriction of the natural surjection π : M −→ M to M [x ] x w(x) x is an isomorphism, which we shall denote with the same symbol unless there is a chance of confusion. Also, M [[x ]] = M [x ]. 123 238 D. Benois, K. Büyükboduk e−1 (b) When x = x ,wehave X(x ) = 0and P ( X) = X . In this case, M = 0 0 x k 0 0 e d ( E [ X ]/ X E [ X ]) as an A-module and e d M = M [[x ]] −→ ( E [ X ]/ X E [ X ]) , k 0 e−1 M [x ]= M [ X]= X M , 0 k k 0 0 M −→ E . If e ≥ 2, the restriction π | : M [x ]→ M of the surjection π is evidently the x 0 x x 0 M [x ] 0 0 zero map. Observe also that in either of the cases (a) or (b) above, M [x ] can be identiﬁed with an A-equivariant image of M under the multiplication-by- P ( X) map. w(x) x 2.3 Specializations Let M be an A-module, and let x ∈ X ( E). For any m ∈ M, we denote by m ∈ M the x x image of m under the specialization map M → M . On the other hand, M = M ⊗ A x A R has the A-module structure via the A-action on the second factor. For any x ∈ X ( E),the specialization of M at x is ( M ) := ( M ⊗ A) ⊗ E = M ⊗ E = M . A R A,x R,w(x) w(x) Deﬁnition 2.2 For an A-module M and x ∈ X ( E),welet sp denote the specialization map M → ( M ) . A A x We need the following lemma. Lemma 2.3 In the notation of Deﬁnition 2.2, suppose that M is a free A-module of rank one. e−1 i e−1−i Let m denote a generator of M and put := X m ⊗ X ∈ M ⊗ A. Then i =0 (i) We have ( X ⊗ 1) = (1 ⊗ X). (ii) The element sp () ∈ M generates the E -vector space M [x ]. w(x) e−1 (iii) We have π ◦ sp () = X(x) m . x x Proof The ﬁrst part is the abstract version of [10, Lemma 4.13] and the second part is that of [10, Proposition 4.14]. Final assertion follows from a direct calculation. Remark 2.4 Suppose in this remark that e = 1, so that A = R. In this scenario, we have M = M and M − → M . Moreover, the specialization map sp is simply the canonical A w(x) x projection M → M . 2.4 Duality Let M and M be two free A-modules of ﬁnite rank equipped with an R-linear pairing (, ) : M ⊗ M → R. Assume that this pairing satisﬁes the following property: Adj) For every m ∈ M and m ∈ M,wehave (Xm , m) = (m , Xm). 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 239 For any x ∈ X ( E), we denote by (, ) : M ⊗ M → E w(x) E w(x) w(x) the specialization of (, ) at w(x). Lemma 2.5 There exists a unique pairing (, ) : M ⊗ M [x]→ E x E such that the restriction of (, ) to the subspace M ⊗ M [x ] factors as w(x) E w(x) (, ) w(x) M ⊗ M [x ] E w(x) (2) π ⊗id (, ) x x M ⊗ M [x ] Proof If x = x , the vector space M is a direct summand of M , and in this case, (, ) 0 x w(x) coincides with the restriction of (, ) on M ⊗ M [x ]. w(x) E e−1 If x = x , then M = M /XM and M [x ]= X M . The property Adj) shows that 0 0 k x 0 (, ) is trivial on XM ⊗ M [x ], and therefore it factorizes uniquely through M ⊗ M [x ]. k 0 E 0 0 k x 0 0 3 Exponential maps and triangulations 3.1 Cohomology of (', 0)-modules 3.1.1. We set K = Q (ζ n), K = ∪ K and = Gal(Q (ζ ∞)/Q ).For any n ≥ 1 n p p ∞ n p p p n=0 we set = Gal(K /K ), and put G = Gal(K /Q ). Let us ﬁx topological generators n ∞ n n n p p p−1 γ ∈ such that γ = γ for all n ≥ 1and γ = γ . For any group G and left n n n+1 1 G-module M we denote by M the right G-module whose underlying group is M and on −1 which G acts by m · g = g m. If G is a ﬁnite abelian group, we denote by X(G) its group of characters. If E is a ﬁxed ﬁeld such that ρ ∈ X( E) takes values in E, we denote by −1 e = ρ (g)g |G| g∈G (ρ) the corresponding indempotent of E [G]. For any E [G]-module M we denote by M = e M (ρ) its ρ-isotypic component. For any map f : N → M we denote by f the compositum [e ] f ρ (ρ) (ρ) f : N − → M − − → M . 3.1.2. In this section, we review the construction of the Bloch–Kato exponential map for crystalline (ϕ, )-modules. Let E be a ﬁnite extension ofQ . For each n ≥ 0, we denote by R the Robba ring of formal power series f (π) = a π coverging on some annulus E m m∈Z of the form r( f ) ≤|π | < 1. Recall that R := R ∩ E [[π ]] is the ring of formal power p E 123 240 D. Benois, K. Büyükboduk series converging on the open unit disk. We equip R with the usual E-linear actions of the Frobenius operator ϕ and the cyclotomic Galois group given by ϕ(π) = (1 + π) − 1, χ(γ ) γ(π) = (1 + π) − 1,γ ∈ . Let ψ denote the left inverse of ϕ deﬁned by ⎛ ⎞ −1 ⎝ ⎠ ψ( f (π)) = ϕ f (ζ(1 + π) − 1) . ζ =1 ψ =0 Then R is a H ()-module of rank one, generated by 1 +π. The differential operator ψ =0 ∂ = (1 + π) is a bijection from R to itself. Furthermore, we have dπ ∂ ◦ γ = χ(γ )γ ◦ ∂. Let t = log(1 + π) denote the “additive generator of Z (1)”. Recall that ϕ(t) = pt ,γ(t) = χ(γ )t,γ ∈ . Let A = E X/ p beaTatealgebra over E as above.Wedenoteby R = A ⊗ R the A E E Robba ring with coefﬁcients in A. The action of , ϕ, ψ and ∂ can be extended to R by linearity. 3.1.3. Recall that a (ϕ, )-module over R is a ﬁnitely generated projective module over R equipped with commuting semilinear actions of ϕ and and satisfying some additional technical properties which we shall not record here (see [12,25] for details). The cohomology H (K , D) of D over K is deﬁned as the cohomology of the Fontaine–Herr complex n n d d 0 1 C (D) : 0 −→ D − → D ⊕ D − → D −→ 0, ϕ,γ where d (x) := ((ϕ − 1)x,(γ − 1)x) and d ( y, z) := (γ − 1) y − (ϕ − 1)z. 0 n 1 n 3.1.4. For any (ϕ, )-module D over R we set D (D) = (D[1/t ]) . Recall that D (D) A cris cris is a ﬁnitely generated free A-module equipped with an A-linear frobenius ϕ and a decreasing ﬁltration Fil D (D) . In [27], Nakamura deﬁned A-linear maps cris i ∈Z exp : D (D) ⊗ K −→ H (K , D), n ≥ 0 cris Q n n D,K p which extends the deﬁnition of Bloch–Kato exponential maps to (ϕ, )-modules. 3.1.5. Let V be a p-adic representation of G with coefﬁcients in A. The theory of (ϕ, )- † † modules associates to V a (ϕ, )-module D (V ) over R . The functor D is fully rig, A rig, A faithul and we have functorial isomorphisms i i H (K , V ) H (K , D (V )). n n rig, A If V is crystalline in the sense of [12], we have a functorial isomorphism between the “clas- sical” ﬁltered Dieudonné module D (V ) associated to V and D (D (V )). Moreover, cris cris rig, A the diagram exp V ,K D (V ) ⊗ K H (K , V ) cris Q n n exp D (V ),K rig, A † † D (D (V )) ⊗ K H (K , D (V )), cris Q n n rig, A p rig, A 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 241 where the upper row is the Bloch–Kato exponential map, commutes. 3.1.6. We refer the reader to [25, Section 4.4] and [30] for the proofs of the results reviewed in this subsection and for further details. Let H denote the algebra of formal power series f (z) = a z with coefﬁcients in E that converges on the open unit disk. We put j =0 H ( ) := { f (γ − 1) | f ∈ H } , H () := E []⊗ H ( ), E 1 1 E E E E 1 H () := A ⊗ H (), A E E where = Gal(Q (ζ )/Q ).Notethat H () contains the Iwasawa algebra := p p p A A A ⊗ Z [[]]. Z p The Iwasawa cohomology H (Q , D) of a (ϕ, )-module D over R is deﬁned as the p A Iw cohomology of the complex ψ −1 D −−→ D 1 ψ =1 concentrated in degrees 1 and 2. In particular, H (Q , D) = D . We have canonical Iw projections 1 1 pr : H (Q , D) → H (K , D), n ≥ 0. p n n Iw If D = D (V ) is the (ϕ, )-module associated to a p-adic representation V over A,we rig, A then have a functorial isomorphism 1 1 † H () ⊗ H (Q , V ) H (Q , D (V )), (3) A p p A Iw Iw rig, A where H (Q , V ) denotes the classical Iwasawa cohomology with coefﬁcients in V.The Iw isomorphism (3) composed with the projections pr coincide with the natural morphisms 1 1 H () ⊗ H (Q , V ) −→ H (K , V ) A p n A Iw 1 1 induced by the Iwasawa theoretic descent maps H (Q , V ) → H (K , V ). p n Iw For each integer m, the cyclotomic twist 1 1 ⊗m Tw : H (Q , V ) → H (Q , V (m)), Tw (x) = x ⊗ ε m p p m Iw Iw can be extended to a map 1 1 Tw : H () ⊗ H (Q , V ) −→ H () ⊗ H (Q , V (m)) m A p A p A Iw A Iw −m f (γ − 1) ⊗ x −→ f (χ (γ )γ − 1) ⊗ Tw (x). 3.2 The Perrin-Riou exponential map 3.2.1. In this subsection, we shall review fragments of Perrin-Riou’s theory of large expo- nential maps. Deﬁne the operators log(γ ) ∇= ,γ ∈ , 1 1 log χ(γ ) = j −∇, j ∈ Z. Note that ∇ does not depend on the choice of γ ∈ . It is easy to check by induction that 1 1 h−1 h h h = (−1) t ∂ (on R ). j E j =0 123 242 D. Benois, K. Büyükboduk Let V be a p-adic crystalline representation of G with coefﬁcients in E satisfying the following condition: ϕ= p LE) D (V ) = 0for all i ∈ Z. cris Note that this assumption can be relaxed (see, for example, [8,31]), but it simpliﬁes the 0 1 presentation. In particular, it implies that H (K , V ) = 0 and therefore also that H (Q , V ) ∞ p Iw is torsion free over the Iwasawa algebra (c.f. [31], Lemme 3.4.3 and Proposition 3.2.1). Let us set ψ =0 D(V ) = R ⊗ D (V ). E cris For any α ∈ D(V ), the equation (1 − ϕ)( F) = α has a unique solution F ∈ R ⊗ D (V ). We deﬁne the maps E cris : D(V ) → D (V ) ⊗ K , n ≥ 0 V ,n cris Q n on setting −n −n p (id ⊗ ϕ) ( F)(ζ n − 1) if n ≥ 1, −1 −1 (α) = V ,n 1 − p ϕ α(0) if n = 0. 1 − ϕ The following is the main result of [31]. Theorem 3.1 (Perrin-Riou) Let V be a crystalline representation which satisﬁes the condition −h LE) above. Then for integers h ≥ 1 such that Fil D (V ) = D (V ), there exists a cris cris H ()-homomorphism Exp : D(V ) −→ H () ⊗ H (Q , V ) E p V ,h E Iw satisfying the following properties: (i) For all n ≥ 0 the following diagram commutes: Exp V ,h D(V ) H () ⊗ H (Q , V ) E p E Iw pr V ,n n (h−1)! exp V ,K D (V ) ⊗ K H (K , V ). cris n n −1 (ii) Let us denote by e := ε ⊗ t the canonical generator of D (Q (−1)).Then −1 cris p Exp =−Tw ◦ Exp ◦ (∂ ⊗ e ). 1 −1 V (1),h+1 V ,h (iii) We have Exp = Exp . V ,h+1 V ,h Let us put Exp = pr ◦ Exp : D(V ) −→ H (K , V ). (4) V ,h,n n V ,h 3.2.2. Set G = Gal(K /K ). Without loss of generality, we can assume that the characters of n n G take values in E. Recall that Shapiro’s lemma gives an isomorphism of E [G ]-modules n n 1 1 ι H (K , V ) −→ H (Q , V ⊗ E [G ] ). n p E n 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 243 On taking the ρ-isotypic components, we obtain isomorphisms 1 (ρ) 1 −1 H (K , V ) −→ H (Q , V (ρ )), ρ ∈ X(G ). n p n We shall make use of the following elementary lemma giving the ρ-isotypic component of the map . V ,n Lemma 3.2 Assume that ρ ∈ X(G ) is a primitive character. Then for any α = f (π) ⊗ d ∈ D(V ) −n −n e ( f (ζ − 1)) ⊗ p ϕ (d) if n ≥ 1, ρ p (ρ) (α) = V ,n −1 −1 −1 (1 − p ϕ )(1 − ϕ) (d) if n = 0. Proof See [13, Lemma 4.10]. 3.2.3. We recall the explicit construction of the Perrin-Riou exponential that was discovered by Berger in [12]. Let, as before, F ∈ R ⊗ D (V ) denote the solution of the equation cris (1 − ϕ) F = α. Deﬁne log χ(γ ) (α) := − ··· ( F(π)). V ,h h−1 h−2 0 ψ =1 It is not difﬁcult to see that (α) ∈ D (V ) and an explicit computation shows that V ,h rig, E satisﬁes properties i–iii in Theorem 3.1. V ,h See also [27] for a generalization of this approach to de Rham representations. 3.2.4. In the remainder of Sect. 3.2, we shall review the construction of the Perrin-Riou exponential maps for families of (ϕ, )-modules of rank one. × × For any continuous character δ : Q → A we denote by D = R e the (ϕ, )-module δ A δ of rank one over the relative Robba ring R , generated by e and such that A δ ϕ(e ) = δ( p)e ,γ(e ) = δ(χ(γ ))e γ ∈ . δ δ δ δ + + Set D := R e . We say that D is of Hodge–Tate weight −m ∈ Z (sic!) if δ δ δ A m × δ(u) = u , ∀u ∈ Z . If D is of Hodge–Tate weight −m, then D (D ) is the free A-module of rank one generated δ cris δ −m by d = t e . It has the unique ﬁltration break at −m, and ϕ acts on D (D ) as the δ δ cris δ −m multiplication by p δ( p) map. 3.2.5. Let D be a (ϕ, )-module of rank one and Hodge–Tate weight −m ∈ Z. The direct analogue of the condition LE) for families of (ϕ, )-modules is the following condition: LE*) For any i ∈ Z,1 − p δ( p) ∈ A does not vanish on X = Spm( A). For any point x ∈ Spm( A), let D denote the specialization of D at x. To facilitate a δ,x δ comparison with LE), we remark that the condition LE*) implies that ϕ= p D (D ) = 0, ∀x ∈ Spm( A), ∀i ∈ Z. cris δ,x ψ =0 Let us put D(D ) = R ⊗ D (D ). We shall explain the construction of a family δ A cris δ of maps (which we will call Perrin-Riou exponential maps) Exp : D(D ) −→ H (Q , D ), h ≥ m δ p δ D ,h Iw modelled on the discussion in §3.2.3. Let us set α(π) := f (π) ⊗ d ∈ D(D ). The equation δ δ (1 − δ( p)ϕ) F (π) = ∂ f (π) 123 244 D. Benois, K. Büyükboduk has a unique solution in R . It is easy to see that ψ =1 F (π) ⊗ e ∈ D = H (Q , D ). −m δ p δ δ Iw We deﬁne log χ(γ ) m−1 Exp (α) := (−1) F (π) ⊗ e , m δ D ,m and set ⎛ ⎞ h−1 ⎝ ⎠ Exp (α) = ◦ Exp (α), h ≥ m + 1. D ,h D ,m δ δ j =m The following result can be extracted from [28, Section 4] or proved directly using Berger’s arguments. Proposition 3.3 (i) For any h ≥ m we have Exp = Exp . D ,h+1 D ,h δ δ (ii) For any h ≥ m the following diagram commutes: Exp δχ,h+1 D(D ) H (Q , D ) δχ p δχ Iw e ⊗∂ −Tw Exp D ,h D(D ) H (Q , D ) δ p δ Iw ψ =0 (iii) Assume that h ≥ m ≥ 1. For any α(π) = f (π)⊗ d ∈ R ⊗ D (D ), the equation δ A cris δ (1 − ϕ)( F) = α(π) has a unique solution F ∈ D (D ) ⊗ R and it veriﬁes cris δ h−1 log χ(γ ) Exp (α) =− ( F). D ,h j =0 Moreover, h−1 ψ =1 h−1 h h h−m ( F) = (−1) t ∂ ( F) ∈ t D . j =0 (iv) Under the conditions and notation of iii), let us put −1 −1 1− p ϕ α(0) if n = 0, 1−ϕ (α) := D ,n (m−1)n −n p δ( p) F(ζ − 1) if n ≥ 1. Then the map : D(D ) → D (D ) ⊗ K is surjective, and the diagram D ,n δ cris δ Q n δ p Exp D ,h D(D ) H (Q , D ) δ p δ Iw pr D ,n n (h−1)! exp D ,K D (D ) ⊗ K H (K , D ) cris δ Q n n δ commutes. 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 245 3.3 Triangulations in families 3.3.1. In Sect. 3.3, we shall work in the setting of Sect. 2 and introduce various objects with which we shall apply the general constructions in Sect. 2. To that end, let us ﬁx an integer e re e ≥ 1 and put A = R[ X ]/( X − Y ) as before, where R = E Y / p is a Tate algebra. Set W = Spm( R), X = Spm( A) and denote by w : X → W the weight map. We ﬁx an integer re−1 k ≥ 2 and identify W with the closed disk D(k , p ) as in Sect. 2.1.Let x ∈ X denote 0 0 0 the unique point such that w(x ) = k . We put 0 0 cl X ={x ∈ X | w(x) ∈ Z,w(x) ≥ 0}. We let V denote a free A-module of rank 2 which is endowed with a continuous action of the Galois group G . In accordance with the notation of Sect. 2,for any y ∈ W( E) Q,S and x ∈ X,weset V = V ⊗ E, V = V /m V and denote by π : V → V the y R,w x x x w(x) x e e−1 e canonical projection. Note that V = V / X V , V = V /XV and V [x ]= X V / X V . k x 0 0 0 3.3.2. We shall assume that V veriﬁes following conditions: cl C1) For each x ∈ X of integer weight w(x) ≥ 0 the restriction of V on the decomposition group at p is semistable of Hodge–Tate weights (0,w(x) + 1). cl ϕ=α(x) C2) There exists α ∈ A such that for all x ∈ X the eigenspace D (V ) is one st x dimensional. cl C3) For each x ∈ X −{x } ϕ=α(x) w(x)+1 D (V ) ∩ Fil D (V ) = 0. st x st x On shrinking X as necessary, it follows from [25] that one can construct a unique (ϕ, )- submodule D ⊆ D (V ) of rank one with the following properties: rig, A × × ϕ ) D = D with δ : Q → A such that δ( p) = α(x) and δ| × = 1. 1 δ ϕ=α(x) cl ϕ ) D (D ) = D (V ) for each x ∈ X ,where D := D ⊗ E. 2 cris x st x x A,x cl ϕ ) For each x ∈ X −{x }, the (ϕ, )-module D is saturated in D (V ). 3 0 x x rig, E sat Let D denote the saturation of the specialization D of the (ϕ, )-module D at x . x 0 x 0 sat Then, as explained in the ﬁnal section of [25], we have D = t D for some m ≥ 0. We 0 x will consider the following two scenarios: ¬ ϕ=α(x ) k +1 0 0 ) D (V ) ∩ Fil D (V ) = 0. st x st x 0 0 sat ϕ=α(x ) ¬ Since D (D ) = D (V ) , the condition ) implies that the Hodge–Tate weight cris st x x 0 ϕ=α(x ) of D (V ) is 0. Since the Hodge–Tate weight of D is also 0, we deduce that in this st x x 0 0 case m = 0and D is saturated in D (V ). x x rig, E ϕ=α(x ) k +1 0 0 ) D (V ) = Fil D (V ). st x st x 0 0 sat k +1 sat Suppose that ) holds. Then D (D ) = Fil D (V ). Therefore D is of Hodge–Tate cris st x x 0 x 0 0 weight k + 1and m = k + 1. Let β(x ) denote the other eigenvalue of ϕ on D (V ). By 0 0 0 st x the weak admissibility of D (V ),weinfer that v (α(x )) ≥ k + 1, v (β(x )) ≥ 0and st x p 0 0 p 0 that v (α(x )) + v (β(x )) = k + 1. Thence, v (α(x )) = k + 1and v (β(x )) = 0, p 0 p 0 0 p 0 0 p 0 ϕ=α(x ) ϕ=β(x ) 0 0 and the eigenspaces D (V ) and D (V ) are weakly admissible. They are st x st x 0 0 2 ¬ In the context of elliptic modular forms, the condition ) (resp., )) translates into the requirement that the corresponding eigenform is non-θ critical (resp., is θ-critical) in the sense of Coleman. 123 246 D. Benois, K. Büyükboduk therefore admissible and the restriction of V to the decomposition group at p decomposes into a direct sum (α) (β) V = V ⊕ V (5) 0 x x 0 0 † (α) (α) (β) of two one-dimensional G -representations. Let us D = D (V ) and D = Q x p 0 rig, E (β) D (V ). Then rig, E 0 (α) (α) (β) (β) D = R e , D = R e , E E where (α) −1−k (α) (α) −1−k (α) 0 0 γ(e ) = χ (γ )e ,ϕ(e ) = α(x ) p e , (β) (β) (β) (β) γ(e ) = e,ϕ(e ) = β(x )e,γ ∈ . We have (α) ϕ=α(x ) D (V ) = D (V ) , cris st x x 0 (β) ϕ=β(x ) D (V ) = D (V ) , (6) cris st x x 0 (α) sat −(k +1) D = D = t D . x 0 (α) We remark that D (D ) and D (V ) areisomorphicas ϕ-modules but not as ﬁltered cris x cris x 0 0 modules: they have Hodge–Tate weights 0 and k + 1, respectively. 4 Two variable p-adic L-functions: the abstract deﬁnitions Suppose that we are given another free A-module V of rank two which is equipped with a continuous G -action, together with a Galois equivariant R-linear pairing Q,S (, ) : V ⊗ V −→ R (7) satisfying the condition Adj) of Sect. 2: For every v ∈ V and v ∈ V , we have ( Xv ,v) = (v , Xv). 4.1 Duality (bis) We have a canonical H ()-bilinear pairing 1 1 ι : H () ⊗ H (Q , V (1)) ⊗ H () ⊗ H (Q , V ) −→ H (), (8) R p R p R R Iw R Iw which has the following explicit description. Let 1 1 ( , ) : H (K , V (1)) × H (K , V ) −→ R, n ≥ 0 n n denote the cup-product pairings induced by (7). Recall from §3.1.6. the projection maps 1 1 pr : H () ⊗ H (Q , V ) −→ H (K , V ), R p n n R Iw 1 1 pr : H () ⊗ H (Q , V (1)) −→ H (K , V (1)), n ≥ 0. R p n n R Iw We then have −1 x, y ≡ τ pr (x), pr ( y) τ mod (γ − 1) (9) n n τ ∈/ 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 247 (see [31], Section 3.6). In particular, for any ﬁnite character ρ ∈ X() of conductor p ,we have ι −1 ι ρ x, y = τ pr (x), pr ( y) ρ(τ) = e (x), e ( y ) , ρ ρ n n n ρ,0 τ ∈/ where ( , ) stands for the cup-product pairing ρ,0 ( , ) ρ,0 1 −1 1 H (Q , V (χρ )) ⊗ H (Q , V (ρ)) − −−→ R. p p Note that the pairing (8) can be recast entirely in terms of the Iwasawa cohomology of associated (ϕ, )-modules; c.f. [25, Section 4.2]. 4.1.1. We apply the formalism of Sect. 2.4 to our situation. Equip the tensor product 1 1 A ⊗ H () ⊗ H (Q , V ) H () ⊗ H (Q , V ) R p A p R R Iw R Iw with the action of A through the ﬁrst factor. We extend the pairing (8) by linearity to the pairing 1 1 ι : H () ⊗ H (Q , V (1)) ⊗ H () ⊗ H (Q , V ) −→ H (), R p H () A p A A R Iw R R Iw (10) which is H ()-linear (respectively, H ()-linear) with respect to the ﬁrst (respectively R A second) factor. For any x ∈ X ( E), the functoriality of the cup-products gives rise to the following commutative diagram: 1 1 ι H () ⊗ H (Q , V (1)) ⊗ H () ⊗ H (Q , V ) H () R p H () A p A R Iw R R Iw sp w(x) x x w(x) 1 1 ι H () ⊗ H (Q , V (1)) ⊗ H () ⊗ H (Q , V ) H (). E p H () E p w(x) E E Iw w(x) E E Iw (11) where the left and the right vertical maps are the specializations at w(x) and x respectively, and the middle vertical map is the specialization deﬁned in40314031 Sect. 2.4. By Lemma 2.5,for any x ∈ X ( E) the pairing (, ) induces a pairing (, ) : V ⊗ V [x]−→ E. x x E The pairing in the bottom row of (11) therefore factors as (12) 4.2 Perrin-Riou-style two variable p-adic L-functions Recall from §3.2.5 the Perrin-Riou exponential Exp : D(V ) −→ H (Q , D). D,h Iw 123 248 D. Benois, K. Büyükboduk Note that we have a canonical injection 1 1 1 H (Q , D) −→ H () ⊗ H (Q , V ) = H () ⊗ H (Q , V ). p R p R p Iw R Iw R Iw Let us ﬁx an R-module generator η ∈ D (D). cris Let us set D(V ) = D(V ) ⊗ A. We have a canonical isomorphism A R ψ =0 D(V ) D (D) ⊗ R A cris R The Perrin-Riou exponential map Exp can be extended by linearity to an A-linear map D,h Exp ⊗id D,h A 1 Exp : D(V ) − −−−−−− → H (Q , D) ⊗ A. A p R D,h Iw Note that we have a canonical injection 1 1 1 H (Q , D) ⊗ A −→ A ⊗ H () ⊗ H (Q , V ) = H () ⊗ H (Q , V ). p R R p A p Iw R R Iw R Iw Fix an A-module generator η ∈ D (D) and set cris e−1 i e−1−i ” := X η ⊗ X ∈ D (D) ⊗ A, cris R i =0 ” := ” ⊗ (1 + π) ∈ D(V ) . We remark that for each x ∈ X ( E), the specialization ” := sp (”) is a generator of D (D)[x ] by Lemma 2.3. Let us put ” := ” ⊗ (1 + π). cris x x Deﬁnition 4.1 (i) For each h ≥ 0 and cohomology class z ∈ H (Q , V (1)),wedeﬁne Iw A 1 L : H (Q , V (1)) −→ H (), p A D,”,1−h Iw A A ι L (z) := z, c ◦ Exp (”) . D,”,1−h D,h (ii) We deﬁne the two-variable p-adic L-function associated to z on setting L (z) := L (z) ∈ H (). p,” A D,”,1 cl For each x ∈ X ( E), we similarly deﬁne the one-variable p-adic L-function L (z ) := z , c ◦ Exp (” ) ∈ H (). p,” x x x E x D[x ],0 (iii) For any x ∈ X ( E), ﬁnite character ρ ∈ X(),wedenoteby L (z, x,ρ) := ρ ◦ x ◦ L (z, X). p,” p,” the value of L (z) at (x,ρ). p,” cl Proposition 4.2 Suppose e = 1. For any x ∈ X ( E) we have L (z)(x) = L (z ). p,” p,” x Proof This is an immediate consequence of the diagram (11), combined with the discussion in Remark 2.4. 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 249 5 Local description of the eigencurve In Sect. 5, we review Bellaïche’s results in [10] on the local description of the eigencurve (most particularly, about a θ-critical eigenform). We also follow his exposition very closely here, particularly of Sects. 2.1.1, 2.1.3, 3.1, 3.2 and 3.4 in op. cit.; and rely also on the notation set therein for the most part (e.g., unless we declare otherwise). 5.1 Modular symbols 5.1.1. Let WS = Hom(Z ,G ) denote the weight space which we consider as a rigid × × analytic space over a ﬁnite extension E ofQ .If y ∈ WS( E), we denote by κ : Z → E p y the associated character. We consider Z as a subset of WS( E) identifying k ∈ Z with the character u → u . Set ∗ p−1 × W = y ∈ WS | v (κ (a) − 1)> ∀a ∈ Z . p y p − 1 ∗ ∗ Note that Z ⊂ W . If U ⊂ W is either an afﬁnoid disk or a wide open disk we will write ◦ ◦ O for the ring of analytic functions on V that are bounded by 1 and set O = O [1/ p]. U U ◦,× We denote by κ : Z → O the universal weight character characterized by the p U property κ = y ◦ κ, ∀ y ∈ U( E). ◦,× We remark that our assumption that U ⊂ W is to guarantee that κ lands in O . ◦ ◦ × ◦ 5.1.2. Let A(U) (resp., A (U) ) denote the space of functions f : Z ×Z → O (resp., p U × ◦ f : pZ ×Z → O ) satisfying the following properties: • f is homogeneous of weight κ , namely f (ax, ay) = κ (a) f (x, y), ∀a ∈ Z ; • The one variable function f (1, z) (resp., f ( pz, 1) is analytic, namely it can be written in the form m ◦ c z , c ∈ O m m m=0 where c goes to zero when m →∞. Note that in the scenario when U is wide open, we work with the m -adic topology. ◦ ◦ ◦ ◦ ◦ SetD(U) := Hom (A(U) , O ),A(U) = A(U) [1/ p] andD(U) := D(U) [1/ p]. O ,cont ◦ ◦ ◦ ◦ Similarly, we setD (U) := Hom (A (U) , O ),A (U) = A (U) [1/ p] andD (U) := O ,cont D (U) [1/ p]. Let ab ( p) := ∈ M (Z ) : p a, p | c, ad − bc = 0 0 2×2 p cd and ab ( p) := ∈ M (Z ) : p d, p | c, ad − bc = 0 . 2×2 p cd 123 250 D. Benois, K. Büyükboduk The monoid ( p) acts naturally on Z ×Z and therefore on A(U) and D(U). Similarly, 0 p the monoid ( p) acts on Z ×Z and therefore on A (U) and D (U). 0 p If W = Spm( R) is an afﬁnoid, ourD(W) corresponds to the space of distributions D( R)[0] in [10]. If U is an open wide disk, ourD(U) andD (U) correspond to D (T ) and D (T ) U,0 0 U,0 of [26]. If U reduces to one point y ∈ WS( E), we write D ( E) (resp., D ( E)) instead of D(U) (resp., D (U))and A ( E) (resp., A ( E)) instead of A(U). 5.1.3. If W ⊂ WS is an afﬁnoid disk we will also work with the space of overconvergent modular symbols D (W) which corresponds to the space D [0]( R) in [10]. To be more precise, we let A[r ](W) denote the space of functions f : Z → O that are analytic on p W the closed disk of radius r and let D[r ](W) = Hom (A[r ](W), O ) denote the space O ,cpt W of compact continuous linear maps A[r ](W) → O . In particular, A[1](W) = A(W) and D[1](W) = D(W). For any r > r the natural map A[r ](W) → A[r ](W) induces a map D[r ](W) → D[r ](W), and one deﬁnes D (W) = limD[r ](W). ← − r>0 If W ⊂ U, where U is a wide open, then we have the restriction map A → A(W), which induces an isomorphism of O -modules D(W) D(U) , D(U) := D(U)⊗ O W W O W (c.f. [3], Lemma 3.8). In particular, we have an injection D (W) −→ D(U) . (13) For any y ∈ W( E), the natural map † † D (W) ⊗ E −→ D ( E) (14) O , y W y is a ( p)-equivariant isomorphism of E-Banach spaces (c.f. [10], Lemma 3.6). We refer the reader to [3,10]and [26, Section 4] for further details pertaining to these objects. 5.1.4. Put = ( N) ∩ ( p) ⊂ SL (Z); note that this congruence subgroup is denoted p 1 0 2 ∗ ∗ by in [10]. We denote by H ( , −) (resp., H ( , −)) the cohomology of (resp., the p p p 0 2 cohomology with compact support). Note that H ( , −) and H ( , −) vanish since p p p is the fundamental group of an open curve. Lemma 5.1 Let W ⊂ W be an afﬁnoid disk. Then: (i) One has {0} if 0 ∈/ W( E), 2 † H ( ,D (W)) Eif 0 ∈ W( E). (ii) For any y ∈ W( E), the specialization at y gives rise to the short exact sequence † † † 0 −→ D (W) −→ D (W) −→ D ( E) −→ 0 (15) (iii) The exact sequence (15) induces an isomorphism 1 † 1 † H ( ,D (W)) ⊗ E H ( ,D ( E)) (16) p O , y p W y and an injection 1 † 1 † H ( ,D (W)) ⊗ E → H ( ,D ( E)) (17) p O , y p c c y 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 251 (iv) If y = 0, the map (17) is an isomorphism. If y = 0, then coker(17) is isomorphic to E. † ∗ (v) The statements i-iv) remain true ifD (W) is replaced byD(U), where U ⊂ W is a wide open. Proof The part (i) is proved in [10, Lemma 3.9] and the same arguments compute H with coefﬁcients in D(U). The exactness of (15) in part (ii) is proved in [10, Lemma 3.6], whereas its analogue for D(U) in [3, Proposition 3.11]. The isomorphism (16) in part (iii) follows from the long exact sequence associated to the 2 † specialization exact sequence and the vanishing of H ( ,D (W)). The inclusion (17)for the cohomology with compact support and the point (iv) follow formally from (i) (c.f. [10], Theorem 3.10), and the proofs of these statements for cohomology with coefﬁcients inD(U) are the same. 5.1.5. Consider the spaces of modular symbols 0 1 † † Symb (D) := Hom (Div (P (Q)),D), D = D(U),D ,D (W), p y 0 1 1 where Div (P (Q)) is the group of divisors of degree 0 onP (Q). Recall the functorial Hecke equivariant isomorphism of Ash and Stevens [4, Proposition 4.2] Symb (D) −→ H ( ,D). (18) ≤ν For a positive real number ν, we denote by Symb (D) the space of modular symbols of slope bounded by ν and by ± ≤ν ≤ν Symb (D) ⊂ Symb (D) p p the ±1-eigenspace for the involution ι given by the action of the matrix (see for 0 −1 example, [10, Section 3.2.4]). Lemma 5.2 Let U be a wide open and W ⊂ U an afﬁnoid disk. Then we have canonical isomorphisms Symb (D(U) ) Symb (D(U)) W W p p 1 1 H ( ,D(U) ) H ( ,D(U)) p W p W where D(U) = D(U)⊗ O and Symb (D(U)) = Symb (D(U))⊗ O . W O W W O W U U p p Proof Only in this proof, A shall denote a general ring. For any complex C of A-modules over a ring A and a morphism of rings A → B, one has the spectral sequence ij A j • i + j • E = Tor ( H (C ), B) ⇒ H (C ⊗ B). 2 −i Let us choose C as the complex computing the cohomology with compact support with ◦ ◦ ◦ n coefﬁcients in D(U) , A = O and B = O / p . Then our spectral sequence induces an U W exact sequence 1 ◦ ◦ n 1 ◦ ◦ n ◦ ◦ 0 −→ H ( ,D(U) ) ⊗ O / p −→ H ( ,D(U) ⊗ O / p ) p O p O c W c W U U U 2 ◦ ◦ n −→ Tor ( H ( ,D(U) ), O / p ). (19) c W 123 252 D. Benois, K. Büyükboduk 2 ◦ It follows from Lemma 5.1(i) (combined with (v) of the same lemma) that H ( ,D(U) ) is a ﬁnitely generated O -module and therefore, there exists a natural number N such that for any n ≥ 1, the O -module U 2 ◦ ◦ n Tor ( H ( ,D(U) ), O / p ) 1 W is annihilated by p . On passing to projective limits in (19) and using [3, Lemma 3.13], we infer that 1 ◦ ◦ 1 ◦ ◦ ◦ ◦ 0 −→ H ( ,D(U) )⊗ O −→ H ( ,D(U) ⊗ O ) −→ C p O p O c W c W U U where C is a ﬁnitely generated module that is annihilated by p . This concludes the proof of the ﬁrst isomorphism. The proof of the second isomorphism is analogous (an even simpler 2 ◦ because H ( ,D(U) ) vanishes). 5.1.6. For a non-negative integer k,welet P ( E) ⊂ E [ Z ] denote the space of polynomials of degree less or equal to k, which is equipped with a left GL (Z )-action; see [10, 3.2.5] 2 p for a description of this action. We let V ( E) denote the E-linear dual of P ( E), which also k k carries an induced GL (Z )-action. 2 p Regarding elements P ( E) as analytic functions, we have an induced map ∗ † ρ : D ( E) −→ V ( E) k k which is a ( p)-equivariant surjection. By Stevens’ control theorem (c.f. [32], Theorem 5.4) it induces a Hecke equivariant isomorphism of E-vector spaces ∗ † <k+1 <k+1 ρ : Symb (D ( E)) − → Symb (V ( E)) (20) k k p p We also have the non-compact version of this isomorphism, proved by Ash and Stevens [1, Theorem 1] : 1 <k+1 1 <k+1 H ( ,D ( E)) − − → H ( , V ( E)) . (21) p p k 5.2 Local description of the Coleman–Mazur–Buzzard eigencurve Our main objective in this subsection is to record Proposition 5.3 and Theorem 5.4 (which is due to Bellaïche; see the second paragraph in [10, §1.5]). 5.2.1. We now brieﬂy recall the Coleman–Mazur–Buzzard construction of the eigencurve. We retain the notation from Sect. 5.1 and continue following the exposition in [10, §2.1]. Let H denotes the Hecke algebra generated over Z by the Hecke operators {T } ,the Np Atkin–Lehner operator U and diamond operators { d } ×. For any afﬁnoid W we d∈(Z/ NZ) set H = H ⊗ O . W Z W We ﬁx a nice afﬁnoid disk W = Spm(O ) (in the sense of [10], Deﬁnition 3.5) of the weight space WS adapted to slope ν in the sense of [10], §3.2.4. Let us denote by M ( , W) Coleman’s space of overconvergent modular forms of level and weight in p p † ≤ν W;and let M ( , W) denote its O -submodule on which U acts with slope at most p W p ν. We similarly let S ( , W) denote the space of cuspidal overconvergent modular forms † ≤ν † of level and weight in the afﬁnoid disk W;and S ( , W) ⊂ S ( , W). p p p cusp Let T (resp., T ) denote the image of H in End M ( , W) (resp., W,ν W O p W,ν W cusp cusp End S ( , W) ). Then C := Spm(T ) (resp., C := Spm(T ) is an open O p W,ν W,ν W W,ν W,ν afﬁnoid subspace of the Coleman–Mazur–Buzzard eigencurve C (resp, cuspidal eigencurve cusp C ) that lies over W. 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 253 ± ± ≤ν We letT denote the image of H in End (Symb (D(W)) ).Wethendeﬁne the W O W,ν W ± ± open afﬁnoids C := Spm(T ).By[10, Theorem 3.30], there exist canonical closed W,ν W,ν immersions cusp ± + − C → C → C , C = C ∪ C . (22) W,ν W,ν W,ν W,ν W,ν W,ν The open afﬁnoids C admissibly cover the Coleman–Mazur–Buzzard eigencurve C as W,ν (W,ν) varies (see [10], §3). 5.2.2. In the remainder of this subsection, we ﬁx a p-stabilization of a newform f and denote by x the corresponding point of C. The following proposition will allow us to apply the formalism of Sects. 3–4 in the context of eigencurves. We denote by W an afﬁnoid neighborhood of k = w(x ) in the weight space, where O is of the form O = E Y / p 0 0 W W Proposition 5.3 (Bellaïche) Up to shrinking W and enlarging E , there exists an afﬁnoid cusp neighborhood X of x ∈ C such that W,ν (a) X is a connected component of C ; W,ν re (b) There exist integers r, e ≥ 1 and an element a ∈ O such that O = E Y / p and X W the map X → a induces an isomorphism of R-algebras O [ X ]/( X − Y ) O . W X Proof By [10, Proposition 4.11], there exists an afﬁnoid neighborhood X of C which W,ν cusp satisﬁes the condition (b). By [10, Corollary 2.17], the eigencurves C and C are locally isomorphic at x . Together with (22), this concludes the proof of (a). 5.2.3. We set ± ≤ν ± † ≤ν Symb (X ) := Symb (D (W)) ⊗ ± O , p p W,ν where X is as in Proposition 5.3. Theorem 5.4 (Bellaïche) ≤ν (i) The O -module Symb (X ) is free of rank one. ≤ν (ii) The O -module Symb (X ) is free of rank 2. Proof The ﬁrst assertion follows as a consequence of the discussion in Section 4.2.1 in op. cit.; see also the ﬁrst paragraph following the proof of [10, Proposition 4.11]. The second assertion follows from the ﬁrst one and the decomposition + − ≤ν ≤ν ≤ν Symb (X ) = Symb (X ) ⊕ Symb (X ) . p p 5.3 A variant of Bellaïche’s construction 5.3.1. Fixanafﬁnoiddisk W and a wide open afﬁnoid U such that W ⊂ U ⊂ W where W is given as in §5.1.1. See [3] as well as §5.1.1 for an explanation of this assumption. Note that this condition translates to the requirement that U is 0-accessible in the sense of [26, Deﬁnition 4.1.1]. We then have a natural (restriction) map O → R. Recall that 123 254 D. Benois, K. Büyükboduk ◦,× κ : Z → O denotes the restriction of the universal weight character to U.Inwhat follows, we will allow ourselves to shrink both U and W as necessary. 5.3.2. Recall that H denotes the Hecke algebra over O . In this subsection, we work with W W the spaces D(U) and A (U). Set ±,≤ν 1 1 ±,≤ν H ( ,D(U)) := H ( ,D(U)) ⊗ O , p p O W c c U ±,≤ν 1 1 ±,≤ν H ( ,A (U)) := H ( ,A (U)) ⊗ O p p O W c W c U and let ± 1 ±,≤ν ± 1 ±,≤ν r : H → End ( H ( ,D(U)) ), r : H → End ( H ( ,A (U)) ) W O p W O p 1 W c W 2 W c W ± ± ± denote the canonical representations of H . We deﬁne the ideals I = ker(r ) and I = 1 1 2 ker(r ). Lemma 5.5 Let k ≥ 0 be an integer such that k ∈ W ⊂ U . For sufﬁciently small W and 0 0 Uthe map (13) together with Lemma 5.2 induce natural Hecke equivariant isomorphisms 1 † ±,≤ν 1 ±,≤ν H ( ,D (W)) H ( ,D(U)) , p p c c W (23) 1 † ±,≤ν 1 ±,≤ν H ( ,D (W)) H ( ,D(U)) . p p In particular, the morphism ± ± T −→ im(r ) W,ν 1 induced from (23) is an isomorphism as well. ±,≤ν 1 † ±,≤ν 1 Proof Let us put M := H ( ,D (W)) and M := H ( ,D(U)) and denote 1 p 2 p c c W by g : M → M the map induced from (13) together with Lemma 5.2. It follows from (18) 1 2 and general results about the slope decomposition that M and M are ﬁnitely generated free 1 2 O -modules. By Lemma 5.1, we have a commutative diagram 1 ±,≤ν 2 † ±,≤ν 0 M /m M H ( ,D ( E)) H ( ,D (W)) [k ] 0 1 k 1 p p 0 0 c k c ±,≤ν 1 ±,≤ν 2 0 M /m M H ( ,D ( E)) H ( ,,D(U)) [k ] 0 2 k 2 p k p 0 0 c 0 c where m ⊂ O is the maximal ideal determined by the weight k .By[32, Lemma 5.3], k W 0 the middle vertical map of the diagram is an isomorphism. According to Lemma 5.1, both ±,≤ν 2 † ±,≤ν 2 H ( ,D (W)) [k ] and H ( ,D(U)) [k ] are both either isomorphic to E or else p 0 p 0 c c W vanish (depending on whether k = 0 or not). Moreover, since the middle vertical arrow is an isomorphism, the right vertical arrow is surjective and therefore an isomorphism as well. We conclude that the left vertical map induced from the vertical isomorphism in the middle is an isomorphism: M /m M −→ M /m M . (24) 1 k 1 2 k 2 0 0 In particular, M and M are free O -modules with the same rank. Let G ∈ O denote the 1 2 W W determinant of g in some bases of M and M . The isomorphism (24) shows that G(k ) = 0 1 2 0 and hence, on shrinking the neighborhood W as necessary, G is non-vanishing on W.This concludes the proof that g is an isomorphism for sufﬁciently small W and U,aswehave asserted in the statement of our lemma. The second isomorphism in (23) can be established by the same argument, using the fact 1 ≤ν 1 ≤ν that the map H ( ,D ( E)) = H ( ,D ( E)) by [1, Theorem 5.5.3]. p p k 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 255 5.3.3. Let us set ,± ,± ,± T = im(r ), C := Spm(T ). W,ν 2 W,ν W,ν ± ± ,± ± Note then that we have canonical isomorphisms T H / I and T H / I . W W W,ν 1 W,ν 2 Theorem 5.6 ,± ± ± ± (i) We have I = I . In particular, there exist canonical isomorphismsT T and 1 2 W,ν W,ν ,± C C . W,ν W,ν ± ± ± ± (ii) Suppose x ∈ Spm(H / I ),where I := I = I . For a sufﬁciently small wide open 0 W 1 2 neighborhood U of k = w(x ) and an afﬁnoid W ⊂ U containing k the following two 0 0 0 assertions hold true. ± ± (a) x belongs to a unique connected component X of Spm(H / I ). 0 W ±,≤ν ±,≤ν 1 1 (b) Both O ± -modules H ( ,D(U)) ⊗ O ± and H ( ,A (U)) ⊗ X p H X p H c W W c W W O ± are free of rank one. (iii) Fix a wide open U as well as an afﬁnoid neighborhood W ⊂ Uof x chosen so as to + − ensure that the conclusions of Part ii) are veriﬁed. If x is cuspidal, then x ∈ C ∩C 0 0 W,ν W,ν + − and X = X . Proof (i) We ﬁrst show that for every integer k ∈ U with k ≥ ν there exists a Hecke equivariant isomorphism 1 ±,≤ν 1 ±,≤ν H ( ,D ( E)) −→ H ( ,A ( E)) . (25) p k p c c k For such k, the natural injection 1 ≤ν 1 ≤ν H ( , P ( E)) −→ H ( ,A ( E)) p k p c c k as well as the natural surjection 1 ≤ν 1 ≤ν H ( ,D ( E)) −→ H ( , V ( E)) p k p k c c of Hecke modules are isomorphisms thanks to the control theorem of Ash and Stevens (c.f. [14], Proposition 4.2.2). We are therefore reduced to proving that we have an Hecke equivariant isomorphism 1 1 H ( , V ( E)) −→ H ( , P ( E)). p k p k c c This follows from the isomorphism V ( E) ⊗ det −− − − → Hom(V ( E), E) =: P ( E) k k k ∗ −1 (det ) of GL (Z )-modules (where det stands for the determinant character of GL (Z )), 2 p 2 p together with the fact that det = 1. Here, det is the isomorphism induced from the perfect pairing −k V ( E) ⊗ V ( E) −→ E ⊗ det k k j j k− j v ⊗ v −→ (−1) v ( Z )v ( Z ) ; 1 2 1 2 j =0 Note here and below that we are not specifying the covariance or the contravariance of the Hecke action that ensures Hecke compatibilities. 123 256 D. Benois, K. Büyükboduk see the paragraph following [14, Proposition 3.2] as well as Remark 3.3.2 in op. cit. This proves (25). In view of [14], Proposition 4.2.1, it follows from the isomorphisms (25) that the rep- resentations r and r verify the conditions of [15, Proposition 3.7]. This implies that 1 2 ± ± I = I and the remaining assertions are immediate from this fact and Lemma 5.5. 1 2 (ii) We will only prove the assertion when the coefﬁcients are A (U), since the proof in the case of D(U) is similar. We remark that one may alternatively deduce our claim 1 ±,≤ν that the O ±-module H ( ,D(U)) ⊗ O ± is free of rank one from [10, p H X c X Proposition 4.5], using the isomorphism (18) of Ash–Stevens and Proposition 5.5. 1 ±,≤ν We will prove that for U sufﬁciently small the O ±-module H ( ,A (U)) ⊗ p H X c O ± is free of rank one. It is clear that one may choose U so as to ensure that the ,± condition a) holds. Let us ﬁx such U.Accordingto[14, Proposition 4.2.1], the T - U,ν ±,≤ν module H ( ,A (U)) is of ﬁnite rank as an O -module (therefore also as a p W ,± T -module). For any y ∈ U, the long exact sequence -cohomology induced from O ,ν the short exact sequence of specialization at y 0 −→ A (U)−→A (U) − → A ( E) −→ 0 shows that 1 0 1 H ( ,A (U))[m ]= im( H ( ,A ( E)) −→ H ( ,A (U))). p y p p c c y c 0 1 Since H ( ,A ( E)) = 0, this proves that H ( ,A (U)) is torsion-free over O and p p U c y c therefore, the O -module H ( ,A (U)) ⊗ O is torsion-free as well. We have W p O W ±,≤ν proved that the O -modules H ( ,A (U)) are ﬁnitely generated and torsion-free W p (therefore free, since O is a PID). ± 1 ±,≤ν Set N = H ( ,A (U)) ⊗ O ±. Now the arguments of Bellaïche apply p H X c W verbatim. More precisely, since O ± are PIDs, [10, Lemma 4.1] tells us that for any x ∈ X , the localization N is a free O ± -module of ﬁnite rank. Moreover, for X ,x (x) any classical point x = x of weight k ≥ ν, the isomorphism (25) together with [10, Lemma 2.8] show that N is of rank one over O ± . By the local constancy of the X ,x (x) rank, the same also holds true for N . We conclude that N is a free O ±-module of (x ) of rank one, when U and W sufﬁciently small. (iii) This portion follows directly from [10, Theorem 3.30]. Corollary 5.7 Let us ﬁx W that ensures the validity of the conclusions of Theorem 5.6(ii) and + − ± suppose x is cuspidal, so that X = X according to Theorem 5.6(iii). Put X = X .Then ≤ν ≤ν 1 1 both O -modules H ( ,D(U)) ⊗ O and H ( ,A (U)) ⊗ O are free of X p H X p H X c c W W W W rank 2. Proof Clear, thanks to Theorem 5.6. 5.3.4. We record the following proposition which will shall use in the proof of Proposition 6.5. It is well known to experts, but we prove it for reader’s convenience. Proposition 5.8 Let x be a classical cuspidal point on the eigencurve C and k = w(x ). 0 0 0 Then the following hold true. 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 257 (i) The natural map 1 1 H ( , V ( E)) ⊗ E −→ H ( , V ( E)) ⊗ E (26) p k H,x p k H,x c 0 0 0 0 is an isomorphism of E -vector spaces of dimension 2. (ii) There exists an afﬁne neighborhood W of k and ν> 0 such that for the connected component X ⊂ C of x ∈ C the map W,ν 0 W,ν j⊗id 1 † ≤ν 1 † ≤ν H ( ,D (W)) ⊗ O −−→ H ( ,D (W)) ⊗ O (27) p H X p H X c W W 1 † ≤ν 1 † ≤ν induced from the natural H -equivariant map H ( ,D (W)) − → H ( ,D (W)) , W p p is an isomorphism. Proof (i) Let us denote by Y the Borel–Serre compactiﬁcation of the modular curve Y = \ H (and ∂ Y its boundary). The sheaf V on Y extends to a sheaf on Y . It follows from [4, Proposition 4.2] that 1 1 coker(26)→ im H (Y , V ) −→ H (∂ Y , V ) [m ] k k x 0 0 0 and the surjectivity of (26) follows once we verify that 1 1 im H (Y , V ) − → H (∂ Y , V ) [m ]= 0. k k x 0 0 0 This follows from the properties of the Eisenstein cohomology, since x is cuspidal. The proof that the map (26) is injective is similar, where one instead relies on the vanishing 0 1 of the x -isotypic subspace of im( H (∂ Y , V ) → H (Y , V )). 0 k k 0 c 0 Our assertion on the dimension of these vector spaces is standard, c.f. [10, Proposition 3.18]. (ii) Fix ν ≥ k + 1. Let W be an afﬁnoid neighborhood of k such that the conclusion of 0 0 1 † ≤ν Theorem 5.4 hold for W. To simplify notation, set M = H ( ,D (W)) ⊗ O c p H X c W 1 † ≤ν and M = H ( ,D (W)) ⊗ O . Then the O -module M is free of rank 2. The p H X X c same argument (that Bellaïche utilizes to prove M is free of rank 2) shows that M is also free of rank 2 over O . Namely, on shrinking W if necessarly, we can assume that M is a ﬁnitely generated free O -module. For any point x of weight w(x) we have 1 † ≤ν M/m M H ( ,D (W)) ⊗ E ⊗ E, (28) x p O ,w(x) H,x Together with Lemma 5.1 this gives 1 † ≤ν M/m M H ( ,D ( E)) ⊗ E. (29) x p H,x w(x) The control theorem of Ash–Stevens (21) reads 1 ≤ν 1 ≤ν H ( ,D ( E)) − − → H ( , V ( E)) p p k The required vanishing statement follows from [18, Corollary 4.7] (see also [19], Theorem 1) combined with the strong multiplicity one for GL2, which follows as a consequence of the classiﬁcation result due to Jacquet–Shalika [20,21] (c.f. the proof of [16], Proposition 4.1). In brief terms, Corollary 4.7 of [18] tells us that the systems of Hecke eigenvalues occurring in the image of the map ∂ are those of Eisenstein series. By the classiﬁcation result of Jacquet–Shalika, the set of systems of Hecke eigenvalues of Eisenstein series is disjoint from those of cusp forms. (We are grateful to Fabian Januszewski for indicating these references.) 123 258 D. Benois, K. Büyükboduk for each positive integer k >ν − 1. Choosing x ∈ X such that w(x)>ν − 1, we deduce that 1 ≤ν M/m M H ( , V ( E)) ⊗ E. x p k H,x Now it follows from the classical Eichler–Shimura isomorphism and multiplicity-one that M/m M has dimension 2 over E,and the O -module M is free of rank 2 as well. x X Let det ∈ O denote the determinant of j ⊗ id : M → M with respect to ﬁxed bases of j X c M and M. If we knew that det (x ) = 0, we could shrink W and the neighborhood X of x ) c j 0 0 to ensure that det is non-vanishing on X and thereby conclude that j ⊗ id is an isomorphism, as required. We have therefore reduced to proving that det (x ) = 0. j 0 Recall the isomorphism (29)for x = x : 1 ≤ν M/m M H ( ,D ( E)) ⊗ E. (30) x p H,x 0 0 The analogous isomorphism (28)for M together with Lemma 5.1 shows that we have an injection 1 † ≤ν M /m M → H ( ,D ( E)) ⊗ E, c x c p H,x 0 c k 0 which is an isomorphism if k = 0. As a matter of fact, it follows from [10, Proposition 3.14] that this map is an isomorphism since x is cuspidal. We therefore have the following com- mutative diagram: 1 ≤ν M /m M H ( ,D ( E)) ⊗ E c x c p H,x 0 c k 0 1 ≤ν M/m M H ( ,D ( E)) ⊗ E. x p H,x 0 0 To conclude with the proof of the asserted isomorphism in our proposition with sufﬁciently small W and X , it sufﬁces to prove that the right vertical map is an isomorphism. Observe that the following diagram commutes: † † 1 ≤ν 1 ≤ν H ( ,D ( E)) ⊗ E H ( ,D ( E)) ⊗ E p H,x p H,x c k 0 k 0 0 0 ∗ ∗ ρ ρ ∼ ∼ k k (31) 0 0 1 ≤ν 1 ≤ν H ( , V ( E)) ⊗ E H ( , V ( E)) ⊗ E p k H,x p k H,x c 0 0 0 0 (i) All objects of this diagram have dimension 2 over E. The surjectivity of the vertical arrow in (31) on the left follows from [32, Lemma 5.1], and we infer that the vertical arrow on the left is an isomorphism. The bottom map is an isomorphism by part i). This implies that all the maps of the diagram are isomorphisms. The proof of our proposition is now complete. 6 Interpolation of Beilinson–Kato elements 6.1 Modular curves and Iwasawa sheaves 6.1.1. For a pair of positive integers M and M , we recall the modular curves Y ( M , M ) 1 2 1 2 from [22,§2].Wethenhave Y ( N, N) = Y ( N) and Y (1, N) = Y ( N), modular curves of level ( N) and ( N), respectively. Recall from §2.8 of op. cit. also the modular curves denoted 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 259 by Y ( M , M ( B)) and Y ( M ( B), M ),where B is also a positive integer. In Sections 6.3- 1 2 1 2 6.5, we will be working over the modular curve Y (1, N( p)) of ( N) ∩ ( p)-level. We 1 0 note that the modular curve Y (1, N( p)) is denoted by Y ( N, p) in [3]. If Y denotes any one of the modular curves above, we denote by λ : E → Y the universal elliptic curve with the appropriate level structure (which depends on Y , but we suppress this dependence from our notation). We let T := R λ Z (1) denote the pro-system (T ) ∗ p n n≥1 of étale lisse sheaves T := R λ — n on the open modular curves Y [1/ p] given as in [22, n ∗ p §§1–2]. The sheaf T has rank 2 and the Poincaré duality identiﬁes it with the p-adic Tate module T (E ) of E . We write T for the associated sheaf of Q -vector spaces. Q p p p For each non-negative integer k and a locally free sheaf F over Y [1/ p],welet TSym F denote the locally free sheaf on Y of symmetric k-tensors, given as in [24, §2.2]. 6.1.2. Let i : D → E be a subscheme. Assume that D is etale over Y and consider the diagram E [ p ] D D,n [ p ] D E where E [ p ] D is the ﬁber product of E and D over E . We deﬁne the pro-etale sheaf (T D ) = ( (T D )) on setting n≥1 (T D ) = λ p (Z/ p Z). D,∗ D,n,∗ We refer the reader to [23] where Kings develops these notions in a general framework. If D = Y and i = s : Y → E is a section of λ, we will write (T s ) instead (T D ). In particular, we denote by (T ) = (T 0 ) the sheaf associated to the identity section 0. For any section s, the sheaf (T s ) is a sheaf of modules of rank one over the sheaf of Iwasawa algebras (T ) (c.f. [23], §2.4). 6.1.3. In the remainder of §6.1, we recall a number of notation and constructions from [24,26]. Fix a positive integer N. We denote by λ : E → Y ( N) the universal elliptic curve N N 1 and by T the associated etale sheaf. Recall that Y ( N) is the moduli space of pairs ( E,β ), N 1 N where E is an elliptic curve and β : Z/ NZ → E [ N ] is an injection. Then the map ( E,β ) → β (1) deﬁnes a section s : Y ( N) → E of λ ,andwedenoteby (T s N N N 1 N N N N the associated sheaf. Let g : Y (Np ) → Y ( N) denote the canonical projection. We deﬁne the pro-etale n 1 1 sheaf on Y ( N) on setting N et ´ = , = g (Z/ p Z). n,∗ N N,n N,n n≥1 If p | N, the moduli description of Y (Np ) shows that there exists a canonical isomorphism n n Y (Np ) E [ p ] s , and therefore we have 1 N (T s ) . (32) N N (c.f. the proof of [24], Theorem 4.5.1). In particular, we can apply the formalism developed in [23] with the sheaf . 123 260 D. Benois, K. Büyükboduk 6.1.4. Assume that N is coprime to p and consider the modular curve Y (1, N( p)) equipped with the universal elliptic curve E and the associated sheaf T . Recall that Y (1, N( p)) N( p) N( p) is the moduli space for the triples ( E,β , C), where E is an elliptic curve, β : Z/ NZ → E N N is an injection and C ⊂ E is a (cyclic) subgroup of order Np that contains β (1 + NZ). Denote by C ⊂ E [ p] the canonical subgroup of order p and set D := E [ p]− C and N( p) N( p) D := C −{0}. Note that both D and D are ﬁnite étale over Y (1, N( p)), of degrees p − p and p − 1, respectively, and we denote by (T D ) and (T D ) the associated N( p) N( p) sheaves. Since both D and D are contained in E [ p], the “multiplication-by- p” morphism N( p) induces the trace map [ p] ? ? (T D ) − − → (T ), D ∈{ D, D } (33) N( p) N( p) of sheaves on Y (1, N( p)). The rule ( E,β ) → ( E,β ( p), im(β )) deﬁnes a morphism pr : Y (1, Np) → Np Np Np Y (1, N( p)), which in turn induces the cartesian square E Y (Np) Np 1 pr E Y (1, N( p)) N( p) For each positive integer r,wehaveanatural morphism [ N ] ◦ pr r r E [ p ] s − −−−− → E [ p ] D . (34) Np Np N( p) where the schemes E [ p ] · are given as in [24, Deﬁnition 4.1.4]. The composition of this map with (32) gives a map ∗ ∗ H (Y (Np), ) −→ H (Y (1, N( p)), (T D )), (35) 1 N( p) et ´ Np et ´ (see also [7, §4.2.5] for further details). 6.2 Big Beilinson–Kato elements 6.2.1. Let us ﬁx positive integers M and N such that M + N ≥ 5. We consider the following objects: • The modular curves Y ( M, N) and Y ( N), which come equipped with the universal ellip- tic curves λ : E → Y ( M, N) and λ : E → Y ( N), as well as the sheaves M, N M, N N N 1 1 1 T = R λ Z (1) and T = R λ Z (1) on Y ( M, N) and Y ( N) , respec- M, N M, N,∗ p N N,∗ p 1 et ´ et ´ tively. ◦ ◦ • The morphisms Y ( M, N) → Y ( N) × μ → Y ( N), where μ = Spec(Z[ζ ]). 1 1 M M M These morphisms induce a canonical map T → T | . M, N Y ( M, N) • The canonical projections n n ◦ ◦ f : Y (Mp , Np ) → Y ( M, N), h : μ → μ n n Mp M cyc n n ◦ ◦ g : Y (Np ) → Y ( N), g : Y (Np ) × μ n → Y ( N) × μ n 1 1 n 1 1 Mp M 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 261 which give rise to the inverse systems of sheaves = ( ) , = f (Z/ p Z), n≥1 n,∗ M, N M, N,n M, N,n cyc cyc cyc ◦ = ( ◦ ) , ◦ = h (Z/ p Z), n≥1 n,∗ μ μ ,n μ ,n M M M = ( ) , = g (Z/ p Z), n≥1 n,∗ N N,n N,n cyc cyc cyc cyc = ( ) , = g (Z/ p Z). ◦ ◦ ◦ n≥1 n,∗ N,μ N,μ ,n N,μ ,n M M M By the ﬂatness of our sheaves, it follows that cyc cyc . (36) ◦ ◦ N,μ μ M M For any integer L, let us denote by prime(L) the set of primes dividing L. For any integer m ≥ 1, set = Gal(Q(ζ )/Q(ζ )) and ( ) = Z [[ ]]. Q(ζ ) mp m Q(ζ ) p Q(ζ ) m m m 6.2.2. Poincaré duality gives a canonical isomorphism between T and the Tate module M, N n n T (E ) of the universal elliptic curve E . The interpretation of Y (Mp , Np ) as the M, N M, N n 2 n moduli space of pairs ( E,α), where E is an elliptic curve and α : (Z/Np Z) E [Np ] is n n n n an isomorphism, shows that the restriction of E [ p ] T (E )/ p on Y (Mp , Np ) M, N M, N p has a canonical basis {e , e } given by e = α( M, 0) and e = α(0, N). Analogously, 1,n 2,n 1,n 2,n n n the interpretation of Y (Mp , Np ) as the moduli space of pairs ( E,β), where E is an elliptic n n curve and β : Z/Np Z → E [Np ] is an injection, shows that the restriction of the sheaf T on Y (Np ) has the canonical section β( N) which we also denote by e to simplify N 1 2,n notation. 6.2.3. One has an isomorphism of continuous Galois modules cyc cyc 1 ◦ 1 H (Y ( N) × μ , ) H (Y ( N) , ) ⊗Z [[Gal(Q(ζ )/Q)]]. 1 ◦ 1 ◦ p M et ´ M N,μ et ´ Q N,μ M M cyc Since is a constant sheaf on Y ( N) , one also has cyc 1 1 H (Y ( N) , ) H (Y ( N) , )⊗ ( ). 1 ◦ 1 Q(ζ ) N M et ´ Q N,μ et ´ Q The commutative diagrams (for each n) n n n ◦ Y (Mp , Np ) Y (Np ) × μ 1 n Mp cyc n g Y ( M, N) Y ( N) × μ Y ( N) × μ show that we have the trace map cyc 1 j 1 ◦ j H (Y ( M, N) , TSym (T ) ⊗ ) −→ H (Y ( N) × μ ) , TSym (T ) ⊗ 1 M, N 1 N ◦ Q M, N M Q et ´ et ´ N,μ cyc 1 ◦ j −→ H (Y ( N) × μ ) , TSym (T ) ⊗ 1 N ◦ et ´ Q N,μ cyc 1 j − → H Y ( N) , TSym (T ) ⊗ ⊗Z [Gal(Q(ζ )/Q]. 1 N p m et ´ Q N,μ (37) for each m dividing M. 123 262 D. Benois, K. Büyükboduk 6.2.4. More generally, let (a, B, m) be a triple of integers such that mB | M. Let us choose L ≥ 1 such that M | L, N | L and prime(L) = prime(MN). The construction of [22, §5.2] provides us with a map 1 j t : H (Y ( M, L) , TSym (T ) ⊗ ) m,a( B) 1 M,L Q M,L et ´ cyc 1 ◦ j −→ H (Y ( N) × μ ) , TSym (T ) ⊗ ◦ 1 N m (38) et ´ Q N,μ cyc 1 j H Y ( N) , TSym (T ) ⊗ ◦ ⊗Z [Gal(Q(ζ )/Q]. 1 N p m et ´ Q N,μ Note that this map coincides with (37)if m = M, a = 0, B = 1and L = N. 6.2.5. We write n n × 1 n n 1 n n ∂ : O(Y (Mp , Np )) −→ lim H (Y (Mp , Np ),— ) = H (Y (Mp , Np ),Z (1)) n p p et ´ et ´ ← − for the Kummer map. For any integer j ≥ 0, consider the chain of maps n n Ch : lim K (Y (Mp , Np )) M, N, j 2 ← − 2 n n −→ lim H (Y (Mp , Np ),Z (2)) et ´ ← − ⊗ j ∪e 1,n 2 n n ∗ j ⊗2 (39) − −− → lim H (Y (Mp , Np ), f (TSym (T )) ⊗— ) M, N,n n p et ´ ← − 2 j −→ H (Y ( M, N), TSym (T ) ⊗ (2)) M, N M, N et ´ 1 1 j −→ H Z[1/MN p], H (Y ( M, N) , TSym (T ) ⊗ (2)) , M, N et ´ Q M, N where the ﬁrst map is induced by the composition of ∂ with the cup product in étale coho- mology, and the very last map is deduced from the Hochschild–Serre spectral sequence. n × n × n n Let g ∈ O(Y (Mp , 1) and g ∈ O(Y (1, Np ) denote Kato’s Siegel c 1/Mp ,0 d 0,1/Np units (see [22, §§1–2] for their deﬁnition). As in op. cit., we put n n z n n := g n ∪ g n ∈ K (Y (Mp , Np )). c,d Mp ,Np c 1/Mp ,0 d 0,1/Np 2 We recall that n n ( z n n) ∈ lim K (Y (Mp , Np )) c,d Mp ,Np n≥1 2 ← − belongs to the source of the map Ch (c.f. [22], Proposition 2.3). M, N, j 6.2.6. Let us ﬁx a positive integer N.Asin[22, §5], let ξ denote either the symbol a( B) with a, B ∈ Z and B ≥ 1oranelement of SL (Z). For each integer m ≥ 1, we denote by S the following set of primes: primes(mBp), if ξ = a( B) S = primes(mN p), if ξ ∈ SL (Z). Let (c, d) be a pair of positive integers satisfying the following conditions: (cd, 6) = 1,(d, N) = 1, prime(cd) ∩ S =∅. 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 263 If ξ = a( B), we choose M and L ≥ 1 such that mB | M, M | L, N | L, prime( M) = S, prime(L) = S ∪ prime( N) and denote by n n Ch : lim K (Y (Mp , Lp )) M,L, j,ξ 2 ← − 1 1 j −→ H Z[1/S], H (Y ( M, L) , TSym (T ) ⊗ (2)) 1 M,L M,L et ´ Q cyc 1 1 j −→ H Z[1/S], H Y ( N) , TSym (T ) ⊗ (2) ⊗Z [Gal(Q(ζ )/Q] 1 N ◦ p m et ´ Q N,μ cyc 1 1 j −→ H Z[1/S,ζ ], H Y ( N) , TSym (T ) ⊗ (2) m 1 N ◦ et ´ Q N,μ 1 1 j ι −→ H Z[1/S,ζ ], H Y ( N) , TSym (T ) ⊗ (2) ⊗ ( ) m 1 N Q(ζ ) N m et ´ Q (40) the composition of the maps (39)and (38). Here, the symbol ι on the ﬁnal line means that the target cohomology group is equipped with a continuous action of = Q(ζ ) Gal(Q(ζ )/Q(ζ )) induced by the right action of this group on ( ) via the canonical mp m Q(ζ ) −1 involution ι(g) = g (c.f. §3.1.1). If ξ ∈ SL (Z), we ﬁx L ≥ 3 such that m | L, N | L, prime(L) = S. n n n The element ξ induces an automorphism of Y (Lp ) = Y (Lp , Lp ). We consider the map Ch : lim K (Y (Lp )) L, j,ξ 2 ← − 1 1 j −→ H Z[1/S], H (Y (L) , TSym (T ) ⊗ (2)) L,L L,L et ´ Q cyc 1 1 j −→ H Z[1/S], H Y ( N) , TSym (T ) ⊗ (2) ⊗Z [Gal(Q(ζ )/Q] 1 N ◦ p m et ´ Q N,μ cyc 1 1 j −→ H Z[1/S,ζ ], H Y ( N) , TSym (T ) ⊗ (2) m 1 N ◦ et ´ Q N,μ 1 1 j ι −→ H Z[1/S,ζ ], H Y ( N) , TSym (T ) ⊗ (2) ⊗ ( ) . m 1 N Q(ζ ) N m et ´ Q (41) Deﬁnition 6.1 (i) If ξ = a( B) we set n n BK ( j,ξ) = Ch ( z ) , c,d N,m M,L, j,ξ c,d Mp ,Lp n≥1 where Ch is the map (40). M,L, j,ξ (ii) If ξ ∈ SL (Z),weset n n BK ( j,ξ) = Ch (ξ ( z )) , c,d N,m L, j,ξ c,d Lp ,Lp n≥1 where Ch is the map (41). L, j,ξ Lemma 6.2 The elements BK ( j,ξ) do not depend on the choice of M and L. c,d N,m 123 264 D. Benois, K. Büyükboduk Proof The proof is similar to that of [22, Proposition 8.7]. Let ( M , L ) be another pair of integers that satisfy (40) and such that M | M , N | N . Since the trace map commutes with cup products, we have the following commutative diagram: n n 1 1 j lim K (Y ( M p , L p )) H Z[1/S], H (Y ( M , L ) , TSym (T ) ⊗ (2)) 2 1 M ,L M ,L et ´ Q ← −n n n 1 1 j lim K (Y (Mp , Lp )) H Z[1/S], H (Y ( M, L) , TSym (T ) ⊗ (2)) 2 1 M,L M,L et ´ Q ← − On the other hand, the left vertical map sends ( z n n) to ( z n n) by c,d M p ,L p n≥1 c,d Mp ,Lp n≥1 [22, Proposition 2.3]. This proves the lemma in the case ξ = a( B); the case ξ ∈ SL (Z) is analogous. 6.2.7. The isomorphism (32) allows us to consider the moment maps [k] mom : → TSym (T ) (42) [k] [k] given as in [23, Section 2.5] (see also [24], §4.4). Let mom denote the map mom N,n N [k] modulo p . Then the map mom can be described as the composition N,n ⊗k ∪e 2,n n n ∗ k g (Z/ p Z) − −− → g Z/ p Z ⊗ g (TSym (T )) n,∗ n,∗ N,n trace ∗ k k g ◦ g (TSym (T )) −−→ TSym (T ). n,∗ N,n N,n On the level of cohomology, one then obtains the commutative diagram 1 1 n n H (Y ( N) , ) H (Y (Np ) ,Z/ p ) 1 1 Q N,n Q et ´ et ´ [k] ⊗k mom ∪e N,n 2,n 1 k 1 n ∗ k H (Y ( N) , TSym (T )) H (Y (Np ) , g (TSym (T ))), 1 N,n 1 N,n Q Q n et ´ et ´ where the bottom horizontal map is the trace map (c.f. [23], Proposition 2.6.8). 6.2.8. For each r ∈ Z, we have the moment maps of sheaves on μ [r,n] cyc mom ◦ : ◦ −→ h ◦ h (Z (−r)), n ≥ 0. n,∗ p μ μ n m m cyc The stalks of are isomorphic to the -module ( ) at geometric points (c.f. [24], Q(ζ ) μ m §6.3) and the moment map coincides with the map ( ) −→ Z [G ](−r), G := Gal(Q(ζ n )/Q(ζ )), Q(ζ ) p n n mp m −r −r g −→ χ (g)g ¯ ⊗ χ where g ¯ ∈ G denotes the image of g ∈ under the natural projection → G . n n Q(ζ ) Q(ζ ) n m m Let us deﬁne [k,r,n] [k] [r,n] cyc mom := mom mom ◦ : −→ TSym (T )(−r). ◦ ◦ N N,μ N N,μ m m 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 265 ( p) Recall the elements z (k, r, r ,ξ, S) which Kato has constructed in [22, §8.9]. Set c,d 1, N,m ( p) BK (k, j, r,ξ) := z ( j + k + 2, r, j + 1,ξ, S) c,d N,m c,d 1, N,m 1 1 j +k ∈ H Z[1/S,ζ ], H Y ( N) , Sym (T )(2 − r) . mp 1 N et ´ Q Note that in his construction Kato makes use of the dual sheaf T together with the canonical isomorphism T T (−1) provided by duality. Proposition 6.3 For all integers j, k ≥ 0 and r ∈ Z, the induced map cyc [k,r,n] 1 1 j mom : H Z[1/S,ζ ], H Y ( N) , TSym (T ) ⊗ (2) ◦ ◦ m 1 N N, j,μ Q N,μ et ´ m m 1 1 j +k −→ H Z[1/S,ζ n ], H Y ( N) , TSym (T )(2 − r) mp 1 N et ´ sends BK ( j,ξ) to BK n(k, j, r,ξ). c,d N,m c,d N,mp Proof This is clear by the construction. 6.3 Overconvergent étale sheaves Until the end of this article, we set Y = Y (1, N( p)) unless we state otherwise and assume that N is coprime to p. 6.3.1. We denote by F the canonical sheaf on Y . Throughout this section, W and U denote an afﬁnoid and a wide open disk such that W ⊂ U ⊂ W . In what follows, we will allow ourselves to shrink both U and W as necessary. We adopt the notation and conventions of Sect. 5. In particular, O denotes the ring of analytic functions on U that are bounded by 1. We let ◦,× χ : Z −→ O denote the composition of the cyclotomic character χ : G → Z with the canonical ◦,× weight character κ : Z → O . p U We review the theory of overconvergent sheaves introduced in [3]. See also [26, §4] and [14, §4] for further details concerning the material in this subsection. Deﬁnition 6.4 ◦ ◦ 0 ◦ (i) We let D (T ) (resp., D (T ),resp., A (T ) ) denote the pro-étale sheaf of O -modules U U U U ∞ ∞ ◦ on Y , whose pullback to the pro-scheme Y ( p , Np ) is the constant pro-sheaf D(U) ◦ 0 (resp., D (U) ,resp., A (U) ). (ii) We set −1 ◦ 1 ◦ ◦ 1 ◦ M (T ) := H Y , D (T ) (1), M (T ) := H Y , D (T ) (χ ), U U U U et ´ Q et ´ Q U ◦ 1 0 N (T ) := H Y , A (T ) U et ´ ,c Q U and ◦ ◦ ◦ M (T ) := M (T )[1/ p], M (T ) := M (T )[1/ p], N (T ) := N (T )[1/ p]. U U U U U U 123 266 D. Benois, K. Büyükboduk The evaluation map ◦ ◦ ◦ ev : A (U) ⊗D (U) −→ O induces a pairing 0 ◦ ◦ A (T ) ⊗ D (T ) −→ O , U U U ◦ ◦ where O is the sheaf associated to O as in [14, §4.2]. Together with the trace map U U 2 ◦ ◦ H (Y , O (1)) → O , this pairing induces the pairing Q U U et ´ ,c ◦ 0 ◦ N (T ) ⊗ M (T ) −→ O . (43) U U U U We ﬁnally recall that Proposition 4.4.5 of [26] supplies us with a morphism of sheaves on Y : (T D ) −→ D (T ). (44) 6.3.2. By GAGA, we have the canonical isomorphisms 1 1 M (T )(χ ) H ( ,D(U)), M (T )(−1) H ( ,D (U)), U U p U p N (T ) H ( ,A (U)) (45) U p (c.f. [3], Proposition 3.18). The isomorphisms in (45) allow us to deﬁne a Hecke action on ≤ν ≤ν the spaces on the left of each isomorphism. The submodules M (T ) , M (T ) and U U ≤ν N (T ) are stable under the action of G and the actions of prime-to- p Hecke operators. U Q ≤ν For an afﬁnoid W contained in a wide open U in the weight space, we set M (T ) := ≤ν ≤ν ≤ν M (T ) ⊗ O . We similarly deﬁne M (T ) and N (T ) . U O W W W Proposition 6.5 Let x denote a cuspidal point of the eigencurve C. There exist a sufﬁciently small wide open U, an afﬁnoid W ⊂ U and a slope ν such that x ∈ C and the following 0 W,ν hold true. (i) Let X denote the connected component of x in C . There is an isomorphism of O - 0 W,ν X modules † ≤ν ≤ν comp : Symb (D (W)) ⊗ O −→ M (T ) ⊗ O H X W H X p W W which interpolates Artin’s comparison isomorphisms between Betti and étale cohomol- ogy. (ii) All three O -modules ≤ν ≤ν ≤ν M (T ) ⊗ O , M (T ) ⊗ O , N (T ) ⊗ O (46) W H X W H X W H X W W W are free of rank 2. We remark that it is absolutely crucial that the afﬁnoid neighborhood X of x falls within the cuspidal eigencurve. Proof (i) This portion follows from (45), Lemma 5.2 and Proposition 5.8. ≤ν (ii) It follows from (i) and Theorem 5.4 that M (T ) ⊗ O is free of rank W H X ≤ν 2 over O . By [26, Proposition 4.4.8.4], M (T ) ⊗ O is isomorphic to X W H X ≤ν M (T ) ⊗ O as O -modules and therefore, they both have the same O -rank. W H X W X ≤ν Finally, N (T ) ⊗ O have rank 2 by the isomorphism (45) and Corollary 5.7. W H X 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 267 6.3.3. Let j ≥ 1 be a positive integer. The map F(x) → F(x − j) deﬁnes an isomorphism 0 0 0 t : O O .If F ∈ A (U − j) and G ∈ P (O ) is a homogeneous polynomial of j j E U − j U 0 0 degree j, then t ◦(FG) ∈ A (U) , and we have a well deﬁned mapA (U − j) ⊗P (O ) → j j E A (U) . Passing to the duals, we obtain a map ∗ ◦ ◦ j 2 β : D (U) −→ D (U − j) ⊗ TSym (O ) j E and the associated morphism of sheaves: ∗ ◦ ◦ j β : D (T ) −→ D (T ) ⊗ TSym (T ). (47) j U U − j Let us denote by ◦ ◦ δ : A (U) −→ A (U − j) ⊗ P (O ) j j E 1 ∂ F(x, y) i m the map given by δ ( F) := ⊗ x y . On transposing this map, we obtain i m j ! ∂ x ∂ y i +m= j ∗ ◦ j 2 ◦ δ : D (U − j) ⊗ TSym (O ) −→ D (U) j E and the induced morphism of sheaves ∗ ◦ j ◦ δ : D (T ) ⊗ TSym (T ) −→ D (T ). (48) j U − j U 6.4 Big Galois representations 6.4.1. For the convenience of the reader, we review Deligne’s construction of p-adic rep- resentations associated to eigenforms. Let T be the dual sheaf of T .Let f denote a p-stabilized eigenform of weight k + 2 ≥ 2and level , which is new away from p (but could be new or old at p). Deligne’s representation associated to f is deﬁned as the f - 1 k ∨ isotypic Hecke submodule V [ f ]⊂ H (Y , Sym T ) for the Hecke operators {T } Np Q Q et ´ ,c p and {U } , which are given as in [22, §4.9] (but denoted in op. cit. by T () for all ). |Np Recall that this is a two dimensional vector space equipped with a continuous action of G , which is de Rham at p (crystalline iff f is p-old) with Hodge–Tate weights (0, k + 1).Bythe 1 k ∨ theory of newforms, the Hecke module H (Y , Sym T ) is f -semisimple. In particular, Q Q et ´ ,c p 1 k ∨ V [ f ] coincides with the generalized Hecke eigenspace V [[ f ]] ⊂ H (Y , Sym T ) in et ´ ,c Q Q 1 k ∨ the sense of [10, §1.2]. We denote by V the f -isotypic quotient of H (Y , Sym T ). Q Q et ´ ,c p Then the natural projection V [ f ]→ V is an isomorphism. 1 k Dually, consider H (Y , TSym T )(1). The Hecke algebra H acts on this space via the Q p et ´ dual Hecke operators the dual Hecke operators {T } and {U } , which are given as |Np Np in [22, §4.9] (denoted by T () for all ). Let V [ f ] (respectively, V ) denote the f -isotypic 1 k Hecke submodule (respectively, the f -isotypic quotient) of H (Y , TSym T )(1). Paral- Q p et ´ 1 k leling the discussion in the previous paragraph, the Hecke module H (Y , TSym T )(1) et ´ is semisimple at f and V [ f ] coincides with the generalized Hecke eigenspace V [[ f ]] 1 k inside H (Y , TSym T )(1). The canonical projection V [ f ]→ V is an isomorphism. Q f et ´ Poincaré duality induces a perfect Galois-equivariant pairing (, ) : V ⊗ V [ f ]−→ E. (49) 6.4.2. For any normalized eigenform f = a q ∈ S ( ),weset n k+2 p n≥1 c n f = a q . n≥1 123 268 D. Benois, K. Büyükboduk Since f is an eigenform for the Hecke operators {T } and {U } with eigenvalues |Np Np {a }, it is also an eigenform for the dual operators {T } and {U } , with eigenvalues Np |Np {a }. The isomorphism T (1) T induces a Hecke equivariant map 1 k ∨ 1 k H (Y , Sym T )(k + 1) −→ H (Y , TSym T )(1), et ´ ,c Q Q et ´ Q p which gives rise to a canonical isomorphism V [ f ](k + 1) V [ f ] (50) on the f -isotypic subspaces for the dual operators {T } and {U } . With obvious |Np Np modiﬁcations, all these constructions also make sense for modular forms of level ( N). 6.4.3. We maintain the notation and the conventions of Section 6.3.2. Fix a p-stabilized eigenform f as in §6.4.1 and denote by x the corresponding point of the eigencurve C.Let α denote the U -eigenvalue on f . 0 p Fix a slope ν ≥ v (α ) a wide open afﬁnoid U and an afﬁnoid W as in Proposition 6.5. p 0 Let X denote the connected component of x in C . 0 W,ν In the remainder of this paper we will work with the O -adic Galois representations ≤ν V := N (T ) ⊗ O , X W H X (51) ≤ν V := M (T ) ⊗ O . W H X X W By Proposition 6.5, both O -modules have rank two. We denote by (, ) : V ⊗ V −→ O (52) X X W the G -equivariant cup-product pairing induced by (43). Q,S cl 6.4.4. For any E-valued classical point x ∈ X ( E) of X,welet f denote the corresponding p-stabilized cuspidal eigenform of weight w(x)+2. In particular, f is our ﬁxed p-stabilized eigenform f . By Proposition 5.3 we have O O [ X ]/( X − Y ) for some e ≥ 1, so the X W formalism of Sect. 2 applies in this context. We deﬁne the representations V , V , V [x ] and V [[x ]] (respectively V , X ,x X ,w(x) X X X ,x V , V [x ] and V [[x ]]) by plugging in M = V (respectively M = V )inDeﬁni- X ,w(x) X X X tion 2.2. In particular, V := V ⊗ E and V := V ⊗ E. We recall X ,x X O ,x X ,w(x) X O ,w(x) X W that e = 1 except when the p-stabilized form f is θ-critical. Let P (O ) denote the space of homogeneous polynomials in two variables of degree k k E with coefﬁcients in E.If k ∈ U, we have a canonical injection P (O )→ A (U) , which k E k ⊕2 ◦ ◦ k ⊕2 induces a map Sym (O )→ A (U) and the dual map D (U) → TSym (O ). We E E have the associated morphisms of sheaves ◦ k k ∨ ◦ D (T ) −→ Sym T , Sym T −→ A (T ) (53) U U which give rise to the pair of Hecke equivariant morphisms 1 k ∨ 1 H (Y , Sym T ) −→ H (Y , A (T )), et ´ ,c Q Q et ´ ,c Q (54) 1 1 k H (Y , D (T )) −→ H (Y , TSym T ). U Q et ´ Q et ´ Q We obtain the following morphisms by the deﬁnitions of M (T ) and N (T ): U U 1 k ∨ H (Y , Sym T ) −→ N (T ), et ´ ,c Q 1 k M (T ) −→ H (Y , TSym T (1)). U Q Q p et ´ 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 269 cl On tensoring with O and taking slope ≤ ν submodules, we deduce for any x ∈ X ( E) that there are canonical morphisms V [ f ] − → V [[x ]] → V , x X X (55) V −→ V − → V . X X ,w(x) f It follows from deﬁnitions that • V and V are E-vector spaces of dimension 2e,where e denotes the ramiﬁ- X ,w(x) X ,w(x) cation degree of the weight map w : X → W at x ; • V [x ] and V [x ] are the f -isotypic Hecke eigenspaces inside V and V (in X x X ,w(x) X ,w(x) ? ? alternative wording, V [x ] is the largest subspace of V annihilated by X − X(x) ∈ X X ,w(x) O ); and these are both E-vector spaces of dimension 2. cl For any x ∈ X ( E), let us also write (, ) for the pairing w(x) (, ) : V ⊗ V −→ E w(x) E X ,w(x) X ,w(x) induced from the pairing (52)by O -linearity. We summarize the basic properties of these objects (that we shall make use of in what follows) in Proposition 6.6 below. cl Proposition 6.6 Suppose x ∈ X ( E) is an E -valued classical point. (i) The Hecke-equivariant map V [[x ]] → V − → V X ,w(x) f is surjective. In particular, it induces an isomorphism V −→ V X ,x f (ii) The Hecke-equivariant map V [ f ] − → V [[x ]], x X is injective. In particular, it induces an isomorphism V [ f ] −→ V [x ]. x X (iii) The pairing (, ) veriﬁes the property Adj). (iv) The restriction of (, ) to V ⊗ V [x ] factors as w(x) X X ,w(x) (, ) w(x) V ⊗ V [x ] E X ,w(x) V ⊗ V [x ] E X ,x We shall denote the induced pairing on V ⊗ V [x ] by (, ) . X x X ,x (v) The diagram (, ) V ⊗ V [x ] E X ,x π ∼ ∼ j x x V ⊗ V [ f ] E (, ) f x 123 270 D. Benois, K. Büyükboduk where the bottom pairing is the Poincaré duality (49), commutes. Namely, for all v ∈ V and v ∈ V [ f ] one has X ,x (π (v ), v) = (v , j (v)) . x f x x Proof (i) The surjectivity of V [[x ]] → V follows from [10, Corollary 3.19], where the X f analogous statement is proved for the spaces of modular symbols, combined with Propo- sition 6.5(i). (ii) This is clear by deﬁnitions. (iii) This portion follows from the [14, §4.2] (see in particular the discussion following (69)). (iv) This is a particular case of the factorization (2). (v) This statement follows from the functoriality of the cup products. 6.5 Beilinson–Kato elements over the eigencurve cusp 6.5.1. We maintain previous notation and conventions in Sects. 5 and 6.Let x ∈ C ( E) be a classical cuspidal point on the eigencurve. Fix a wide open U , an afﬁnoid W and a slope ν ≥ 0 such that x ∈ W ⊂ U and the conditions of Proposition 6.5 are satisﬁed. Let X denote the connected component of x in C . Consider the following maps: 0 W,ν • For each non-negative integer k ∈ U, the map (54) which we notate as [k] 1 1 k ρ : H (Y , D (T )) −→ H (Y , TSym (T )). U Q U Q Q Q p et ´ p et ´ • For each classical x ∈ X , the map (55), which we shall notate as [x ] ρ : V → V . X X f Moreover, we have the following commutative diagram, which follows from the slope decomposition: 1 w(x) M (T ) H (Y , TSym T (1)) W Q Q et ´ (56) V V . X f Here, by deﬁnition, M (T ) := H (Y , D (T )) . W U W et ´ • The cyclotomic moment maps [r,n] mom : ( ) → E [G ](−r), r ∈ Z, n ≥ 0, ◦ n Q(ζ ) μ m [k] [x ] which have deﬁned deﬁned in §6.2.8. Together with the maps ρ and ρ , the cyclotomic U X moment maps give rise to the morphisms [k,r,n] cyc 1 1 ◦ ρ : H (Z[1/S,ζ ], H (Y , D (T )⊗ (2))) m ◦ U,m et ´ Q U μ 1 1 k −→ H (Z[1/S,ζ ], H (Y , TSym (T )(2 − r))) (57) mp et ´ Q and [x,r,n] 1 ι 1 ρ : H (Z[1/S,ζ ], V ⊗ ( ) (1)) −→ H (Z[1/S,ζ n ], V (1 − r)). m Q(ζ ) mp X ,m X m f (58) 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 271 It follows from the commutativity of the diagram (56) that the maps (57)and (58)are compatible in the evident sense. 6.5.2. The map (44) induces a morphism 1 1 ◦ H (Y , (T D )) −→ H (Y , D (T )). et ´ Q et ´ Q U The composition of this map with (35) gives rise to the map 1 1 ◦ ◦ H (Y (Np) , (2)) −→ H (Y , D (T )(2)) =: M (T )(1). (59) Q Np Q U U et ´ et ´ More generally, for each integer j ≥ 0, the morphism (48) together with the trace map induce 1 j 1 j ◦ H (Y (Np) , TSym (T ) ⊗ (2)) −→ H (Y , TSym (T ) ⊗ D (T )(2)) 1 Np et ´ Q Np et ´ Q U − j 1 ◦ ◦ − → H (Y , D (T )(2)) =: M (T )(1). Q U U et ´ (60) Deﬁnition 6.7 Let S,ξ, j, c, d, m be as in Deﬁnition 6.1. (i) We let [U ] 1 ◦ ι BK ( j,ξ) ∈ H Z[1/S,ζ ], M (T )⊗ ( ) (1) c,d m Q(ζ ) N,m et ´ U m denote the image of the Beilinson–Kato element BK ( j,ξ) under the map c,d N,m 1 1 j ι H Z[1/S,ζ ], H (Y (Np), TSym (T ) ⊗ ⊗ ( ) (2) m 1 Np Q(ζ ) Np m et ´ et ´ 1 ◦ ι −→ H Z[1/S,ζ ], M (T )⊗ ( ) (1) m Q(ζ ) et ´ U m induced by the map (60). (ii) Let x be a cuspidal point of the eigencurve C. Let W ⊂ U be an afﬁnoid disk centered at x such that W, U and ν ≥ 0 satisfy the conditions of Proposition 6.5.Wedenoteby [W,≤ν] 1 ≤ν ι BK ( j,ξ) ∈ H Z[1/S,ζ ], M (T ) ⊗ ( ) (1) c,d m W Q(ζ ) N,m [U ] the image of BK ( j,ξ) under the map induced by the natural projection M (T ) → c,d N,m U ≤ν M (T ) . (iii) Let X denote the connected component of x . We denote by [X ] 1 ι BK ( j,ξ) ∈ H (Z[1/S,ζ ], V ⊗ ( ) (1)) c,d m Q(ζ ) N,m X m [W,≤ν] the image of BK ( j,ξ) under the map induced by the natural projection c,d N,m ≤ν M (T ) → V . 6.5.3. The following is the interpolation property for O -adic Beilinson–Kato elements [X ] BK ( j,ξ). c,d N,m Theorem 6.8 In the setting of Deﬁnition 6.7, the following are valid: (i) For any k ≥ 0, the diagram [k] mom Np, j 1 j 1 j +k H (Y (Np) , TSym (T ) ⊗ ) H (Y (Np) , TSym (T )) 1 Np 1 Np Np et ´ Q et ´ Q (60) pr [ j +k] 1 ◦ 1 j +k H (Y , D (T )) H (Y , TSym (T )) Q U Q et ´ et ´ (61) 123 272 D. Benois, K. Büyükboduk [k] commutes. Here, mom is the moment map induced from (42), and the right vertical Np, j map pr is the trace map associated to the projection pr : Y (Np) → Y. ii) For any m ≥ 1 and k ≥ 0, the diagram (61) induces the commutative diagram [k,r,n] mom Np, j,μ cyc m 1 1 j 1 1 j +k H (Z[1/S,ζ ], H (Y (Np) , TSym (T ) ⊗ (2)) H (Z[1/S,ζ n ], H (Y (Np) , TSym (T )(2 − r))) m 1 Np ◦ mp 1 Np Q Np,μ Q et ´ et ´ pr pr ∗ ∗ [ j +k,r,n] U,m cyc 1 1 ◦ 1 1 j +k H (Z[1/S,ζ ], H (Y , D (T )⊗ (2))) H (Z[1/S,ζ n ], H (Y , TSym (T )(2 − r))), m ◦ mp Q U μ Q et ´ m et ´ (62) [k,r,n] where mom is the map deﬁned in Proposition 6.3. Np, j,μ cl (iii) For any x ∈ X ( E), the diagram (62) induces the following commutative diagram: [x,r,n] mom Np, j,μ cyc m 1 1 j 1 1 w(x) H (Z[1/S,ζm ], H (Y1(Np) , TSym (TNp) ⊗ ◦ (2)) H (Z[1/S,ζmp ], H (Y1(Np) , TSym T (2 − r))) f Q Np,μ Q x et ´ et ´ pr [x,r,n] X ,m 1 ι 1 H (Z[1/S,ζ ], V ⊗ ( ) (1)) H (Z[1/S,ζ ], V (1 − r)), m Q(ζm) mp X f x where the right vertical isomorphism is induced by the canonical isomorphism between 1 w(x) 1 w(x) f -isotypic components of H (Y (Np) , TSym (T )) and H (Y , TSym x 1 Np,Q Q Q et ´ et ´ [x,r,n] (T )) and the horizontal map mom on the ﬁrst row is the composition of Q ◦ Np, j,μ [w(x),r,n] mom with the projection to the f -component. Np, j,μ cl iv) For any x ∈ X ( E), one has [x,r,n] [X ] c ρ BK ( j,ξ) = BK n( f , j, r,ξ) c,d c,d Np,mp X ,m N,m x where c 1 n n c BK ( f , j, r,ξ) ∈ H (Z[1/S,ζ ], V (w(x) + 2 − r)) c,d Np,mp mp f is the projection of the Beilinson–Kato element BK (w(x) − j, j, r,ξ) intro- c,d Np,mp duced in §6.2.8 to the f -isotypic eigenspace for the action of Hecke operators {T } and{U } . Here we have used the canonical isomorphism (50) to identify Np |Np c 1 BK n( f , j, r,ξ) with an element of H (Z[1/S,ζ n ], V (1 − r)). c,d N,mp mp Proof (i) Consider the following diagram: [k] mom Np, j 1 j H (Y (Np) , TSym (T ) ⊗ ) 1 Np et ´ Q Np [Np] [k] ∗ mom 1 j 1 j 1 j +k H (Y (Np) , TSym (T ) ⊗ (T s )) H (Y (Np) , TSym (T ) ⊗ (T )) H (Y (Np) , TSym (T )) 1 Q Np Np Np 1 Q Np Np 1 Q Np et ´ et ´ et ´ pr 1 pr 2 pr ∗ ∗ ∗ [k] [ p]∗ mom 1 j 1 j 1 j +k H (Y , TSym (T ) ⊗ (T D )) H (Y , TSym (T ) ⊗ (T )) H (Y , TSym (T )) et ´ Q et ´ Q et ´ Q [k] ρ [ j +k] 1 j ◦ 1 ◦ H (Y , TSym (T ) ⊗ D (T )) H (Y , D (T )) et ´ Q U − j ∗ et ´ Q U (60) 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 273 It follows from deﬁnitions that the map (60) decomposes as explained in the diagram. [k] The factorization of the moment map mom follows from its deﬁnition (c.f. [24], Np, j §4) and the fact that it agrees with the description given in Section 6.2.7 proved in [24, Theorem 4.5.2.2]. Therefore, it sufﬁces to prove that this diagram commutes. The commutativity of the squares 1and 2 follow from the functoriality of moment maps. The commutativity of the triangle 3 follows from [26, Proposition 4.2.10]. The commutativity of 4 is clear from deﬁnitions. (ii–iii) These assertions follow immediately from (i). (iv) This portion follows from (iii) and Proposition 6.3. Corollary 6.9 We retain the notation and hypotheses of Theorem 6.8.Welet c 1 ι 1 ι BK ( f , j,ξ) ∈ H (Z[1/S], V c(w(x) + 2) ⊗ ) = H (Z[1/S], V ⊗ (1)) c,d Np,Iw f x f x x c 1 n n denote the unique class which specializes to BK ( f , j, r,ξ) ∈ H (Z[1/S,ζ ], c,d Np, p p V c(w(x) + 2 − r)) for all n ∈ N under the evident morphisms. Then, [X ] c BK ( j,ξ, x) = BK ( f , j,ξ) c,d c,d Np,Iw N,1 x [X ] [X ] where BK ( j,ξ, x) is the specialization of BK ( j,ξ) to x . c,d c,d N,1 N,1 cl Remark 6.10 We retain the notation and hypotheses of Theorem 6.8. Suppose that x ∈ X ( E) is a classical point and f is p-old, arising as the p-stabilization of a newform g of level N coprime to p, with respect to the root a ( f ) of the Hecke polynomial of g at p.Then f is p x the p-stabilization of the newform g with respect to the root a ( f ) of the Hecke polynomial p x c c of g at p. In this remark, we shall explain the relation between BK ( f , j,ξ) and c,d Np,Iw c c the Beilinson–Kato element BK (g , j,ξ) that one associates to the newform g .The c,d N,Iw pr natural projection Y (1, N( p)) =: Y − → Y ( N) induces the trace map 1 k ∨ 1 k ∨ pr : H (Y , Sym T ) −→ H (Y ( N) , Sym T ), ∗ Q Q Q Q et ´ ,c p et ´ ,c p which restricts to an isomorphism (since this map commutes with the action of Hecke oper- ators {T } ) Np c c pr : V V . (63) f g Then by construction, the image of the Beilinson–Kato element BK (g , j,ξ) under the c,d N,Iw natural morphism induced from the isomorphism (63) coincides with BK ( f , j,ξ) c,d Np,Iw (c.f. [22], Proposition 8.7). 6.6 Normalized Beilinson–Kato elements In Sect. 6.6, we normalize the O -adic Beilinson–Kato elements (in a weak sense). These normalized elements interpolate Kato’s normalized elements in [22, §13]. 6.6.1. We recall that ( ) := Z [[ ]], where = Gal(Q (ζ )/Q (ζ )) and = 1 p 1 1 p p p p Z []⊗ ( ), where = Gal(Q(ζ )/Q). One then has p Z 1 p = ( )e . 1 η η∈ X() Let us set ( ) = O ⊗ ( ) and put = O ⊗ . X 1 X Z 1 X X Z p p 123 274 D. Benois, K. Büyükboduk We start with the following auxiliary lemma. Lemma 6.11 Let M be a ﬁnitely generated ( )-module. Suppose that M := M/XM X 1 x is torsion-free as a ( )[1/ p]-module and that M [ X]= 0. Then there exist an afﬁnoid disk X ⊂ X such that M := M ⊗ O is torsion-free as a ( )-module. O O 1 X X X X Proof Suppose on the contrary that 0 = m ∈ M is a torsion element. For sufﬁciently tor small X ⊂ X , the afﬁnoid domain O is a PID. Let us choose X with that property and assume until the end of our proof that X = X .Since is a UFD, we may choose m so that m us annihilated by an irreducible element 0 = f ∈ . Note that since M [ X]= 0, the irreducible element f cannot be an associate of X and moreover, we can assume without loss of generality that m ∈ M \ XM. Let us write f = a ( X)(γ − 1) where a ( X) ∈ O .Since M is torsion-free n 1 n X x n=0 0 as a ( )[1/ p]-module, we infer that a (0) = 0. Indeed, since we have f · (m + XM) = 1 0 a (0)(m + XM) ∈ ( M ) ={0} and as m + XM = 0 as an element of the torsion-free 0 x tor module M/XM = M , the required conclusion that a (0) = 0 follows. This in turn shows x 0 that X | f .Since f is irreducible, this in turn implies that f is an associate of X, contrary to the discussion in the previous paragraph. This concludes the proof of our lemma. 6.6.2. Fix a p-stabilized normalized eigenform f ∈ S ( ( N) ∩ ( p)) where ( N, p) = k +2 1 0 1. Let us denote by x the corresponding point of the cuspidal eigencurve and set 1 1 ι H Z[1/S], V (1) := H Z[1/S], V ⊗ (1) , Iw X X p 1 1 ι H Z[1/S], V (1) := H Z[1/S], V ⊗ (1) . Iw f f p Lemma 6.12 There exists a neighborhood X of x such that H Z[1/S], V (1) is torsion- Iw X free as a ( )-module. X 1 Proof We apply Lemma 6.11 to M = H Z[1/S], V (1) . The long exact sequence of Iw X Galois cohomology associated to the short exact sequence ι ι ι 0 → V ⊗ (1) − → V ⊗ (1) → V ⊗ (1) → 0 Z Z Z X p X p f p showsthatwehaveaninjection M/XM → H (Z[1/S], V (1)). Iw f By Kato’s fundamental result (c.f. [22], Theorem 12.4), the [1/ p]-module H (Z[1/S], Iw V (1)) is free of rank one. Therefore M = M/XM is also torsion-free. The same long exact sequence together with the fact that H (Z[1/S], V (1)) = 0gives Iw f 1 0 1 M [ X]= H Z[1/S], V (1) [ X]= im H (Z[1/S], V (1)) −→ H (Z[1/S], V (1)) = 0. Iw X Iw f Iw X Thus the hypotheses of Lemma 6.11 are satisﬁed and one can shrink X so as to ensure that M is torsion-free. Let Q( ) (resp., Q( )) denote the total quotient ﬁeld of (resp., of ). It follows X X from Lemma 6.11 that on shrinking X as necessary, one can ensure that the natural 1 1 H (Z[1/S], V (1)) −→ H (Z[1/S], V (1)) ⊗ Q( ) X X is an injection. 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 275 6.6.3. We consider the product −1 E (x) = (1 − a (f)σ ) ∈ (64) N X | N as a two variable function in the weight and cyclotomic variables. Lemma 6.13 For each classical point x and integer m, we have E (x,χ ) = 0 provided w(x) w(x) w(x)+1 +1 that m ∈{ / , }. In particular, we have E (x,χ ) = 0. 2 2 Proof Accordingto[33, Corollary 2] (the Weight-Monodromy Conjecture for modular w(x) w(x)+1 2 2 forms), a ( f ) is an algebraic integer with complex absolute value either or .In w(x) w(x)+1 m −m particular, we have E (x,χ ) = (1 − a ( f ) ) = 0 whenever m ∈{ / , }. N x 2 2 6.6.4. For an integer c,welet σ ∈ denote the element deﬁned by the requirement that χ(σ ) = c. We shall consider σ also as an element of in the obvious manner. Following c c Kato, we deﬁne the partial normalization factors 2 − j μ (c, j) = c − c σ , 1 c 2 j −w(x) (65) μ (d, j, x) = d − d σ , 2 d μ (c, d, j, x) := μ (c, j)μ (d, j, x) ∈ . 0 1 2 X w(x) k w(x)−k w(x)−k 0 0 0 Here, for any x ∈ X,wedeﬁne d := d d ,where d := exp((w(x) − w(x)−k w(x)−k cl 0 0 k ) log m ). We remark that d = d whenever x ∈ X ,since w(x) − k is 0 0 divisible by p − 1. We also deﬁne the full normalization factor: μ(c, d, j, x) := μ (c, d, j, x)E (x) ∈ . (66) 0 N X [X ] [X ] 6.6.5. To simplify notation, we shall write BK ( j,ξ) in place of BK ( j,ξ). When c,d c,d N N,1 c, d and j are understood, we will write μ in place of μ (c, d, j, x) for ? ∈{0, ∅}.We ? ? −1 −1 denote by [μ ]⊂ Q( ) the -subalgebra generated by μ (c, d, j, x) .Wealso X X X 0 −1 −1 denote by [μ ]⊂ Q( ) the [ ]-subalgebra generated by μ (c, d, j, x ) . 0 0 x p Deﬁnition 6.14 Let us ﬁx W so that H Z[1/S], V (1) is torsion-free as a -module (we Iw X can choose such W thanks to our discussion in §6.6.2). Let us ﬁx integers c ≡ 1 ≡ d mod N 2 2 with (cd, 6 p) = 1and c = 1 = d , and an integer j ∈[0, k ]. We deﬁne the partially normalized Beilinson–Kato elements on setting [X ] [X ] −1 1 −1 BK ( j,ξ) := μ (c, d, j, x) BK ( j,ξ) ∈ H Z[1/S], V (1) ⊗ [μ ], 0 c,d X Iw X X N N 0 ξ ∈ SL (Z). The integrality properties of the partial normalizations of the Beilinson–Kato elements are established in Proposition 6.15, whose proof has been relegated to Appendix A. Proposition 6.15 Suppose c, d and j are as in Deﬁnition 6.14. i) The partially normalized Beilinson–Kato element [X ] [X ] −1 BK ( j,ξ) := μ (c, d, j, x) BK ( j,ξ) 0 c,d N N is independent of the choice of c and d . [X ] ii) There exists an afﬁnoid neighborhood W of k so thatBK ( j,ξ) ∈ H Z[1/S], V (1) . N Iw X Proof The ﬁrst assertion is Proposition A.2, whereas the second is Proposition A.4 combined with (i). 123 276 D. Benois, K. Büyükboduk 6.6.6. We check that μ(c, d, j, x) agrees with the normalization factor considered in [22, cl §13.9], whenever x ∈ X . For each newform g of weight k Kato normalizes the elements that he denotes by ( p) z (g, k, j,ξ, prime( pN)) ∈ H (Z[1/S], V ). c,d n g p Iw n≥1 ( p) These elements are given as the Hecke equivariant projection of the elements z (k, k, j, c,d n ξ, prime( pN)) recalled at the end of Sect. 6.2 to the g-isotypic Hecke eigenspace. He n≥1 introduces the normalization factor 2 k+1− j 2 j +1 −k −1 μ (c, d, j, g) := (c − c σ )(d − d σ ) (1 − a (g) σ ), Kato c d | N (0) Kato Kato see [22, §13.9]. cl Suppose x ∈ X ( E) is a classical point. As we have explained in Corollary 6.9 (see also [X ] Remark 6.10), our big Beilinson–Kato element BK ( j,ξ) interpolates the elements c,d N,1 c 1 cl BK ( f , j,ξ) ∈ H (Z[1/S], V (w(x) + 2)), x ∈ X ( E) c,d Np,Iw f x Iw which coincide with the f -isotypic projection of Kato’s element z (k + j +2, 0, j +1, c,d x p ξ, prime( pN)) ,where w(x) = k + j. Kato’s element z (k + j + 2, 0, j + 1, c,d n≥1 ξ, prime( pN)) differs from n≥1 z (k + j + 2, k + j + 2, j + 1,ξ, prime( pN)) (67) c,d n≥1 w(x)+2 k+ j +2 by the twist by χ = χ . Recall that we denote by Tw : → the twisting −m morphism which is given on the group like elements by γ → χ (γ )γ . Since f is new away from p, it follows from [10, Lemma 2.7] that one can shrink X as cl necessary to ensure that f is new away from p and p-old for all x ∈ X ( E) \{x }.There x 0 are two scenarios: (i) f is a newform (so x = x and f = f , owing to our choice of X ). Then f is a x 0 x newform too. (ii) f is the p-stabilization of a newform f =: g of level N coprime to p with respect to the root a ( f ) of the Hecke polynomial of g at p. As we have explained p x c c in Remark 6.10, f is a p-stabilization of the newform g and one naturally c c identiﬁes BK ( f , j,ξ) with BK (g , j,ξ).Inthisscenario,wehave c,d Np,Iw c,d N,Iw c c μ (c, d, j, f ) = μ (c, d, j, g ). Kato Kato We then have, in both scenarios, Tw (μ (c, d, j, f )) k+ j +2 Kato ⎛ ⎞ 2 k+2 2 j +2 −k− j −2 −1 ⎝ ⎠ = Tw (c − c σ )(d − d σ ) (1 − a ( f ) σ ) k+ j +2 c d x | N −1 2 − j 2 j −w(x) =(c − c σ )(d − d σ ) (1 − a ( f )σ ) =: μ(c, d, j, x). c d x | N 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 277 c 1 In [22, Section 13], Kato shows that there exists an element z(g , j,ξ) ∈ H Z[1/S], V Iw (w(x) + 2) such that c c BK (g , j,ξ) = μ(c, d, j, x) z(g , j,ξ). c,d N,Iw ( p) We remark that in Kato’s notation, we have z(g , j,ξ) = Tw (z ),where γ is the w(x)+2 γ cohomology class associated to j and ξ. Combining this fact with the preceding remarks, we deduce that [X ] BK ( j,ξ, x) = E (x) z(g , j,ξ). (68) 7 “Étale” construction of p-adic L-functions Our objective in Sect. 7 is to use the O -adic Beilinson–Kato element we have introduced in Deﬁnition 6.7(iii) and 6.14 to give an “étale” construction of Stevens’ two-variable p-adic L- function in neighborhoods of non-θ-critical cuspidal points on the eigencurve. In particular, we assume throughout Sect. 7 that X is étale over W (i.e. e = 1); see our companion article [5] for the treatment of the complementary case (which concerns the neighborhoods of θ-critical points on the eigencurve). Recall the representations V and V givenasin(51). We will always assume that X is sufﬁciently small to satisfy the conditions of Proposition 6.15. For each classical cl x ∈ X ( E),wedenoteby f the corresponding eigenform. The O -adic Beilinson-Kato x X [X ] elements BK ( j,ξ) take coefﬁcients in the cohomology of the universal cyclotomic twist of V . Recall also that these representations are equipped with the O -linear pairing (, ) : V ⊗ V −→ O X X (c.f. (52), bearing in mind that we have assumed e = 1), which satisﬁes the formalism of Sect. 2. We will use repeatedly the properties of this pairing summarized in Proposition 6.6. Let f = f be a p-stabilized non-θ-critical cuspidal eigenform of weight k +2 ≥ 2and as before, which corresponds to the point x on the cuspidal eigencurve. The eigencurve is étale at x over the weight space (c.f. [10], Lemma 2.8). In other words, e = 1and O = O . 0 X W As we have explained in §3.3.2, the simpliﬁcation of the local behaviour of the eigencurve in this scenario exhibits itself also in the the p-local study (namely, the properties of the triangulation over the eigencurve). The reader might ﬁnd it convenient to identify X with a Coleman family f through f , so that the overconvergent eigenform f that corresponds to a point x ∈ X ( E) is simply the specialization f(x) of the Coleman family f. To simplify notation, we write R for the relative Robba ring R and H () for X O X the relative Iwasawa algebra H ().The p-adic representation V (given as in (51)) O X veriﬁes the properties (C1)–(C3) and therefore comes equipped with a triangulation D ⊂ D (V ) over the relative Robba ring R , verifying the properties ϕ )–ϕ ). Fix an X X 1 3 rig,O O -basis {η} of D (D) and let η ∈ D (D ) = D (D[x ]) denote its specialization at X cris x cris x cris x ∈ X ( E) (equivalently, at weight w(x)). Since O = O in this setting (therefore also H () = H ()), the G - X W X O Q,S equivariant pairing (, ) in (52)is O -linear. Recall from (8) that this pairing induces a canonical H ()-linear pairing 1 1 ι : H () ⊗ H (Q , V (1)) ⊗ H () ⊗ H (Q , V ) −→ H (). X p X p X X X Iw X X Iw (69) 123 278 D. Benois, K. Büyükboduk Recall the Perrin-Riou exponential map ψ =0 Exp : R ⊗ D (D) −→ H (Q , D) O cris p D,0 Iw X X 1 1 as well as the canonical embedding H (Q , D)→ H () ⊗ H (Q , V ) that the p X p X Iw Iw Fontaine–Herr complex gives rise to, through which we treat the image of Exp as a D,0 submodule of H () ⊗ H (Q , V ). X p X X Iw Consider the action of the complex conjugation on H (Z[1/S], V (1)). For all integers Iw X [X ],± c, d as in Deﬁnition 6.14,0 ≤ j ≤ k and ξ ∈ SL (Z),wedenoteby BK ( j,ξ) ∈ 0 2 c,d [X ] H (Q , V (1)) the ±-parts of the Beilinson–Kato element BK ( j,ξ). We similarly p X c,d Iw N [X ],± deﬁne BK ( j,ξ) ∈ H (Q , V (1)) (c.f. Proposition 6.15(ii)) for the partially normal- p X N Iw [X ] [X ] −1 ized Beilinson–Kato element BK ( j,ξ) := μ (c, d, j, x) BK ( j,ξ) 0 c,d N N Let us also denote by !( j) the sign of (−1) . Deﬁnition 7.1 (Arithmetic non-θ-critical p-adic L-function in two-variables)We set (sic!) [X ],!(k −1) + 0 ι L (f ; j,ξ) := res BK ( j,ξ) , c ◦ Exp ( η) ∈ H () p X p,η D,0 [X ],!(k ) − 0 ι L (f ; j,ξ) := res BK ( j,ξ) , c ◦ Exp ( η) ∈ H (), p X p,η D,0 where η = η ⊗ (1 + π). ± ± Deﬁnition 7.2 For any x ∈ X ( E), we put L (f ; j,ξ, x) = x ◦ L (f ; j,ξ, x) ∈ H (). p,η p,η ± ± For a character ρ ∈ X(),weset L (f ; j,ξ, x,ρ) := ρ ◦ L (f ; j,ξ, x). p,η p,η For x above,welet f denote the newform associated to f . We write α(x) for the U - x p eigenvalue on f , so that when f is p-old, it is the p-stabilization of the newform f with x x ◦ ◦ respect to the root α(x) of the Hecke polynomial of f at p.Welet L ( f ) denote the p,α(x) x x Manin–Višik p-adic L-function associated to an unspeciﬁed choice of (apairof) Shimura periods (see, however, item c) in the proof of Theorem 7.3 below, where this choice is made explicit). The following result (in varying level of generality) has been previously announced in the independent works of Hansen, Ochiai and Wang. Theorem 7.3 There exists a neighborhood X of x such that the following hold true. (i) There exist ( j ,ξ ) and ( j ,ξ ) such that L (f ; j ,ξ , x ) are nonzero elements of + + − − ± ± 0 p,η H (). cl (ii) Assume that ( j ,ξ ) satisfy the conditions in (i). Then for each x ∈ X ( E) such that ± ± v (α(x)) < w(x) + 1, the p-adic L -functions L (f ; j ,ξ , x) agree with the Manin– p ± ± p,η ◦ ± ± × Višik p-adic L -function L ( f ), up to multiplication by D E (x) where D ∈ E p,α(x) N and E (x) is the product of Euler factors at bad primes, given as in (64). (iii) Assume that v (α ) = k + 1, but f = f(x ) is not θ -critical. Then L (f ; j ,ξ , x ) p 0 0 0 ± ± 0 p,η agree with the one variable p-adic L -functions of Pollack–Stevens [32] up to multipli- ± ± × cation by D E (x),where D ∈ E is a constant. (iv) Let L (f, ) denote the two-variable p-adic L -functions of Bellaïche and Stevens associated to modular symbols ∈ Symb (X ) (c.f. [10], Theorem 3). Then there exist functions u (x) ∈ O such that ± ± ± L (f ; j ,ξ ) = u (x)E (x)L (f, ). ± ± N p p,η 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 279 cl Proof Let x ∈ X ( E) be a classical point as in the statement of the theorem. Let g denote the unique newform which admits f as a p-stabilization if f is p-old and g = f if f x x x x is new. By (68), the specialization of the big Beilinson–Kato element at x can be compared with the normalized Beilinson–Kato element for g: [X ] c BK ( j,ξ, x) = E (x) z(g , j,ξ). Since Perrin-Riou’s exponential map commutes with base-change, the specializations ± ± L (f ; j,ξ, x) of p-adic L-functions L (f ; j,ξ) coincide with one variable Perrin-Riou’s p,η p,η [X ] L-functions constructed using the element BK ( j,ξ, x): [X ],!(k −1) + 0 ι L (f ; j,ξ, x) := res BK ( j,ξ, x) , c ◦ Exp ( η ) ∈ H (), p x E p,η D ,0 N x − [X ],!(k ) ι L (f ; j,ξ, x) := res BK ( j,ξ, x) , c ◦ Exp ( η ) ∈ H (). p x E p,η N D ,0 c 1 We identify z(g , j,ξ) with an element of H (Z[1/S], V (1)) via the isomorphism V (1) x x Iw V (w(x) + 2) and deﬁne + c !(k −1) ι L (g, j,ξ) := res z(g , j,ξ) , c ◦ Exp ( η ) ∈ H (), p x E p,η D ,0 − c !(k ) ι L (g, j,ξ) := res z(g , j,ξ) , c ◦ Exp ( η ) ∈ H (). p x E p,η D ,0 Therefore ± ± L (f ; j,ξ, x) = E (x)L (g, j,ξ). (70) p,η p,η The following statements concerning L (f ; j,ξ, x) areprovedbyKato: p,η + − a) There exist ( j ,ξ ) and ( j ,ξ ) such that L (f ; j ,ξ , x) and L (f ; j ,ξ , x) are + + − − + + − − p,η p,η non-zero functions. This follows from Kato’s explicit reciprocity law [22, Theorem 6.6] together with Ash–Stevens theorem (see [22], Theorem 13.6), the non-vanishing results of Jacquet–Shalika and Rohrlich (see [22], Theorem 13.5) and the fact that the product E (x) ∈ H () of bad Euler factors is a non-zero-divisor (c.f. Lemma 6.13). N E ± ± ± b) For any pair ( j,ξ), there exist constants C = C ( j,ξ) ∈ E such that L (g; j,ξ, x) = p,η ± ± C L (g; j ,ξ , x). This follows from Kato’s explicit reciprocity law and the fact that ± ± p,η H (Z[1/S], V (1)) is free of rank one over [1/ p]. See [22, §13.9]. Iw c) If v (α(x)) < w(x) + 1, the functions L (g; j ,ξ ) agree with the Manin–Višik p ± ± p,η p-adic L-functions associated to the Shimura periods that come attached to the choice of the classes δ(g, j ,ξ ) (see [22] for precise deﬁnitions). ± ± Plugging in x = x in (70)wehave ± ± L (f ; j,ξ, x ) = E (x)L (g, j,ξ). 0 N p,η p,η Applying a) and b), and noticing that E (x ) ∈ H () is a non-zero-divisor (c.f. N 0 E Lemma 6.13), the proof of Part (i) follows. Let us choose ( j ,ξ ) as in (i), so that L (f ; j ,ξ , x ) is non-zero. In particu- + + + + 0 p,η lar, for each even integer 1 ≤ i ≤ p − 1, there exists a character χ ∈ X( ) so that i 1 + i + i L (f ; j ,ξ , x ,ω χ ) = 0. Since L (f ; j ,ξ , x,ω χ ) is analytic in the variable + + 0 i + + i p,η p,η x (where ω is the Teichmüller character), there exists a neighborhood X of x such that + + L (f ; j ,ξ , x) is a non-zero-divisor of H () ,for all x ∈ X . Similarly, one checks + + E p,η − − that L (f ; j ,ξ , x) is a non-zero-divisor of H () . Combining these observations with − − E p,η (a), (b) and (c) above, we conclude the proof of (ii). 123 280 D. Benois, K. Büyükboduk We next prove Part (iv). Using the Amice transform, we can consider the functions L (f, ) as elements of H (). It follows from [10, Theorem 3] for classical non-critical p X points x ∈ X ( E) that L (f, , x) coincides, up to multiplication by a non-zero constant, with the Manin–Višik p-adic L-function. We infer from (i) that for classical non-critical cl points x ∈ X ( E),wehave + + L (f ; j ,ξ , x) = u E (x)L (f, , x) + + x N p p,η for some constant u ∈ E, with u = 0 by the choice of ( j ,ξ ). On expressing x x + + + + L (f ; j ,ξ , x) and E (x)L (f, , x) as power series with coefﬁcients in O ,we + + N p X p,η + + immediately deduce that there exists a function u (x) ∈ O such that u (x ) = 0and X 0 + + + L (f ; j ,ξ , x) = u (x)E (x)L (f, , x). + + N p p,η On shrinking X as necessary, we can ensure that u (x) does not vanish on X . The same − − − − argument proves that L (f ; j ,ξ , x) = u (x)L (f, , x) for some u (x) ∈ O . This − − p X p,η concludes the proof of (iv). Part (iii) is a direct consequence of (iv) (as per the deﬁnition of the Pollack–Stevens p-adic L-function) and our theorem is proved. We note that in the particular case when f is the Coleman family passing through a newform 0 k /2 f = f of level (Np) and weight k + 2 with α = a ( f ) = p , the conclusions of 0 0 0 0 p 0 Theorem 7.3 play a crucial role in [6]. When f = f has critical slope but not θ-critical, it is also crucially used in the proof of a conjecture of Perrin-Riou in [11]. Acknowledgements The ﬁrst named author (D.B.) wishes to thank the second author (K.B.) for his invitation to University College Dublin in April 2019 and Bogaziçi ˇ University of Istanbul in December 2019. This work was started during these visits. D.B. was also partially supported by the Agence National de Recherche (grant ANR-18-CE40-0029) in the framework of the ANR-FNR project “Galois representations, automorphic forms and their L-functions”. We thank the anonymous referees for carefully reading our work and their very helpful comments, which guided us towards many technical and stylistic improvements to the earlier versions of our article. Funding Open Access funding provided by the IReL Consortium. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix A: Integrality of normalizations w(x) k w(x)−k 0 0 For any x ∈ X , let us deﬁne d := d d . We deﬁne the partial normalization factor 2 − j 2 j −w(x) μ (c, d, j, x) := (c − c σ ) (d − d σ ) ∈ (71) 0 c d X μ (c, j) μ (d, j,x) 1 2 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 281 so that we have −1 μ(c, d, j, x) = μ (c, d, j, x) (1 − a (f)σ ) | N for the normalization factor μ(c, d, j, x) given as in (66). A.1 The normalization factors revisited × −1 Let ﬄ : G → denote the universal cyclotomic character, so that ﬄ gives the cyc Q cyc action on . We also deﬁne the universal cyclotomic character parametrized by the weight space ﬄ as the compositum wt × × × × ﬄ : G − → Z → Z [[Z ]] → O . wt Q p p p W Let γ ∈ denote a ﬁxed topological generator. Lemma A.1 Suppose c and d are integers coprime to p. (i) We have − j j +2 −c μ (c, j) = σ − χ (σ ). 1 c c In particular, if c is a primitive root modulo p (so that σ generates ), then j +2 μ (c, j) = ˙ γ − χ (γ ),where“= ˙ ” means equality up to a unit multiple. (ii) We have j −w(x) − j +2 d μ (d, j, x) = σ −ﬄ χ (σ ). 2 d wt d In particular, if d is a primitive root modulo p (so that σ generates ), then − j +2 μ (d, j, x) = ˙ γ −ﬄ χ (γ ). 2 wt 2 cl (iii) Suppose c and d are primitive roots modulo p .For each y ∈ X ( E) with w( y) = 2 j, the elements μ (c, j) and μ (d, j, y) of ⊗ O are coprime. 1 2 Z E Proof Direct calculation. A.2 Dependence on c and d In this subsection, we prove the ﬁrst part of Proposition 6.15. Proposition A.2 Assume that X is sufﬁciently small. (i) Suppose c, c , d, d , j are auxiliary integers so that (c, d, j) and (c , d , j) are as in Deﬁnition 6.1.Then [X ] [X ] μ (c , d , j, x) BK ( j,ξ) = μ (c, d, j, x) BK ( j,ξ). 0 c,d 0 c ,d N N (ii) For c, d, j as above, the partially normalized Beilinson–Kato element [X ] [X ] −1 1 −1 BK ( j,ξ) := μ (c, d, j, x) BK ( j,ξ) ∈ H (Z[1/S], V (1)) ⊗ [μ ] 0 c,d X Iw X N N X 0 is independent of the choice of c and d . Here and elsewhere, μ is a shorthand for μ (c, d, j) when c, d, j are understood. 123 282 D. Benois, K. Büyükboduk Before we proceed with the proof of Proposition A.2, we record the following auxiliary lemma. Lemma A.3 Suppose R is an integral domain and M is a ﬁnitely generated torsion-free R- module. Let { P } be an inﬁnite collection of prime ideals of R such that P ={0}. i i ∈ I i i ∈ I Then, P M ={0}. i ∈ I Proof of Lemma A.3 Let r be the rank of M. Then M contains a free submodule M := ⊕ Re 0 k k=1 of rank r,and M/ M is annihilated by multiplication by some a ∈ R. Let m ∈ P M 0 i i ∈ I be any element. Then am belongs to P M .Since am can be written in a unique way i 0 i ∈ I in the form am = α e,α ∈ R, k k k k=1 we see that α ∈ P ={0} for all k. k i i ∈ I Proof of Proposition A.2 We will proceed in two steps. 1 cl (a) We consider H (Z[1/S], V (1)) as a module over ( ). For any x ∈ X ( E), let X 1 Iw X m ⊂ ( ) denote the kernel of the evaluation-at-x map x X 1 ∞ ∞ i i ( ) −→ ( ), a ( X)(γ − 1) −→ a (x)(γ − 1) . X 1 E 1 i 1 i 1 i =0 i =0 It is clear from its deﬁnition that m is the principal ideal generated by ( X − x). Then cl m = 0. Combining Lemmas 6.12 and A.3, we deduce for sufﬁciently small x ∈X ( E) X that m H (Z[1/S], V (1)) ={0}. Iw X cl x ∈X ( E) b) It follows from [22, §13.9] that [X ] [X ] μ (c , d , j, x) BK ( j,ξ) − μ (c, d, j, x) BK ( j,ξ) 0 c,d 0 c ,d N N 1 1 ∈ ker H (Z[1/S], V (1)) → H (Z[1/S], V (1)) Iw X Iw x = m H (Z[1/S], V (1)) Iw X cl for all x ∈ X . Now the part (i) of the proposition follows from (a). The part (ii) is clear. A.3 Integrality of partial normalizations A.3.1. We will next analyze the regularity of the partially normalized Beilinson–Kato [X ] [X ] element BK ( j,ξ). This amounts to an analysis of the divisibility of BK ( j,ξ) by c,d N N μ (c, d, j, x). In view of Proposition A.2, we may (and henceforth will) work with c and d which are both primitive roots mod p , without any loss. 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 283 Proposition A.4 Suppose c and d are primitive roots modulo p . There exists an afﬁnoid [X ] neighborhood W of k and a unique element BK ( j,ξ) ∈ H Z[1/S], V (1) so that N Iw X [X ] [X ] BK ( j,ξ) = μ (d, j, x) ·BK ( j,ξ). c,d 0 N N [X ] The uniqueness ofBK ( j,ξ) follows from Lemma 6.12 (on shrinking X as necessary). The existence will be proved in the §A.3.2 and §A.3.2 below. A.3.2. We ﬁrst prove that there exists an afﬁnoid neighborhood W of k such that [X ] BK ( j,ξ) ∈ μ (c, j) · H Z[1/S], V (1) (72) c,d 1 Iw X for all ξ ∈ SL (Z). Considering the long exact sequence of Galois cohomology induced from 0 −→ V (− j − 1) − → V (− j − 1) −→ V (− j − 1) −→ 0 X X f together with the vanishing of H (Z[1/S], V (− j − 1)) (which follows from the fact that V is irreducible), we infer that the O -module H (Z[1/S], V (− j − 1)) has no X-torsion. f X On shrinking X as necessary, we may therefore ensure that it is torsion free, and therefore free (since O is a PID) as a O -module. As a matter of fact, on shrinking X , we can ensure X X c cl that the L-value L( f ,w(x) − j) is non-central for any x ∈ X ( E) \{x }. For all such x, it follows from [22, Theorem 14.5] and the isomorphism V (− j − 1) V c(w(x) − j) that H (Z[1/S], V (− j − 1)) is a 1-dimensional E-vector space. This in turn shows that H (Z[1/S], V (− j − 1)) is a free O -module of rank one. j +2 The specialization map γ → χ (γ ) induces an exact sequence j +2 γ −χ (γ ) ι ι 0 −→ V (1)⊗ − −−−−− → V (1)⊗ − → V (− j − 1) −→ 0, Z Z X p X p X which gives rise to an exact sequence j +2 γ −χ (γ ) 1 1 1 H (Z[1/S], V (1)) − − −−−− → H (Z[1/S], V (1)) −→ H (Z[1/S], V (− j − 1)). Iw X Iw X X cl For each x ∈ X ( E), let m ⊂ O denote the maximal ideal ( X −x). We have a commutative x X diagram m H (Z[1/S], V (− j − 1)) j +2 γ −χ (γ ) 1 1 1 H (Z[1/S], V (1)) H (Z[1/S], V (1)) H (Z[1/S], V (− j − 1)) Iw X Iw X X sp sp sp x x x j +2 γ −χ (γ ) 1 1 1 H (Z[1/S], V (1)) H (Z[1/S], V (1)) H (Z[1/S], V (− j − 1)), Iw x Iw x x with exact rows and columns, where the vertical maps that are denoted by sp are induced by the specialization at x. Let us denote by [X ] BK ( j,ξ)| ∈ H (Z[1/S], V (− j − 1)) c,d s= j +2 123 284 D. Benois, K. Büyükboduk [X ] the image of BK ( j,ξ) under the indicated cyclotomic specialization, and by c,d [X ] BK ( j,ξ, x) ∈ H (Z[1/S], V (1)) c,d Iw x [X ] its image under sp . Note that by Corollary 6.9, BK ( j,ξ, x) coincides with the classical c,d Beilinson–Kato element BK ( f , j,ξ). Now Kato’s integrality results in [22, §13.12] c,d Np,Iw (combined with the discussion in Remark 6.10 when f is p-old) show that [X ,x ] 1 1 BK ( j,ξ) ∈ μ(c, d, j, x) · H (Z[1/S], V (1)) ⊂ μ (c, j) H (Z[1/S], V (1)) c,d 1 Iw x Iw x j +2 for all c, d as before, integers j ∈[0, k ] and ξ ∈ SL (Z).Since μ (c, j) = ˙ γ − χ (γ ) 0 2 1 by Lemma A.1, an easy diagram chase shows that [X ] BK ( j,ξ)| ∈ m H (Z[1/S], V (− j − 1)). c,d s= j +2 x [X ] It follows from Lemma A.3 that BK ( j,ξ)| = 0. Therefore c,d s= j +2 [X ] j +2 1 BK ( j,ξ) ∈ (γ − χ (γ )) · H Z[1/S], V (1) , c,d N Iw X and (72) is proved. ( j,ξ) [X ] A.3.3. Let X ∈ H Z[1/S], V (1) denote an element such that BK ( j,ξ) = c,d 1,X Iw X N ( j,ξ) μ (c, j) X . We shall prove that there exists an afﬁnoid neighborhood W of k such that 1 0 1,X ( j,ξ) X ∈ μ (d, j, x) · H Z[1/S], V (1) . (73) 1,X Iw X The proof of this part is very similar to the veriﬁcation of (72) in §A.3.2, after obvious −1 j −1 modiﬁcations. We consider the Galois representation V (ﬄ χ ), in place of V (− j −1). X wt X −1 j −1 Note that the specialization of V (ﬄ χ ) at x is X wt −w(x)+ j −1 V (χ ) V ( j). Employing the same argument as in the ﬁrst paragraph of our proof of (72), one observes that −1 1 j −1 the O -module H (Z[1/S], V (ﬄ χ )) is free of rank one, on shrinking X if necessary. X wt 2− j The cyclotomic specialization γ → ﬄ χ(γ ) gives an exact sequence wt − j +2 γ −ﬄ χ (γ ) wt ι ι −1 j −1 0 −→ V (1)⊗ − −−−−−−−−→ V (1)⊗ − → V (ﬄ χ ) −→ 0. (74) Z Z X p X p X wt Consider the following commutative diagram with exact rows and columns: −1 1 j −1 m H (Z[1/S], V (ﬄ χ )) wt − j +2 γ −ﬄ χ (γ ) wt 1 1 1 −1 j −1 H (Z[1/S], V (1)) H (Z[1/S], V (1)) H (Z[1/S], V (ﬄ χ )) Iw X Iw X X wt sp sp sp x x x − j +2 γ −ﬄ χ (γ ) wt −1 1 1 1 j −1 H (Z[1/S], V (1)) H (Z[1/S], V (1)) H (Z[1/S], V (ﬄ χ )). x x x wt Iw Iw 123 Interpolation of Beilinson–Kato elements and p-adic L-functions 285 ( j,ξ) ( j,ξ) cl Let X (x) denote the specialization of X at x ∈ X ( E). Since 1,X 1,X ( j,ξ) [X ,x ] μ (c, j) X (x) = BK ( j,ξ) ∈ H (Z[1/S], V (1)) 1 c,d 1,X x coincides with the classical Beilinson–Kato element c 1 BK ( f , j,ξ) ∈ H (Z[1/S], V (1)). (75) c,d Np,Iw y x Iw Kato’s integrality results in [22, §13.12] (combined with the discussion in Remark 6.10 when f is p-old) show for such x that [X ] BK ( j,ξ, x) ∈ μ (c, d, j, x) · H (Z[1/S], V (1)), c,d 0 N Iw for all c, d as before, integers j ∈[0, k ] and ξ ∈ SL (Z). Since 0 2 μ (c, d, j, x) = μ (c, j)μ (d, j, x), 0 1 2 and H (Z[1/S], V (1)) is torsion-free (and even free) [1/ p]-module, we deduce that Iw ( j,ξ) X (x) ∈ μ (d, j, x) · H (Z[1/S], V (1)). 1,X Iw x ( j,ξ) −1 1 j −1 Let X | ∈ H (Z[1/S], V (ﬄ χ )) denote the cyclotomic specialization of s=w− j +2 1,X X wt ( j,ξ) · − j +2 X . Recall that μ (d, j, x) = γ −ﬄ χ (γ ). Now a diagram chase similar to the one 2 wt 1,X employed in §A.3.2 shows that ( j,ξ) 1 −1 j −1 X | ∈ m H (Z[1/S], V (ﬄ χ )). s=w− j +2 x 1,X X wt Since the intersection of any inﬁnite family of ideals m is zero, we deduce using Lemma A.3 that ( j,ξ) 1 −1 j −1 X | ∈ m H (Z[1/S], V (ﬄ χ )) ={0} s=w− j +2 x 1,X X wt cl x ∈X ( E) This shows that ( j,ξ) − j +2 1 X ∈ (γ −ﬄ χ (γ )) H (Z[1/S], V (1)), wt Iw X 1,X and (73) is proved. This also completes the proof of Proposition A.4. 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Annales mathématiques du Québec – Springer Journals
Published: Oct 1, 2022
Keywords: Eigencurve; θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-criticality; Triangulations; Beilinson–Kato elements; p-adic L-functions; 11F11; 11F67 (primary ); 11R23 (secondary )
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