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This article proves a case of the p-adic Birch and Swinnerton-Dyer conjecture for Garrett p-adic L-functions of [6], in the exceptional zero setting of extended analytic rank 2. Résumé Cet article prouve un cas de la conjecture p-adique de Birch et Swinnerton-Dyer pour les fonctions Lp-adiques de Garrett formulée dans [6], dans le cadre de zéros exceptionnels de rang analytique étendu égal à 2. Keywords Birch and Swinnerton-Dyer Conjecture · p-adic L-functions · Exceptional zeros Mathematics Subject Classiﬁcation 11F67 (11G40 11G35) Introduction Let A be an elliptic curve deﬁned over Q, having ordinary reduction at a rational prime p > 3. Let and be odd, irreducible, two-dimensional Artin representations of the 1 2 absolute Galois group of Q, which are unramiﬁed at p and satisfy the self-duality condition −1 det( ) = det( ) . 1 2 By modularity, the triple ( A, , ) arises from a triple ( f , g, h) of cuspidal p-ordinary 1 2 newforms of weights w = (2, 1, 1).Let f be the ordinary p-stabilisation of f ,and ﬁx o α B Rodolfo Venerucci rodolfo.venerucci@unimi.it Massimo Bertolini massimo.bertolini@uni-due.de Marco Adamo Seveso seveso.marco@gmail.com Universität Duisburg-Essen Fakultät für Mathematik, Mathematikcarrée Thea-Leymann-Straße 9, 45127 Essen, Germany Università degli Studi di Milano Dipartimento di Matematica Federigo Enriques, Via Cesare Saldini, 50, 20133 Milano, Italy 123 304 M. Bertolini et al.. p-stabilisations g and h of g and h respectively. Set = ⊗ . In the recent paper [6] α α 1 2 we proposed a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for the leading αα term at w of the 3-variable Garrett–Hida p-adic L-function L ( A,) = L ( f , g , h ) o p α p α associated with the triple ( f , g , h ) of Hida families specialising to ( f , g , h ) at w .In α α α α α o this article we verify our conjecture in the analytic rank-zero exceptional cases, viz. when the complex Garrett L-function L( A,, s) = L( f ⊗ g ⊗ h, s) does not vanish at s = 1 αα and L ( A,) has an exceptional zero at w in the sense of Mazur–Tate–Teitelbaum (cf. αα Theorem 2.1 and Sect. 2.1 below). Moreover, when L( A,, 1) = 0and L ( A,) has an exceptional zero, we propose a conjecture relating the value at w of the fourth partial αα derivative of L ( A,) along the f -direction to the p-adic logarithms of two global points on A rational over the number ﬁeld cut out by (cf. Conjecture 2.3). 1 Setting and notations ¯ ¯ Fix algebraic closures Q and Q of Q and Q respectively, and ﬁeld embeddings p p ¯ ¯ ¯ i : Q −→ Q and i : Q −→ C. With the notations of the Introduction, let p p ∞ ξ = a (ξ) · q ∈ S ( N ,χ ) n u ξ ξ ¯ n≥1 denote one of the cuspidal newforms f , g and h.Here u and N are the weight and the conductor of ξ respectively, and S ( N ,χ ) is the space of cuspidal modular forms of u ξ ξ F level ( N ), weight u, character χ and Fourier coefﬁcients in the subﬁeld F of Q .Fix a 1 ξ ξ number ﬁeld Q() containing for any ξ the Fourier coefﬁcients a (ξ), as well as the roots α and β of the pth Hecke polynomials P = X − a (ξ) · X + χ ( p) · p.Let V be a ξ ξ ξ, p p ξ two-dimensional Q()-vector space affording the representation ,and let K be a Galois number ﬁeld such that factors through Gal(K /Q).Set V = V ⊗ V and V ( A,) = V ( A) ⊗ V , Q() p p Q 1 2 where V ( A) = H ( A , Q (1)) is the p-adic Tate module of A with Q -coefﬁcients. p ¯ p p ét Q Throughout this note we make the following Assumption 1.1 1. (Self-duality) The characters χ and χ are inverse to each other. g h 2. (Local signs) The conductors N and N arecoprimeto p · N . g h f 3. (Étaleness) The forms g and h are cuspidal, p-regular and do not have RM by a real quadratic ﬁeld in which p splits. The ﬁrst condition is a reformulation of the self-duality condition mentioned in the Intro- −1 duction, namely det( ) = det( ) . Recall that the form ξ is p-regular if P has distinct 1 2 ξ, p roots. Moreover, one says that a weight-one eigenform has RM (real multiplication) if it is the theta series associated with a ray class character of a real quadratic ﬁeld. Assumption 1.1.3 is equivalent to require that V is irreducible, not isomorphic to Ind χ for a ﬁnite order character χ : G −→ Q() of a real quadratic ﬁeld K in which p splits, and that an arith- metic Frobenius at p acts on V with distinct eigenvalues. For ξ = g, h, this assumption guarantees that the p-adic Coleman–Mazur–Buzzard eigencurve of tame level N is étale over the weight space at the points corresponding to the p-stabilisations of ξ (cf. [2]). It is used in [6] to construct the Garrett–Nekovár ˇ height ⟪·, ·⟫ which appears in the main fg h α α result of this note. To explain the relevance of Assumptions 1.1.1 and 1.1.2, let α be the unit root of P andﬁxroots α and α of P and P respectively. Fix a ﬁnite extension f , p g h g, p h, p 123 On exceptional zeros of Garrett–Hida... 305 L of Q containing Q() and the roots of unity of order lcm( N , N , N ).Let ξ be one of f g h f , g and h,and let u be the weight of ξ. According to the results of [2,10,18], there exists a unique Hida family ξ = a (ξ ) · q ∈ O [[q ]] α α ξ n≥1 which specialises at u to the p-stabilised newform u−1 χ ( p) p ξ = ξ(q) − · ξ(q ) ∈ S ( p · M ,χ ) . α u ξ ξ L ord ( N ) p ξ Here M = N / p is the tame level of ξ (so that M = N if ξ = g, h), and O is the ξ ξ ξ ξ ξ ring of bounded analytic functions on a (sufﬁciently small) connected open disc U in the p-adic weight space over L. For each classical weight u in U ∩ Z , the weight-u ξ ≥3 specialisation ξ = a (ξ )(u) · q ∈ L [[q ]] of ξ is the q-expansion of the ordi- α,u n α α n≥1 nary p-stabilisation of a newform ξ in S ( M ,χ ) .Since f has a unique p-ordinary u u ξ ξ L p-stabilisation f , we simply write f for f . Assumption 1.1.1 guarantees that for each classical triple w = (k, l, m) in the set = U × U × U ∩ Z f g h ≥1 the complex Garrett L-function L( f ⊗ g ⊗ h , s) admits an analytic continuation to all k l m of C and satisﬁes a functional equation relating its values at s and k + l + m − 2 − s, with root number ε(w) = ε (w) equal to +1orto −1. Assumption 1.1.2 implies ≤∞ that all the local signs ε (w) are equal to +1for every w in the f -unbalanced region ={w = (k, l, m) ∈ : k ≥ l + m} (cf. [11]). Under these assumptions, [12] associates with ( f , g , h ) an analytic function αα L ( A,) = L ( f , g , h ) p α p α ˆ ˆ in the ring O = O ⊗ O ⊗ O , whose square fgh f L g L h αα 2 L ( A,) = L ( f , g , h ) = L ( f , g , h ) p α p α p α α satisﬁes the following interpolation property. For each w = (k, l, m) in ,the valueof αα L ( A,) at w is an explicit non-zero complex multiple of 2 2 2 2 β α α β β α β α β β β β k l m k l m k l m k l m 1 − 1 − 1 − 1 − · L( f ⊗ g ⊗h , c ). k l m w c c c c w w w w p p p p (1) k+l+m−2 Here c = ,and for ξ = f , g , h one denotes by α the unit root of P and w α u ξ , p α u u−1 sets β · α = χ ( p) · p ,where χ is the prime-to- p part of χ (so that χ = χ for u u ξ ξ ξ ξ ξ ξ = g, h,and χ is the trivial character modulo M ). We refer to Theorem A of loc. cit. αα for the precise interpolation formula. We call L ( A,) = L ( f , g , h ) the Garrett–Hida p α p α p-adic L -function associated with ( A,) (or with ( f , g , h )). α α 2 Exceptional zero formulae The p-adic variant of the Birch and Swinnerton-Dyer conjecture formulated in [6]predicts αα that the leading term of L ( A,) at w = (2, 1, 1) is encoded by the discriminant of the 123 306 M. Bertolini et al.. Garrett– Nekovár ˇ height pairing † † 2 ·, · : A (K ) ⊗ A (K ) −→ I /I (2) ⟪ ⟫ Q() fg h constructed in Section 2 of loco citato, where I is the ideal of functions in O which fgh vanish at w and the p-extended Mordell–Weil group A (K ) is deﬁned as follows. When A has good reduction at p, one sets A (K ) = A(K ) ,where A(K ) is a shorthand for the Gal(K /Q)-invariants of A(K ) ⊗ V .If A has multiplicative reduction at p, then α = a ( f ) =±1 and the maximal p-unramiﬁed quotient V ( A) of V ( A) is a f p p p 1-dimensional Q -vector space on which an arithmetic Frobenius acts as multiplication by α .Let q in pZ be the p-adic Tate period of the base change A of A to Q (cf. Chapter f A p Q Vof[15]), and let Q be the quadratic unramiﬁed extension of Q . The Tate uniformisation yields a rigid analytic morphism rig ℘ : G −→ A Tate Q m,Q 2 2 p with kernel q and unique up to sign. Set − − q( A) = p (℘ ( q )) ∈ V ( A) , Tate A n≥1 p − − where p denotes the projection V ( A) −→ V ( A) and ( q ) is any compatible p p A n≥1 system of p -th roots of q , and deﬁne A (K ) = A(K ) ⊕ Q ( A,) to be the direct sum of A(K ) and the Q()-submodule Q ( A,) = H (Q , Q() · q( A) ⊗ V ) p Q() 0 − of H (Q , V ( A) ⊗ V ). The Garrett–Nekovár ˇ height⟪·, ·⟫ depends on the choice p Q fg h α α of suitably normalised G -equivariant embeddings γ : V −→ V (g) and γ : V −→ V (h), (3) g h 1 2 where V (ξ) = V (ξ ) ⊗ L (for ξ = g, h) is the weight-one specialisation of the big Galois representation V (ξ ) associated with ξ . (We refer to Sect. 3.1 below for precise α α deﬁnitions.) More precisely, denote by V ( f ) the f -isotypic component of the cohomology group H ( X ( N , p) , Q (1)),where X ( N , p) is the base change to Q of the compact 1 f ¯ p 1 f ¯ ét Q Q modular curve X ( N , p) of level ( N ) ∩ ( p) over Q,and set 1 f 1 f 0 V ( f , g, h) = V ( f ) ⊗ V (g) ⊗ V (h). Q L Section 2 of [6] constructs a canonical Garrett–Nekovárp ˇ -adic height pairing † † 2 ⟪·, ·⟫ : Sel (Q, V ( f , g, h)) ⊗ Sel (Q, V ( f , g, h)) −→ I /I (4) fg h on the naive extended Selmer group of V ( f , g, h) over Q, deﬁned as the direct sum of the Bloch–Kato Selmer group Sel(Q, V ( f , g, h)) of V ( f , g, h) over Q and the module 0 − − H (Q , V ( f , g, h) ) of G -invariants of the maximal p-unramiﬁed quotient V ( f , g, h) of V ( f , g, h). (The deﬁnition of ⟪·, ·⟫ is brieﬂy recalled in Sect. 3.2.3 below.) Fix a fg h modular parametrisation ℘ : X ( N , p) −→ A, under which one identiﬁes V ( f ) and ∞ 1 f V ( A). The embeddings γ and γ and the global Kummer map on A(K ) then induce p g h † † an embedding γ : A (K ) −→ Sel (Q, V ( f , g, h)). The pairing (2)isdeﬁnedtobe gh ⊗2 composition of the canonical Garrett–Nekovár ˇ height and γ . The pairings (2)and (4) gh † † r ( A,) r ( A,)+1 ∗2 are skew-symmetric, and the discriminant of (2)in (I /I )/Q() ,where 123 On exceptional zeros of Garrett–Hida... 307 † † r ( A,) = dim A (K ) , is independent of the choice of ℘ , γ and γ .Werefer to Q() ∞ g h [6] for more details. If ξ denotes either g or h, then the restriction to G of the Artin representation V (ξ) is the direct sum of the submodules V (ξ) and V (ξ) on which an arithmetic Frobenius acts α β as multiplication by α and β respectively (cf. Assumption 1.1.3). The G -representation ξ ξ Q V ( f , g, h) then decomposes as the direct sum of the subspaces − − V ( f ) = V ( f ) ⊗ V (g) ⊗ V (h) , Q i L j ij p where (i, j) is apairofelementsof {α, β}.If ξ denotes either g or h, Sect. 3.1.1 below recalls the deﬁnition of canonical weight-one differentials G G nr nr Q Q p p ω ∈ (V (ξ) ⊗ Q ) and η ∈ (V (ξ) ⊗ Q ) , (5) ξ α Q ξ β Q α p p α p p nr where Q is the maximal unramiﬁed extension of Q .If A is multiplicative at p,set p p −1 − q( f ) = ℘ (q( A)) ∈ V ( f ) , − − where one denotes again by ℘ : V ( f ) V ( A) the isomorphism arising form the ﬁxed ∞ p modular parametrisation ℘ : X ( N , p) −→ A. ∞ 1 f Under the running assumptions, the Q()-module Q ( A,) (resp., the L-module 0 − H (Q , V ( f , g, h) )) is non-zero precisely A is multiplicative at p and α = α · α or α = β · α , f g h f g h in which case it has dimension 2 and one says that ( A,) is exceptional at p. More precisely, note that α = β by Assumptions 1.1.3, hence only one of the previous identities can be g g satisﬁed. Moreover α = α · α (resp., α = β · α ) if and only if α = β · β (resp., f g h f g h f g h α = α · β ) by Assumption 1.1.1. Fix an auxiliary integer m such that p splits (resp., is f g h p √ √ inert) in Q m if α =+1 (resp., α =−1), so that G acts trivially on m · q( f ) p f f Q p − nr − − in V ( f ) ⊗ Q .If α = α · α ,then G acts trivially on V ( f ) and V ( f ) , hence Q f g h Q p αα p p ββ the p-adic periods √ √ q = m · q( f ) ⊗ ω ⊗ ω and q = m · q( f ) ⊗ η ⊗ η αα p g h ββ p g h α α α α − − can naturally be viewed as elements of V ( f ) and V ( f ) respectively, which generate αα ββ 0 − H (Q , V ( f , g, h) ). Similarly, if α = β · α , then the periods f g h √ √ q = m · q( f ) ⊗ ω ⊗ η and q = m · q( f ) ⊗ η ⊗ ω αβ p g g βα p g h α h α α 0 − can naturally be viewed as generators of H (Q , V ( f , g, h) ). αα Equation (1) shows that the value of the square-root Garrett–Hida L-function L ( A,) at w is a non-zero multiple of α α β α α β β β g h g h g h g h 1 − 1 − 1 − 1 − · L( A,, 1), α α α α f f f f where L( A,, s) = L( f ⊗ g ⊗ h, s). The previous discussion then shows that ( A,) is exceptional at p precisely if one of the Euler factors which appear in the previous expression αα αα is zero, id est if L ( A,) (or L ( A,)) has an exceptional zero in the sense of Mazur– p p αα Tate–Teitelbaum [13]. In this case Lemma 9.8 of [7] proves that the restriction L ( A,)| αα of L ( A,) to the improving line L deﬁned by the equations m = 1and k = l + 1 admits the factorisation αα αα L ( A,)| = E · E · L ( A,) L f g p p 123 308 M. Bertolini et al.. in the ring O(L) of analytic functions on L,where a ( f ) a (g ) p p E = 1 − and E = 1 − χ ( p) · . f g h a (g ) · a (h ) a ( f ) · a (h ) p p α p p α L L αα Moreover, the value at w of the improved p-adic L-function L ( A,) is an explicit algebraic number in Q(), equal to zero precisely if L( A,, s) vanishes at s = 1. We refer to the proof of Proposition 8.3 of [12] for details. The following is the main result of this note. Theorem 2.1 Assume that ( A,) is exceptional at p. Let (q , q ) denote either the pair (q , q ) or (q , q ), depending on whether α = α · α or α = β · α respectively. αα ββ αβ βα f g h f g h Then the following equality holds in I /I up to sign. deg(℘ ) · (1 − β /α ) ∞ h h αα 2 αα L ( A,) (mod I ) = · L ( A,) (w ) · ⟪q , q ⟫ p p fg h m · ord (q ) p p A Theorem 2.1 is proved in Sect. 4 below. More precisely, Sects. 3.3 and 3.4 below prove that the following equality holds in I /I up to sign: 2 · deg(℘ ) an an an an · ⟪q , q ⟫ = L − L · (l − 1) + ε · L − L · (m − 1), f g f h fg h α α α α m · ord (q ) p p A (6) where ε =+1if α = α · α and ε =−1if α = β · β ,and where f g h f g h an − · L = d log a (ξ) (7) p u=u ξ o is the value at the centre u of U of the logarithmic derivative of the p-th Fourier coefﬁcient o ξ of the Hida family ξ = f , g , h . In Sect. 4 we then deduce Theorem 2.1 from Eq. (6)and α α αα the study carried out in [7, Section 9] of the linear term of L ( A,) at w in the exceptional case. It should be possible to extend Theorem 2.1 (and Conjecture 2.3 below) to the case of p-new eigenforms of even weight k ≥ 2 and trivial character (cf. Section 1.1 of [6]). We have not checked the details. 2.1 The rank-zero exceptional case of [6, Conjecture 1.1] Assume in this section that ( A,) is exceptional at p, and that the Garrett complex L-function L( A,, s) = L( f ⊗ g ⊗ h, s) does not vanish at s = 1: L( A,, 1) = 0. According to the main result of [8] (see also Theorem B of [3]), one has A(K ) = 0, † αα hence A (K ) = Q ( A,).The Garrett– Nekovár ˇ p-adic regulator R ( A,),viz.the discriminant of the p-adic height⟪·, ·⟫ on A (K ) , is then given by fg h αα R ( A,) = det ⟪q , q ⟫ = ⟪q , q ⟫ i j 1 2 p fg h fg h 1≤i, j ≤2 α α α 2 3 ∗2 in (I /I )/Q() ,where (q , q ) is a Q()-basis of Q ( A,). 1 2 p 123 On exceptional zeros of Garrett–Hida... 309 − − Let γ : V ( A,) −→ V ( f , g, h) be the G -equivariant embedding deﬁned by the gh Q − − tensor product of the isomorphism V ( A) V ( f ) induced by ℘ , γ and γ (cf. Eq. (3)). p ∞ g h The normalisation imposed on the embeddings γ and γ (and described in Sect. 3.1.1 below) g h implies that the matrix M in GL (L) deﬁned by the identity (q q ) · M = (γ (q )γ (q )) 2 gh 1 gh 2 has determinant in Q() . In light of the above discussion, Theorem 2.1 then proves the following corollary, which together with Eq. (6) establishes [6, Conjecture 1.1] in the present setting. Corollary 2.2 If L( A,, s) does not vanish at s = 1,then A (K ) = Q ( A,) and the 2 3 ∗2 following equality holds in the quotient of I /I by the action of Q() . αα 3 αα L ( A,) (mod I ) = R ( A,) p p 2.2 Exceptional zeros and rational points (cf. [14]) Assume in this section that ( A,) is exceptional at p, and that the Garrett complex L-function L( A,, s) vanishes at the central critical point s = 1: L( A,, 1) = 0. Set {, }={αα, ββ} of {, }={αβ, βα}, depending on whether α = α · α orα = β · α . f g h f g h αα 2 The p-adic L-function L ( A,) belongs to I (cf. Theorem 2.1) and Conjecture 2.3 of 2 3 ∗ [6] predicts that its image in (I /I )/Q() equals ⟪q , q ⟫ ⟪ P, Q⟫ −⟪q , P⟫ ⟪q , Q⟫ +⟪q , Q⟫ ⟪q , P⟫ fg h fg h fg h fg h fg h fg h α α α α α α α α α α α α for two rational points P and Q in A(K ) . (Recall that the p-adic height ⟪·, ·⟫ is fg h α α skew-symmetric, hence the previous expression is a square root of its discriminant on the Q()-submodule of A (K ) generated by q , q , P and Q.) One has ⟪q , q ⟫ (k, 1, 1) = 0 fg h α α by Eq. (6). Moreover, Sect. 3.5 below proves that ⟪q , x⟫ (k, 1, 1) = · log (res (x)) · (k − 2) (8) fg h for each Selmer class x in Sel(Q, V ( f , g, h)),where log =log (·), q : H (Q , V ( f , g, h)) −→ L. fgh p ﬁn p 1 0 Here log : H (Q , V ( f , g, h)) D (V ( f , g, h))/Fil is the Bloch–Kato p-adic loga- dR p p ﬁn ⊗2 rithm (cf. Lemma 9.1 of [7]), and ·, · : D (V ( f , g, h)) −→ L is the pairing induced dR fgh ⊗2 by the natural Kummer duality π : V ( f , g, h) −→ L(1) deﬁned in Sect. 3.1.1 below fgh (cf. Eq. (11)). We are then led to the following Conjecture 2.3 Assume that A(K ) is a 2-dimensional Q()-vector space. Then for any Q()-basis ( P, Q) of A(K ) , the equality 2 αα ∂ L ( A,) (w ) = log ( P) · log ( Q) − log ( P) · log ( Q) ∂ k holds in L up to multiplication by a non-zero scalar in Q() . 123 310 M. Bertolini et al.. As explained in [5], the main result of [1] can be used to prove cases of Conjecture 2.3 when g and h are theta series associated with certain ray class characters of the same imaginary quadratic ﬁeld in which p is inert (and P and Q are Heegner points). By combining this with an extension of the height computations carried out in [16,17], the article [4] proves instances of Conjecture 1.1 of [6] in this setting. Remark 2.4 In light of the aforementioned results of [5], Rivero proposes in [14, Conjecture 4.5] a variant of Conjecture 2.3. He also asks (cf. Question 5.3 of [14]) if one can expect 2 αα ∂ L ( A,) a similar description of (w ) when A has good reduction at p. The previous ∂ k discussion places Rivero’s conjecture within a conceptual framework and sheds some light on this question. 3 Height computations Throughout the rest of this note we assume that ( A,) is exceptional at p. In particular A has multiplicative reduction at p,idest p divides exactly N . 3.1 Setting and notations This subsection brieﬂy recalls the needed deﬁnitions and notations from our previous articles [6,7]. 3.1.1 Galois representations Set N = lcm( N , N , N ) and let G be the Galois group of the maximal extension f g h Q, N of Q contained in Q and unramiﬁed outside N ∞.If ξ denotes one of f , g and h ,let V (ξ) be the big Galois representation associated with ξ (cf. Section 5 of [7]). It is a free O -module of rank two, equipped with a continuous linear action G . For each u in ξ Q, N U ∩ Z the base change V (ξ) ⊗ L of V (ξ) along evaluation at u on O is canonically ξ ≥2 u ξ isomorphic to the homological p-adic Deligne representation of ξ with coefﬁcients in L (cf. loco citato for more details). In particular if ξ = f and u = 2 there is a natural specialisation isomorphism ρ : V ( f ) ⊗ L V ( f ).If ξ = g , h and u = 1set V (ξ) = V (ξ) ⊗ L (cf. 2 2 α 1 Sect. 1). It is a two-dimensional L-vector space affording the dual of the p-adic Deligne– Serre representation of ξ = g, h with coefﬁcients in L. In order to have a uniform notation, in this case one deﬁnes ρ : V (ξ) ⊗ L −→ V (ξ) to be the identity. 1 1 The restriction of V (ξ) to G (via the embedding i ﬁxed at the outset) ﬁts into a short Q p + − ± exact sequence of O [G ]-modules V (ξ) −→ V (ξ) − V (ξ) with V (ξ) free of rank one over O . More precisely, let χ : G −→ Z be the p-adic cyclotomic character, and ξ cyc Q let a ˇ (ξ) : G −→ O be the unramiﬁed character sending an arithmetic Frobenius to the p Q p ξ p-th Fourier coefﬁcients a (ξ) of ξ.Then + u−1 −1 − V (ξ) O χ · χ a ˇ (ξ) and V (ξ) O (a ˇ (ξ)), (9) ξ p p ξ ξ cyc u−1 ∗ u−1 u−1 where χ : G −→ O satisﬁes χ (σ)(u) = χ (σ) for each u in U ∩ Z. Q cyc ξ cyc cyc (The freeness of V (ξ) is guaranteed by Assumption 1.1.3, cf. Section 5 of [7].) If ξ = f and u = 2 the specialisation isomorphism ρ identiﬁes V ( f ) ⊗ L with the maximal 2 2 − + unramiﬁed quotient V ( f ) of V ( f ).If ξ = g , h and u = 1weset V (ξ) = V (ξ) ⊗ L α β 1 123 On exceptional zeros of Garrett–Hida... 311 − Frob =γ p ξ and V (ξ) = V (ξ) ⊗ L. One has V (ξ) = V (ξ) ⊕ V (ξ) ,where V (ξ) = V (ξ) α 1 α β γ for γ = α, β is the submodule of V (ξ) on which an arithmetic Frobenius Frob acts as multiplication by γ = α ,β (cf. Assumption 1.1.3). ξ ξ ξ There is a natural G -equivariant skew-symmetric perfect pairing u−1 π : V (ξ) ⊗ V (ξ) −→ O (χ · χ ), ξ O ξ ξ ξ cyc ± ∓ u−1 inducing perfect dualities π : V (ξ) ⊗ V (ξ) −→ O (χ · χ ).(SeeSection5cf.[7] O ξ ξ ξ cyc for the deﬁnitions). (4−k−l−m)/2 Denote by = χ : G −→ O the character whose composition with fgh cyc fgh (4−k−l−m)/2 evaluation at (k, l, m) in U × U × U ∩ Z on O equals χ .If · denotes one f g h fgh cyc of the symbols ∅, + and −,deﬁne · · ˆ ˆ V = V ( f ) ⊗ V (g )⊗V (h ) ⊗ . L α O fgh α fgh ± ± Then V = V ( f , g , h ),resp. V = V ( f , g , h ) is a free O -module of rank 8, resp. α α fgh α α 4, equipped with a continuous action of G ,resp. G .As χ ·χ = 1 (cf. Assumption 1.1), Q, N Q g h the product of the perfect dualities π ,for ξ = f , g , h , yields a perfect skew-symmetric ξ α Kummer duality π : V ⊗ V −→ O (1), inducing a perfect local Kummer duality O fgh fgh ± ∓ π : V ⊗ V −→ O (1). After setting O fgh fgh · · · V = V ( f , g, h) = V ( f ) ⊗ V (g) ⊗ V (h) L L ˆ ˆ and w = (2, 1, 1), the product ρ = ρ ⊗ρ ⊗ρ gives natural isomorphisms o w 2 1 1 · · ρ : V ⊗ L V (10) w w o o (where ·⊗ L denotes the base change along evaluation at w on O ). Let w o fgh π : V ⊗ V −→ L(1) (11) fgh L ± ∓ be the specialisation of π via ρ , and deﬁne π : V ⊗ V −→ L(1) similarly. w L nr nr − 0 − ˆ ˆ Weight one differentials Deﬁne D(ξ) = H (Q , V (ξ) ⊗ Q ),where Q is the p Q p p p p-adic completion of the maximal unramiﬁed extension of Q (and as usual ξ denotes one of f , g and h ). For each u in U ∩ Z there is a natural comparison isomorphism between α ξ ≥2 D(ξ) ⊗ L and the ξ -isotypic component of the space of cuspidal modular forms of weight u,level ( N p) and Fourier coefﬁcients in L. Assumption 1.1.3 guarantees that D(ξ) is 1 ξ free (of rank one) over O , and admits a basis ω whose image in D(ξ) ⊗ L corresponds ξ ξ u to ξ under the aforementioned comparison isomorphism, for each u in U ∩ Z . (We refer u ξ ≥2 to Section 3.1 of [6] and the references therein for more details.) For ξ = g , h ,the holomorphic weight-one differential α α nr ω ∈ (V (ξ) ⊗ Q ) ξ α Q α p p mentioned in Eq. (5) is deﬁned to be the weight-one specialisation of ω ,viz.the imageof ω ξ ξ in the quotient D(ξ) ⊗ L = D(ξ) . The weight-one specialisation of π yields a perfect 1 α G -equivariant skew-symmetric pairing π : V (ξ) ⊗ V (ξ) −→ L(χ ). ξ L ξ nr Q Let c be the common conductor of χ and χ , and identify (L(χ ) ⊗ Q ) with L via g h ξ Q p p i −1 2π ia/c the Gauß sum G(χ ) = (−c) ∗ χ (a) ⊗ e ,where i = 0and i = 1(so ξ ξ g h a∈(Z/cZ) 123 312 M. Bertolini et al.. that G(χ ) · G(χ ) = 1byAssumption1.1.1). Thepairing π then induces a perfect duality g h ξ nr ·, · : D(ξ) ⊗ D(ξ) −→ L,where D(ξ) = (V (ξ) ⊗ Q ) . One deﬁnes the ξ α L β γ γ Q p p antiholomorphic weight-one differential (cf. Eq. (5)) nr Q η ∈ (V (ξ) ⊗ Q ) ξ β Q α p to be the dual of ω under ·, · , viz. the element satisfying ω ,η = 1. ξ ξ ξ ξ α α α The embeddings and With the notations of Sect. 1,set V = V and V = V . g h g h 1 2 Let ξ denote either g or h. As recalled above, the Artin representation V (ξ) = V (ξ) ⊗ L affords the dual of the p-adic Deligne representation of ξ with coefﬁcients in L,idest is isomorphic to V ⊗ L. Enlarging L if necessary, we normalise the G -equivariant ξ Q() Q embedding γ : V −→ V (ξ) (introduced in Eq. (3)) by requiring that the composition ξ ξ ¯ ¯ π ◦ (γ ⊗ γ ) takes values in the number ﬁeld Q() (via the embedding i : Q −→ Q ξ ξ ξ p ﬁxed at the outset). 3.1.2 Selmer complexes Let R (Q, V ) be the Nekovár ˇ Selmer complex associated with (V , V ) (cf. Section 2.2 of [6]). It is an element of the derived category D (L) of cohomologically bounded complexes ft of L-modules with cohomology of ﬁnite type over L, sitting is an exact triangle p ◦res R (G , V ) −→ R (G , V ) −→ R (Q, V )[1], (12) cont Q, N cont Q f where R (G, ·) is the complex of continuous non-homogeneous cochains of G with cont ¯ ¯ values in ·,res is the restriction map (induced by the embedding i : Q −→ Q ﬁxed at p p − − the outset) and p is the map induced by the projection V −→ V .Denoteby · · ˜ ˜ H (Q, V ) = H (R (Q, V )) the cohomology of R(Q, V ),let Sel(Q, V ) be the Bloch–Kato Selmer group of V over + + Q,and let i : V −→ V be the natural inclusion. Then there is a commutative and exact diagram of L-vector spaces (cf. loc. cit.) 0 − 1 0 H (Q , V ) H (Q, V ) Sel(Q, V ) 0 (13) + res 1 + 1 H (Q , V ) H (Q , V ) p p where the ﬁrst line arises from the exact triangle (12). In addition there is a unique section 1 + ı : Sel(Q, V ) −→ H (Q, V ) of the above projection such that ı (x) belongs to the ur ur 1 + Bloch–Kato ﬁnite subspace H (Q , V ) for each x in Sel(Q, V ).Weoften use j and ı to p ur ﬁn identify Nekovár ˇ’s extended Selmer group H (Q, V ) with the naive extended Selmer group † 0 − Sel (Q, V ) = H (Q , V ) ⊕ Sel(Q, V ) (cf. Sect. 1). One similarly associates with (V, V ) a Selmer complex R (Q, V) ∈ D (O ) f fgh ft sitting in an exact triangle analogous to (12). (We refer to loc. cit. for more details.) 123 On exceptional zeros of Garrett–Hida... 313 3.2 Preliminary lemmas This section gives a concrete description of the functionals⟪q, ·⟫ : Sel (Q, V ) −→ L fg h 0 − for q in H (Q , V ) (cf. Lemma 3.4 below). 3.2.1 Bockstein maps Let (C, C) denote one of the pairs − − ˜ ˜ (R (V ), R (V )), (R(V), R(V )) and (R (Q, V), R (Q, V )), p p f f where R (·) and R(·) are shorthands for R (Q , ·) = R (G , ·) and p cont p cont Q R (G , ·) respectively (cf. Sect. 3.1.2). The specialisation maps ρ (cf. Eq. (10)) cont Q, N w induce isomorphisms L L 2 2 ρ : C ⊗ L C and ρ ⊗ id : C ⊗ I /I [1] C ⊗ I /I [1]. (14) w w L o O ,w o O fgh o fgh Applying C ⊗ · to the exact triangle fgh 2 2 2 I /I −→ O /I −→ L −→ I /I [1] fgh (arising from evaluation on w ) then yields a derived Bockstein map β : C −→ C ⊗ I /I [1], C/C which in turn induces in cohomology a Bockstein map i i +1 2 β : H (C) −→ H (C) ⊗ I /I . C/C L If no risk of confusion arises, we simply write β for β .Let C/C i +1 i − j : H (Q , V ) −→ H (Q, V ) be the maps arising from the exact triangle (12). Lemma 3.1 The following diagram commutes. 0 − 1 − 2 H (Q , V ) H (Q , V ) ⊗ I /I p p j ⊗I /I 1 2 2 ˜ ˜ H (Q, V ) H (Q, V ) ⊗ I /I f f Proof For M = V , V one has an exact triangle (cf. Equation (12)) p ◦res j : R (G , M)[−1] −→ R (Q , M )[−1] −→ R (Q, M). M cont Q, N cont p f Moreover is obtained by applying ·⊗ L to (cf. Eq. (14)). It follows from V V O ,w fgh o − − the deﬁnition of the derived Bockstein maps β and β on R (Q , V ) and R(Q, V ) cont 2 − respectively that j ⊗ I /I [1]◦ β is equal to β ◦ j . Since by deﬁnition the maps j are V V the ones induced in cohomology by j , the lemma follows. The following lemma gives a concrete description of β . C/C 123 314 M. Bertolini et al.. Lemma 3.2 Let (C, C) be as above, let z be a 1-cocycle in C,let Z bea 1-cochain in C, and let Z , Z and Z be 2-cochains in C such that k l m ρ ( Z) = z and d Z = Z · (k − 2) + Z · (l − 1) + Z · (m − 1). w k l m Then z = ρ ( Z ) is a 2-cocycle for ·= k, l, m, and one has the equality · w · −β (cl(z)) = cl(z ) · (k − 2) + cl(z ) · (l − 1) + cl(z ) · (m − 1) C/C k l m 2 2 i in H (C) ⊗ I /I ,where cl(·) is the class in H (C) represented by the i -cocycle ·. Proof The proof is very similar to that of [16, Lemma 5.5]. We omit it. 3.2.2 Local and global duality Nekovár ˇ’s generalised Poitou–Tate duality associates with the perfect duality π introduced fgh in Eq. (11) a global cup-product pairing (cf. Section 2.4 of [6]) 2 1 ˜ ˜ ·, · : H (Q, V ) ⊗ H (Q, V ) −→ L. (15) Nek f f − + The pairing π induces a Kummer duality V ⊗ V −→ L(1) and we denote by fgh L 1 − 1 + ·, · : H (Q , V ) ⊗ H Q , V −→ L (16) Tate p p the induced local Tate duality pairing. Recall ﬁnally the map + 1 1 + · : H (Q, V ) −→ H (Q , V ) introduced in diagram (13). 1 − 1 Lemma 3.3 For each ζ in H (Q , V ) and ξ in H (Q, V ) one has j(ζ),ξ =ζ, ξ . Tate Nek Proof This is proved as in [16, Lemma 5.7]. 3.2.3 The Garrett–Nekováˇrp-adic height pairing Set 1 2 2 ˜ ˜ β = β : H (Q, V ) −→ H (Q, V ) ⊗ I /I . fg h ˜ ˜ L α f f α R (Q,V)/R (Q,V ) f f 1 † After identifying H (Q, V ) with Sel (Q, V ) (cf. Sect. 3.1.2), the canonical height ⟪·, ·⟫ introduced in Sect. is deﬁned by (cf. [6, Section 2]) fg h ⟪x, y⟫ = β (x), y fg h fg h α Nek α α 1 2 for each x and y in H (Q, V ), where we write again ·, · for the I /I -base change of Nek Nekovár ˇ’s cup-product (15). Lemmas 3.1 and 3.3 give the following 0 − Lemma 3.4 For each q in H (Q , V ) one has − + ⟪j( q), ·⟫ = β (q), · fg h α fg h Tate α α α 2 1 as I /I -valued maps on H (Q, V ),where β = β − (and we write f R (V )/R (V ) fg h p p α α again ·, · for the I /I -base change of the local Tate pairing (16)). Tate 123 On exceptional zeros of Garrett–Hida... 315 3.3 Computation of⟪q , q ⟫ ˇˇ ˛˛ fg h 0 − Assume in this subsection α = α · α ,sothat H (Q , V ) is generated over L by the f g h periods √ √ q = m · q( f ) ⊗ ω ⊗ ω and q = m · q( f ) ⊗ η ⊗ η . αα p g h ββ p g h α α α α Recall that χ : G −→ Z denotes the p-adic cyclotomic character. Fix a lift q in cyc Q p ββ V of q under ρ . Since (cf. Sect. 3.1.1) ββ w − − q ∈ V ( f ) ⊗ V (g) ⊗ V (h) −→ V ββ Q β L β and V (ξ) = V (ξ ) ⊗ L for ξ = g, h, we can choose q in the G -submodule β 1 Q α ββ − + + − ˆ ˆ V ( f ) ⊗ V (g) ⊗ V (h) ⊗ −→ V L L O fgh fgh (cf. Sect. 3.1.1). By Eq. (9) one has d q = · q , (17) ββ ββ where d denotes the differentials of the complex R (Q , V ) and cont a ˇ ( f ) (l+m−k)/2 = · χ − 1 : G −→ O . fgh cyc p a ˇ (g ) ·ˇ a (h ) p p α The assumption α = α · α implies that takes value in I , and that its composition f g h with the projection I −→ I /I is of the form = ϕ · (k − 2) + ϕ · (l − 1) + ϕ · (m − 1) k l m 1 1 with ϕ in H (Q , Q ) for u = k, l, m. Identify H (Q , Q ) with the Q -vector space p p p p p ∗ ∗ Hom(Q , Q ) of continuous morphisms of groups from Q to Q via the local reciprocity p p p p ∗ ab −1 map rec : Q −→ G , normalised by requiring rec ( p ) to be an arithmetic Frobenius. p p p Q By local class ﬁeld theory, for each p-adic unit u one has ∂ 1 (l+m−k)/2 ϕ (u) = u − 1 =− · log (u), ∂ k w 2 where · : Z −→ 1 + pZ denotes the projection to principal units, and ∂ a (g ) · a (h ) 1 p p α α an ϕ ( p) = − 1 = · L ∂ k a ( f ) 2 (cf. Eq. (7)). As a consequence −2 · ϕ is equal to an 1 log = log −L · ord ∈ H (Q , Q ) f p f p p (where the p-adic valuation ord : Q −→ Q is normalised by ord ( p) = 1). Similarly p p p p an one shows that 2 · ϕ and 2 · ϕ are equal to the logarithms log = log −L · ord and l m p g p g α α an log = log −L · ord . It then follows from Eq. (17) and Lemma 3.2 that h p g 2 · β (q ) = log ·(k − 2) − log ·(l − 1) − log ·(m − 1) ⊗ q (18) ββ ββ f g h fg h α α α 1 − 2 in H (Q , V ) ⊗ I /I , where (with the notations introduced in Sect. 3.2.1) one writes β for the Bockstein map β associated with C = R (V ).Notethat C/C p fg h − − V ( f ) = V ( f ) ⊗ V (g) ⊗ V (h) Q β L β ββ p 123 316 M. Bertolini et al.. is an L[G ]-direct summand of V on which G acts trivially, so that log ⊗q (for Q Q ββ p p ξ = f , g , h ) belongs to the direct summand α α 1 − 1 − H (Q , V ( f ) ) = H (Q , Q ) ⊗ V ( f ) p p p Q ββ p ββ 1 − of the local cohomology group H (Q , V ). Similarly + + V ( f ) = V ( f ) ⊗ V (g) ⊗ V (h) Q α L α αα p is an L[G ]-direct summand of V isomorphic to Q (1), hence 1 + 1 + H (Q , V ( f ) ) = H (Q , Q (1)) ⊗ V ( f ) (−1) (19) p p p Q αα p αα 1 + is a direct summand of H (Q , V ). The local Tate pairing ·, · introduced in Sect. 3.2.2 p Tate 1 − induces a perfect duality (denoted by the same symbol) between H (Q , V ( f ) ) and ββ 1 + 1 ∗ H (Q , V ( f ) ), and identifying H (Q , Z (1)) with the p-adic completion Q of Q p αα p p p via the local Kummer map, local class ﬁeld theory gives − + + − ϕ ⊗ v , u ⊗ v = ϕ(u) · π (−1)(v ⊗ v ) (20) Tate fgh 1 1 − + + for each ϕ in H (Q , Q ), u in H (Q , Q (1)), v in V ( f ) and v in V ( f ) .Here p p p p ββ αα π (−1) : V ( f ) (−1) ⊗ V ( f ) −→ L fgh L αα ββ is the composition of π ⊗ Q (−1) with the evaluation pairing L(1) ⊗ L(−1) −→ L. fgh L 0 − 1 Recall that we identify H (Q , V ) with a submodule of H (Q, V ) via the embedding j introduced in Diagram (13). Lemma 3.4 and Eqs. (18)and (20)give Lemma 3.8 − + 2 · ⟪q , z⟫ = 2 · β (q ), z ββ ββ fg h fg h α Tate α α α Equation (18) u + = (−1) ·log ⊗q , z · (u − u ) ββ Tate o Equation (20) u + = (−1) · log (z ) · (u − u ) (21) ξ αα for each z in H (Q, V ),where ξ = f , g , h , u = 2, 1, 1 is the centre of U ,and α o ξ + 1 z ∈ H (Q , Q (1)) = Q ⊗ Q αα p p p p p 1 + is deﬁned as follows. Let pr denote the projection onto the direct summand H (Q , V ( f ) ) αα αα 1 + + of the local cohomology group H (Q , V ),and let q be the generator of V ( f ) (−1) ββ αα dual to q under π (−1), namely satisfying ββ fgh π (−1)(q ⊗ q ) = 1. fgh ββ ββ Then z is deﬁned (via the natural isomorphism (19)) by the identity αα + + ∗ pr (z ) = z ⊗ q . (22) αα αα ββ We now determine z for z = j( q ). By deﬁnition j( q ) is represented by αα αα αα c = (0, d q ˜ , q ˜ ) ∈ C (Q, V ), αα αα αα 123 On exceptional zeros of Garrett–Hida... 317 where q ˜ in V is a lift of q under the the projection V −→ V ,and where αα αα d q ˜ : G −→ V αα Q is its image under the differential in R (Q , V ). By construction d q ˜ represents the cont p αα + + 1 + class q = j( q ) in H (Q , V ).Since V (ξ) is the direct sum of V (ξ) and V (ξ) for αα α β αα ξ = g, h, we can (and will) choose q ˜ of the form αα q ˜ = m ·˜ q( f ) ⊗ ω ⊗ ω αα p g h α α for a lift q ˜( f ) of q( f ) under the projection V ( f ) −→ V ( f ) ,sothat d q ˜ represents the αα image of q under the connecting morphism αα − 1 + δ : V ( f ) −→ H (Q , V ( f ) ) αα αα αα arising from the short exact sequence of G -modules + − 0 −→ V ( f ) −→ V ( f ) −→ V ( f ) −→ 0, αα αα αα · · · where V ( f ) is the L[G ]-direct summand V ( f ) ⊗ V (g) ⊗ V (h) of V .Let q Q Q α L α A αα p p in pZ be the Tate period of A . Tate’s theory gives a rigid analytic isomorphisms between p Q rig the base change E 2 of theTatecurve E = G /q to the quadratic unramiﬁed extension Q m,Q A p p Q of Q and A .Set V ( E) = H ( E , Q (1)) and let ℘ : V ( E) V ( A) be the p Q p ¯ p Tate p p p 2 K et Q p p isomorphisms of G -modules induced by the Tate uniformisation. There is a short exact sequence of Q [G ]-modules a b 0 −→ Q (1) −→ V ( E) −→ Q −→ 0, (23) p p Z n where a(ζ ∞) = (ζ n · q ) for each compatible system ζ ∞ = (ζ n) of p -th roots p p n≥1 p p n≥1 of unity, and b is the Q -linear extension of the inverse limit of (canonical) maps Z n ¯ ¯ n n b : E(Q ) = (Q /q ) −→ Z/ p Z n p p p p A p ·ord (x) p − − Z n deﬁned by b (x · q ) = + p · Z. By deﬁnition q( A) = ℘ (1),where ℘ ◦ b Tate Tate A ord (q ) p A is the composition of ℘ and the projection V ( A) −→ V ( A) onto the maximal G - Tate p p Q unramiﬁed quotient, and −1 q ˜( f ) = ℘ ◦ ℘ ( q ) Tate A is the image of a compatible system q of p -th roots of the Tate period q under the A A composition of ℘ and the inverse of the isomorphism ℘ : V ( f ) V ( A) induced by Tate ∞ p the ﬁxed modular parametrisation ℘ : X ( N ) −→ A. As a consequence 1 in Q maps to ∞ 1 f 1 ∗ ˆ ˆ q ⊗1 under the connecting map Q −→ H (Q , Q (1)) = Q ⊗Q associated with the A p p p p short exact sequence (23), hence + −1 + j( q ) = cl(d q ˜ ) = δ (q ) = m · (℘ ◦ ℘ ) (q ⊗1) ⊗ ω ⊗ ω (24) αα αα αα αα p Tate A g h ∞∗ ∗ α α in 1 + 0 1 + H (Q , V ( f ) ) = H Gal(Q 2/Q), H (Q 2, V ( f ) ) ⊗ V (g) ⊗ V (h) , Q α L α p αα p p where −1 + ∗ 1 + (℘ ◦ ℘ ) : Q ⊗Q H (Q 2, V ( f ) ) Tate 2 ∞ ∗ p p 123 318 M. Bertolini et al.. −1 is the map induced in cohomology by the composition of ℘ and ℘ = ℘ ◦ a. Tate Tate If A denotes either A or E, denote by π : V (A)(−1) ⊗ V (A) −→ Q A p Q p the composition of the evaluation pairing Q (1) ⊗ Q (−1) −→ Q with the base change p p p of the Weil pairing on V (A) by Q (−1).Set p p ∗ + ∗ + q( A) = ℘ (ζ ∞) ⊗ ζ ∈ V ( A) (−1), p p Tate p where ζ is a generator of Q (1) and ζ in Q (−1) is its dual basis, and set p ∞ p p ∗ −1 ∗ + q( f ) = deg(℘ ) · ℘ (q( A) ) ∈ V ( f ) (−1). As π ((a( y) ⊗ z) ⊗ x) = b(x) · z( y) for each x in V ( E), y in Q (1) and z in Q (−1),the E p p p functoriality of the Poincaré duality under ﬁnite morphisms yields ∗ ∗ ∗ π (q( f ) ⊗ q( f )) = π (q( A) ⊗ q( A)) = π (a(ζ ∞) ⊗ ζ ) ⊗ q = 1, f A E p A then (by the deﬁnition of the weight-one differentials η , cf. Sect. 3.1.1) ∗ ∗ q = · q( f ) ⊗ ω ⊗ ω . √ g h ββ α α Together with Eq. (24)thisgives + ∗ j( q ) = · (q ⊗1) ⊗ q , (25) αα A ββ deg(℘ ) id est j( q ) = · q ⊗1. (26) αα A αα deg(℘ ) log (q ) p A an According to Theorem 3.18 of [9] L = ,sothat f ord (q ) p A 2 · deg(℘ ) an an an an − · ⟪q , q ⟫ = (L − L ) · (l − 1) + (L − L ) · (m − 1) ββ αα f g f h fg h α α α α m · ord (q ) p p A (27) by Eqs. (21)and (26). 3.4 Computation of⟪q , q ⟫ ˛ˇ ˇ˛ fg h ˛ ˛ 0 − Assume in this subsection α = β · α ,sothat H (Q , V ) is generated by the p-adic f g h periods √ √ q = m · q( f ) ⊗ ω ⊗ η and q = m · q( f ) ⊗ η ⊗ ω . αβ p g h βα p g h α α α α · · For γδ = αβ, βα and ·=∅, ±,deﬁne V ( f ) = V ( f ) ⊗ V (g) ⊗ V (h) .Then Q γ δ γδ p 0 − − − H (Q , V ) = V ( f ) ⊕ V ( f ) , αβ βα + + G acts on V ( f ) and V ( f ) via the p-adic cyclotomic character, and the local Tate p αβ βα pairing ·, · introduced in Sect. 3.2.2 induces a perfect duality (denoted by the same Tate 123 On exceptional zeros of Garrett–Hida... 319 1 − 1 + symbol) between H (Q , V ( f ) ) and H (Q , V ( f ) ). The argument of the proof of Eq. p p αβ βα (25) shows that + ∗ j( q ) = · (q ⊗1) ⊗ q (28) βα A αβ deg(℘ ) + ∗ + 1 1 + in the direct summand H (Q , V ( f ) ) = Q ⊗V ( f ) (−1) of H (Q , V ),where p p βα p βα ∗ ∗ ∗ q = · q( f ) ⊗ η ⊗ ω satisﬁes π (−1)(q ⊗ q ) = 1. (29) √ g h fgh αβ αβ α α αβ 1 − 1 − Let pr : H (Q , V ) −→ H (Q , Q ) ⊗ V ( f ) denote the projection, and write αβ p p p p αβ 2 − pr ⊗ I /I ◦ β (q ) = γ ⊗ q · (u − u ) (30) αβ u αβ o αβ fg h 1 ∗ with γ in H (Q , Q ) = Hom(Q , Q ) for u = k, l, m, where (with the notations intro- u p p p duced in Sect. 3.2.1) β is a shorthand for fg h 0 − 1 − 2 β − − : H (Q , V ) −→ H (Q , V ) ⊗ I /I , R (Q ,V )/R (Q ,V ) p p cont p cont p and u = 2if u = k and u = 1if u = l, m.Then(cf.Eq. (21)) o o Lemma 3.4 − + ⟪q , q ⟫ = β (q ), j ( q ) αβ βα αβ βα fg h fg h α α Tate α α Eqs. (28) and (30) = · γ ⊗ q ,(q ⊗1) ⊗ q · (u − u ) u αβ A Tate o αβ deg(℘ ) = · γ (q ) · (u − u ), (31) u A o deg(℘ ) where the last equality follows from Eq. (29) and the analogue of Eq. (20) obtained by replacing αα and ββ with βα and αβ respectively. It then remains to compute γ for u equal to k, l and m. For ξ = f , g , h ,ﬁx O -bases b of V (ξ) . After identifying V (ξ) with O ⊕ O via α ξ ξ ξ + − the O -basis (b , b ), the action of G on V (ξ) is given by (cf. Eq. (9)) ξ Q ξ ξ ⎛ ⎞ −1 u−1 χ ·ˇ a (ξ) · χ c ξ p cyc ⎝ ⎠ : G −→ GL (O ) Q 2 ξ 0 a ˇ (ξ) for a continuous map c : G −→ O . Without loss of generality, assume that ξ Q ξ − − + ˆ ˆ q = b ⊗b ⊗b ⊗ 1 αβ g f h α α − − ˆ ˆ in V = V ( f ) ⊗ V (g )⊗ V (h ) ⊗ maps to L L α O fgh fgh q ∈ V ( f ) = V ( f ) ⊗ V (g) ⊗ V (h) αβ Q α L β αβ p − − under ρ : V −→ V . (Recall that V (ξ) = V (ξ ) ⊗ L is the direct sum of the modules w 1 − + V (ξ) = V (ξ ) ⊗ L and V (ξ) = V (ξ ) ⊗ L for ξ = g, h, cf. Sect. 3.1.1.) Then α 1 β 1 α α d q = · q + · q , (32) αβ αβ ββ 123 320 M. Bertolini et al.. − + + ˆ ˆ where q = b ⊗b ⊗b ⊗ 1, where ββ g f h α α a ˇ ( f ) ·ˇ a (g ) p p (m−k−l+2)/2 = · χ · χ − 1 cyc a ˇ (h ) p α and where −1 (m−k−l+2)/2 =ˇ a ( f ) ·ˇ a (h ) · χ · χ · c . p p α h g cyc The exceptional zero condition α = β ·α and the self duality condition χ ·χ = 1imply f g h g h that takes values in I . Moreover, since the G -module V (g) = V (g ) ⊗ L splits as Q α 1 + − the direct sum of V (g) = V (g ) ⊗ L and V (g) = V (g ) ⊗ L,the map c takes β 1 α 1 g α α values in (l − 1) · O , hence takesvaluesin I . Because by construction q maps to an ββ − − element of V ( f ) under the specialisation map ρ : V −→ V , Lemma 3.2 and Eqs. ββ o (30)and (32) yield the identities γ =− (·)(w ), u o ∂ u hence (as in the previous subsection) a direct computation gives 1 1 1 γ = · log ,γ = · log and γ =− · log . (33) l m k f g h α α 2 2 2 Recalling that log (q ) = 0by [9, Theorem 3.18], Eq. (31) ﬁnally proves 2 · deg(℘ ) an an an an · ⟪q , q ⟫ = (L − L ) · (l − 1) − (L − L ) · (m − 1). αβ βα fg h f g f h α α α m · ord (q ) p p A (34) 3.5 Proof of equation (8) Assume in this subsection that ( A,) is exceptional at p,and ﬁx aSelmerclass x in Sel(Q, V ( f , g, h)).Let x˜ = ı (x) ∈ H (Q, V ( f , g, h)) ur be the corresponding extended Selmer class (cf. Sect. 3.1.2). By construction x˜ belongs 1 + + to the ﬁnite subspace of H (Q , V ), and its image under the natural map i : 1 + 1 H (Q , V ) −→ H (Q , V ) equals the restriction of x at p: p p ﬁn ﬁn + + res (x) = i (x˜ ). (35) The Galois group G acts on V ( f ) via the p-adic cyclotomic character, hence + + 1 ∗ H (Q , V ( f ) ) = Z ⊗ V ( f ) (−1) ﬁn p p p by Kummer theory. If q in V ( f ) denotes (as in the previous subsections) the dual basis of q in V ( f ) under the pairing π , and if one writes fgh + + ∗ 1 + pr (x˜ ) =˜ x ⊗ q ∈ H (Q , V ( f ) ) ﬁn p + ∗ for some x˜ in Z ⊗ L, then Eq. (35) yields the equality p p + + + ∗ + log (res (x)) =log (x˜ ), q =log (x˜ ) ⊗ q , q = log (x˜ ), (36) p fgh fgh p p p 123 On exceptional zeros of Garrett–Hida... 321 + 1 + + where log : H (Q , V ) D (V ) is the Bloch–Kato logarithm and (with a slight dR p p ﬁn abuse of notation) we denote again by log : Z ⊗ L −→ L the L-linear extension of p p p the p-adic logarithm. In the previous equation we used the functoriality of the Bloch–Kato logarithm and the fact that (by construction) the linear form ·, q on D (V ) factors dR fgh + + through the projection onto D (V ( f ) ) = V ( f ) (−1). dR Assume (α = α · α and) q = q .AccordingtoEqs.(21)and (36) f g h ββ 2 · ⟪q , x⟫ = log (res (x)) · (k − l − m), (37) ββ p αα fg h thus proving Eq. (8) in this case. Assume q = q . Since (with the notations of Section 3.4) takes values in (l −1)·O , αβ fgh it follows from Lemma 3.2 and Eqs. (32)and (33)that 2 · β (q ) = ε · log ⊗q · (u − u ) + ϑ · (l − 1) (38) αβ ξ αβ o fg h α α for some cohomology class ϑ in H (Q , V ( f ) ),where ε =−1and ε =+1for p h ξ ββ α ξ = f , g . One has then Lemma 3.4 − + ⟪q , x⟫ (k, 1, 1) = β (q ), x˜ (k, 1, 1) αβ αβ fg h fg h α α Tate α α Equation (38) + ∗ = ·log ⊗q , x˜ ⊗ q · (k − 2) αβ Tate βα αβ + ∗ = · log (x˜ ) · π (q ⊗ q ) · (k − 2) fgh αβ f αβ αβ Equation (36) 1 = · log (res (x)) · (k − 2), (39) αβ thus proving Eq. (8)when q = q . Switching the roles of the Hida families g and h , αβ α this also proves Eq. (8)when q = q . βα + − Assume ﬁnally q = q . With the notations of Sect. 3.4,let (b , b ) be O -bases αα ξ ξ − − ˆ ˆ of V (ξ) such that q = b ⊗b ⊗b ⊗ 1 is a lift of q under the specialisation map αα αα f g h α α − − ρ : V −→ V .Since c takes values in (u − u ) · O for ξ = g , h , one has w ξ o ξ α o α (4−k−l−m)/2 1 − d q ≡ χ · a ˇ (ξ) − 1 · q mod (l − 1, m − 1) · C (Q , V ) , αα cyc αα cont p hence Lemma 3.2 and a direct computation give 2 · β (q ) = log ⊗q · (k − 2) + ϑ · (l − 1) + ϑ · (m − 1) (40) αα αα fg h f 1 − for some local cohomology classes ϑ and ϑ in H (Q , V ).Asin(39) one deduces Eq. (8)for q = q from Lemma 3.4 and Eqs. (36)and (40). αα 4 Proof of theorem 2.1 Let , and be the improving planes in U × U × U deﬁned respectively by the f g h f g h equations k = l + m, k = l − m + 2and k = m − l + 2. For ξ = f , g, h deﬁne a (ξ) E = 1 −¯ χ ( p) · ξ ξ a (ξ ) · a (ξ ) p p 123 322 M. Bertolini et al.. in O ,where {ξ, ξ , ξ }={ f , g , h }. Lemma 9.8 of [7] implies that fgh α αα αα L ( A,)| = E | · L ( A,) (41) p ξ ξ p ξ αα for an improved p-adic L-function L ( A,) in O( ). Indeed loc. cit. (together with its p ξ analogue obtained by switching the roles of g and h) proves that the meromorphic function αα L ( A,) on deﬁned by the previous equation is (bounded, hence) regular at w . ξ o p ξ Shrinking the discs U if necessary, we then conclude that the improved p-adic L-function αα L ( A,) is analytic on , as claimed. p ξ Assume ﬁrst α = α · α ,sothat f g h 2 an an an 2 · E (mod I ) = L · (k − 2) − L · (l − 1) − L · (m − 1). (42) f h α α αα According to Theorem A and Proposition 9.3 of [7], the partial derivative of L ( A,) with respect to k vanishes at w , hence αα 2 2 · L ( A,) (mod I ) is equal to an an an an αα (L − L ) · (l − 1) + (L − L ) · (m − 1) · L ( A,) (w ) f g f h p f α α by Eqs. (41)and (42). Moreover, with the notations introduced before the statement of Theorem 2.1, one has L = ∩ and E = E | , thus f g f f L αα αα L ( A,) (w ) = E (w ) · L ( A,) (w ). o g o o p f p Noting that E (w ) = 1 − β /α (when α = α · α ), the previous discussion and Eq. (27) g o h h f g h conclude the proof of Theorem 2.1 when α = α · α . f g h Assume now α = β · α . In this case, for ξ = g, h, one has f g h 2 an an an 2 · E (mod I ) = L · (u − 1) − L · (k − 2) − L · (u − 1), (43) ξ f where {(ξ , u), (ξ , u )}={(g , l), (h , m)},and α α α αα αα αα − L ( A,) (w ) = L ( A,) (w ) = E (w ) · L ( A,) (w ). (44) o o f o o p h p g p The second equality in the previous equation follows as above from the deﬁnitions, according to which L = ∩ and E = E | . The ﬁrst equality follows by noting that the restrictions f g g g L of E and E to the line ∩ satisfy g h g h χ ¯ ( p) · a (g ) g p E | =− · E | g ∩ h ∩ g h g h a ( f ) · a (h ) p p α g h −1 (as a ( f )| = α = α and χ · χ = 1 by Assumption 1.1.1) with p ∩ f g h g h f χ ¯ ( p) · a (g ) g p − (w ) =−1. a ( f ) · a (h ) p p α (In other words E | and −E | have the same leading term at w , which together g ∩ h ∩ o g h g h αα αα with the equality E · L ( A,) | = E · L ( A,) | implies the ﬁrst identity g ∩ h ∩ p g g p g h h h in Eq. (44).) Write αα 2 2 · L ( A,) (mod I ) = a · (k − 2) + b · (l − 1) + c · (m − 1) 123 On exceptional zeros of Garrett–Hida... 323 with a, b and c in L. Equations (41)and (43) with ξ = g and Eq. (44)give an an an an a + b = E (w ) · L − L · L (w ) and c − a = E (w ) · L − L · L (w ), f o o f o o g p p f f h α α αα where L is a shorthand for L ( A,) . Similarly p p an an an an b − a = E (w ) · L − L · L (w ) and a + c = E (w ) · L − L · L (w ) f o o f o o g f p f h p by Eqs. (41)and (43) with ξ = h and Eq. (44). As a consequence αα 2 −2 · L ( A,) (mod I ) equals an an an an αα E (w ) · (L − L ) · (l − 1) − (L − L ) · (m − 1) · L ( A,) (w ). f o o f g f h p Noting that E (w ) = 1 − (when α = β · α ), the previous discussion and Eq. (34) f o f g h prove Theorem 2.1 when α = β · α . f g h Funding Open access funding provided by Universitá degli Studi di Milano within the CRUI-CARE Agree- ment. 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/. References 1. Massimo Bertolini and Henri Darmon. Hida families and rational points on elliptic curves. Invent. Math., 168(2):371–431, 2007. 2. Joël Bellaïche and Mladen Dimitrov. On the eigencurve at classical weight 1 points. Duke Math. J., 165(2):245–266, 2016. 3. Massimo Bertolini, Marco Adamo Seveso, and Rodolfo Venerucci. Diagonal classes and the Bloch-Kato conjecture. Münster J. Math., 13(2):317–352, 2020. 4. M. Bertolini, M. A. Seveso, and R. Venerucci. 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Rodolfo Venerucci. Exceptional zero formulae and a conjecture of Perrin-Riou. Invent. Math., 203(3):923– 972, 2016. 17. Rodolfo Venerucci. On the p-converse of the Kolyvagin–Gross–Zagier theorem. Comment. Math. Helv., 91(3):397–444, 2016. 18. A. Wiles. On ordinary λ-adic representations associated to modular forms. Invent. Math., 94(3):529–573, Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.
Annales mathématiques du Québec – Springer Journals
Published: Oct 1, 2022
Keywords: Birch and Swinnerton-Dyer Conjecture; p-adic L-functions; Exceptional zeros; 11F67 (11G40 11G35)
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