P-adic L-functions in universal deformation families

David Loeffler

doi:10.1007/s40316-021-00187-1

P-adic L-functions in universal deformation families

Loeffler, David 2023-04-01 00:00:00 We construct examples of p-adic L-functions over universal deformation spaces for GL .We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions and Euler systems are not the usual eigenvarieties (parametrising nearly-ordinary families of automorphic representations), but other, larger spaces depending on a choice of a parabolic subgroup, which we call ‘big parabolic eigenvarieties’. Résumé Nous construisons des exemples de fonctions Lp-adiques déﬁnies sur les espaces des défor- mations universels de GL . Nous formulons une conjecture qui prédit que les espaces (2) naturels des paramétres pour les fonctions Lp-adiques et les systémes d’Euler ne sont pas les variétés de Hecke usuelles, mais d’autres espaces plus grands, qui dépendent d’un choix de sous-groupe parabolique. Mathematics Subject Classiﬁcation 11F67 · 11F85 1 Introduction It is well known that many interesting automorphic L-functions L(π, s) have p-adic coun- terparts; and that these can often be extended to multi-variable p-adic L-functions, in which the automorphic representation π itself also varies in a p-adic family of some kind. In the literature so far, the p-adic families considered have been Hida families, or more generally Coleman families—families of automorphic representations which are principal series at p, together with the additional data of a “ p-reﬁnement” (a choice of one among the Weyl-group orbit of characters from which π is induced). In Galois-theoretic terms, this corresponds to a full ﬂag of subspaces in the local Galois representation at p (or in its (ϕ, )-module, for Coleman families). The parameter spaces for these families are known as eigenvarieties. The aim of this note is to give an example of a p-adic L-function varying in a family of a rather different type: it arises from a family of automorphic representations of GL × GL ,but 2 2 the parameter space for this family (arising from Galois deformation theory) has strictly bigger Supported by Royal Society University Research Fellowship “L-functions and Iwasawa theory”. B David Loefﬂer d.a.loefﬂer@warwick.ac.uk Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 123 D. Loefﬂer dimension than the eigenvariety for this group—it has dimension 4, while the eigenvariety in this case has dimension 3. We also sketch some generalisations of the result which can be proved by the same methods. This corresponds to the fact that a p-reﬁnement is a little more data than is actually needed to deﬁne a p-adic L-function: rather than a full ﬂag, it sufﬁces to have a single local subrepresentation of a speciﬁc dimension (a Panchishkin subrepresentation), which is a weaker condition and hence permits variation over a larger parameter space. This weaker condition is also sufﬁcient to interpolate Selmer groups in families, and hence to formulate an Iwasawa main conjecture. We conclude with some speculative conjectures whose aim is to identify the largest param- eter spaces on which p-adic L-functions and Euler systems can make sense. We conjecture that, given a reductive group G and parabolic subgroup P (and appropriate auxiliary data), there should be two natural p-adic formal schemes, the big and small P-nearly-ordinary eigenvarieties. These coincide if P is a Borel subgroup, but not otherwise; if G = GL and P is the whole of G, then the big eigenvariety is the 3-dimensional Galois deformation space of a modular mod p representation (with no local conditions at p). In general, we expect that the “natural home” of p-adic L-functions – and also of Euler systems – should be a big ordinary eigenvariety for an appropriate parabolic subgroup. 2 Families of Galois representations 2.1 The Panchishkin condition Let L be a ﬁnite extension of Q and let V be a ﬁnite-dimensional L-vector space with a continuous linear action of = Gal(Q/Q). Recall that V is said to be geometric if it is unramiﬁed at all but ﬁnitely many primes and de Rham at p; in particular it is Hodge–Tate at p, so we may consider its Hodge–Tate weights. (In this paper, we adopt the common, but not entirely universal, convention that the cyclotomic character has Hodge–Tate weight +1.) We recall the following condition introduced in [29, Deﬁnition 7.2] (generalising the more familiar case r = 0, which is the condition originally studied by Panchishkin in [33, Deﬁnition 5.5]). Deﬁnition 2.1 Let r ≥ 0 be an integer. We say V satisﬁes the r-Panchishkin condition if it is geometric, and the following conditions hold: (1) We have (c=+1) (number of Hodge–Tate weights ≥ 1ofV ) = dim V − r , where c ∈ is (any) complex conjugation. + + (2) There exists a subspace V ⊆ V stable under such that V has all Hodge–Tate weights ≥ 1, and V /V has all Hodge–Tate weights ≤ 0. Remark 2.2 (i) Note that V is unique if exists; we call it the Panchishkin subrepresentation of V at p. (ii) Condition (1) is equivalent to requiring that the Tate dual V (1) be “r-critical”, in the sense of [29, Deﬁnition 6.4]. This generalises the notion of “critical values” of an L- function due to Deligne: if V is the p-adic realisation of a motive M, then condition (1) for r = 0 is equivalent to requiring that L( M , 0) is a critical value of the L-function L( M , s) in the sense of [15]. (Note that this use of the word “critical” is unrelated to the usage in the theory of eigenvarieties, as in [2] for instance.) 123 P-adic L-functions in universal... (iii) The Panchishkin condition is closely related to the concept of near-ordinarity:arep- resentation V is said to be nearly-ordinary if it is geometric with distinct Hodge–Tate weights, and there exists a full ﬂag of subspaces of V such that the Hodge–Tate weights of the graded pieces are in strictly increasing order. Clearly, if V is nearly-ordinary and r-critical, then it is also r-Panchishkin; but the condition of near-ordinarity is much more restrictive, and we want to emphasise here that near-ordinarity is an unnecessarily strong hypothesis for the study of p-adic L-functions. 2.2 Panchishkin families By a “Panchishkin family”, we mean a family of p-adic Galois representations equipped with a family of Panchishkin subrepresentations, in the sense of the following deﬁnition. For simplicity, we shall suppose throughout this paper that p > 2, so that we can diagonalise the action of complex conjugation without introducing denominators. Let O be the ring of integers of L,and F its residue ﬁeld. We let CNL be the category of complete Noetherian local O-algebras with residue ﬁeld F. Deﬁnition 2.3 Let R be an object of CNL ,and r ≥ 1aninteger.An r-Panchishkin family of Galois representations (V, V ) over R consists of the following data: • a ﬁnite free R-module V with an R-linear continuous action of , unramiﬁed at almost all primes. + c=1 • an R-direct-summand V ⊆ V stable under ,of R-rank equal to that of V . These are required to satisfy the following condition: • The set (V, V ) of maximal ideals x of R[1/ p] such that V satisﬁes the r-Panchishkin condition and V is its Panchishkin subrepresentation is dense in Spec R[1/ p]. + + + We call V an r-Panchishkin submodule of V,and (V, V ) the interpolation region of V . It is important to note that V is not uniquely determined by V (orevenbythe pair (V, r )), in contrast to the case of Q -linear representations, as the examples of the next section will show. Remark 2.4 The natural notion of a nearly-ordinary family in this case would be a ﬁnite free R-module V with -action as above, together with a ﬁltration by -stable submodules Q Q 0 1 V = F V ⊃ F V ⊃ ... with graded pieces free of rank 1 over R. Our main focus in the present work will be in examples of r-Panchishkin families which do not admit a nearly- ordinary ﬁltration. 2.3 Examples from character twists We brieﬂy illustrate the above deﬁnitions using families of representations arising by twisting a ﬁxed, geometric representation by a family of characters. Example 2.5 (Cyclotomic twists) The original examples of Panchishkin families are those of the following form. Let V be an L-linear representation of satisfying the r-Panchishkin ◦ + condition, and V a O-lattice in V stable under .Let V be the Panchishkin subrepre- ◦+ + ◦ sentation, and V = V ∩ V . × × × We let denote the Iwasawa algebra O[[Z ]],and j the canonical character Z → . p p c=1 c=−1 If dim V = dim V , then we can take R to be the localisation of at any of its ( p −1) 123 D. Loefﬂer × × maximal ideals, corresponding to characters Z → F ; otherwise, we need to assume our maximal ideal corresponds to a character trivial on −1. We can then let V = V ⊗ R(χ ), O cyc × × where χ : → R denotes the composite of j and the p-adic cyclotomic cyc Q + ◦+ ◦+ + ◦ + character. If we let V = V ⊗ R(χ ),where V = V ∩ V ,then V is an R-direct- O cyc summand of V stable under . By construction, the set (V, V ) contains all points of Spec R[1/ p] corresponding to characters of the form j + χ,where χ is of ﬁnite order and j is an integer in some interval + + containing 0 (depending on the gap between the Hodge–Tate weights of V and V /V ). In particular, it is Zariski-dense, as required; so (V, V ) is an r-Panchishkin family over R. Example 2.6 (Varying r) More generally, if V is a geometric L-linear -representation and ◦ c=+1 V a lattice in V,wecan deﬁne R(V ) to be the set of integers 0 ≤ r ≤ dim V such + c=+1 that V posesses a -stable subrepresentation V of dimension dim V − r with all p r + + Hodge–Tate weights of V strictly larger than those of V /V . Note that R(V ) is never r r c=+1 + empty, since it always contains dim V (with V ={0}); and if V is nearly ordinary, c=+1 then R(V ) is the whole interval [0, dim V ]. For r ∈ R(V ), deﬁning R and V as in the previous example, we can deﬁne V = ◦+ + V ⊗ R(χ ).Then (V, V ) is an r-Panchishkin family, with interpolation region O cyc r r (V, V ) consisting of all j + χ with j in some nonempty interval of integers depend- ing on V and r. Example 2.7 (Twisting by imaginary quadratic Grössencharacters) As a ﬁnal “character- twist” example, we consider the following setting: let K be an imaginary quadratic ﬁeld with p = p p split in K,and let V be a geometric representation of ,ofdimension d.We 1 2 K let K be the unique Z -extension of K , with Galois group G ,and R = O[[G ]].We ∞ K K consider the rank 2d family over R deﬁned by V = Ind V ⊗ R(χ ) O univ ◦ × for some lattice V ⊂ V , with χ the composite map G → R (the “universal univ K K character” valued in R). We now equip this with Panchishkin submodules. Choosing decomposition groups at the p , we obtain an isomorphism Suppose u , u are integers ≥ 0 with u + u ≤ d. We assume that for each i = 1, 2, the 1 2 1 2 representation V has subrepresentations V stable under the decomposition group at p ,of + + dimension u , with each V having strictly larger Hodge–Tate weights than V /V .Ifwe i p p i i + ◦+ deﬁne V to be the image of V ⊗ R(χ )| under the isomorphism (†),then O univ i i Kp + + V is an r-Panchishkin submodule of V where r = d − u − u ;the set (V, V ) consists of 1 2 all locally-algebraic characters of G whose Hodge–Tate weights lie in a certain non-empty region of Z , depending on V and the u . See [29, §10] for an example of this kind with d = 2; in this case, we obtain 6 different Panchishkin submodules, one with r = 2, two with r = 1 and three with r = 0. The interpo- lation regions for these submodules are illustrated in Figure 1 of op.cit.. (More generally, if 1 j We use additive notation for characters, so j + χ is a shorthand for the character z → z χ(z). 123 P-adic L-functions in universal... V is nearly-ordinary of any dimension d ≥ 1, this construction gives rise to (d + 1)(d + 2) different Panchishkin submodules, d + 1 of which have r = 0.) 2.4 Conjectures on p-adic L-functions The following conjecture is due to Coates–Perrin-Riou [12] and Panchishkin [33]inthe case of cyclotomic twists of a ﬁxed representation. The generalisation to families as above is “folklore”; we have been unable to locate its ﬁrst appearance, but it is a special case of more general conjectures of Fukaya and Kato [18] (who have also investigated the case of non-commutative base rings R, which we shall not attempt to consider here). Conjecture 2.8 Suppose (V, V ) is an 0-Panchishkin family. There exists an element + + L(V, V ) ∈ Frac R such that for all x ∈ (V, V ) we have L( M , 0) L(V, V )(x ) = (Euler factor) · , (period) where M is the (conjectural) motive whose realisation is V . x x If V is semistable at p, the expected form of the Euler factor is −1 −1 + + det (1 − p ϕ ) : D (V ) · det (1 − ϕ) : D (V /V ) . cris cris We refer to [18] for more details of the interpolation factors involved. 2.5 Euler systems In [29], Zerbes and the present author conjectured that when V is the family of cyclotomic twists of a ﬁxed representation, the r-Panchishkin condition was the “correct” condition for a family of Euler systems of rank r to exist, taking values in the r-th wedge power of the Galois ∗ + cohomology of the Tate dual V (1), and satisfying a local condition at p determined by V . This extends the conjectures formulated by Perrin-Riou in [34], which correspond to taking c=1 r to be the maximal value dim V (in which case the r-Panchishkin condition is automatic, as we have seen). It is also consistent with the above conjectures of Coates–Perrin-Riou and Panchishkin for r = 0, if we understand a “rank 0 Euler system” to be a p-adic L-function. It seems natural to expect that an analogue of Conjecture 2.8 should hold for arbitrary r-Panchishkin families, predicting the existence of families of rank r Euler systems over R; and, as in the rank 0 case, one can show that this would follow as a consequence of the very general conjectures of [18]. Remark 2.9 There are a number of (unconditional) results concerning the variation of Euler systems in Hida families of automorphic representations, which are examples of nearly- ordinary families; see e.g. [32] for Kato’s Euler system, and [26]for theGL × GL 2 2 Beilinson–Flach Euler system. However, the above conjecture predicts that Euler systems should vary in more general families, which are not nearly-ordinary but are still r-Panchishkin. Some examples of cyclo- tomic twist type for r = 1 are discussed in [29]. A much more sophisticated example due to Nakamura, in which R is the universal deformation space of a 2-dimensional modular Galois representation, is discussed in §3.5 below. 123 D. Loefﬂer 3 Examples from GL Notation: Suppose f is a modular cusp form which is a normalised eigenform for the Hecke operators, with a chosen embedding of its coefﬁcient ﬁeld Q( f ) into L.Wedenoteby ρ the unique Galois representation → GL (L) characterised as follows: we have f , p Q 2 −1 tr ρ (Frob ) = a ( f ) for almost all primes , where Frob denotes an arithmetic Frobe- f , p 1−k nius at . Thus det ρ is χ up to ﬁnite-order characters, where k is the weight of f .(We f , p cyc warn the reader that some references use the notation ρ for the dual of this representation.) f , p 3.1 The universal deformation ring Let ρ¯ : → GL(V ) GL (F) be a 2-dimensional, odd, irreducible (hence, by Khare– Q 2 Wintenberger, modular) representation. Hypotheses 3.1 We shall assume ρ¯ satisﬁes the following: (a) ρ¯| is irreducible, where K = Q(ζ ) (Taylor–Wiles condition). (b) if ρ¯| is not absolutely irreducible, with semisimpliﬁcation χ ⊕ χ (after possibly 1, p 2, p Q p ±1 extending F), then we have χ /χ ∈{ / 1,ε } where ε is the mod p cyclotomic 1, p 2, p p character. (c) ρ¯ is unramiﬁed away from p (i.e. its tame level is 1). Remark 3.2 Note that the ﬁrst two assumptions are essential to our method, because they are hypotheses for major theorems which we need to quote. On the other hand, the third is much less fundamental and is imposed chieﬂy in order to simplify the calculations in Sect. 4 below. See Sect. 5.1 below for further discussion. Deﬁnition 3.3 Let R(ρ) ¯ ∈ CNL be the universal deformation ring over O parametrising deformations of ρ¯ as a -representation, and ρ : → GL (R(ρ) ¯ ) the universal Q,{ p} Q,{ p} 2 deformation. Let X(ρ) ¯ = Spf R(ρ) ¯ . Theorem 3.4 (Böckle, Emerton) • The ring R(ρ) ¯ is a reduced complete intersection ring, and is ﬂat over O of relative dimension 3. • We have a canonical isomorphism R(ρ) ¯ = T (ρ) ¯ ,where T (ρ) ¯ is the localisation at the maximal ideal corresponding to ρ¯ of the prime-to- p Hecke algebra acting on the space S(1, O) of cuspidal p-adic modular forms of tame level 1. Proof This is proved in [5] assuming that ρ¯| has a twist which is either ordinary, or Q p irreducible and ﬂat. This was extended to the setting described above (allowing irreducible but non-ﬂat ρ¯) by Emerton, see [17, Theorem 1.2.3]. Remark 3.5 If ρ¯ is unobstructed in the sense that H , Ad(ρ) ¯ = 0, then R(ρ) ¯ is Q,{ p} isomorphic to a power-series ring in 3 variables over O. It is shown in [37]thatif f is a ﬁxed newform of weight ≥ 3, then for all but ﬁnitely many primes p of the coefﬁcient ﬁeld Q( f ), the mod p representation ρ¯ is unobstructed. f ,p Deﬁnition 3.6 (i) If f is a classical modular newform of p-power level (and any weight) such that ρ¯ =¯ ρ,then ρ is a deformation of ρ¯ and hence determines a Q -point of f , p f , p p X(ρ) ¯ . We shall call these points classical. 123 P-adic L-functions in universal... (ii) More generally, a Q -point of X(ρ) ¯ will be called nearly classical if the corresponding −t Galois representation ρ has the form ρ ⊗ (χ ) , for some (necessarily unique) f , p cyc newform f and t ∈ Z. −t In the setting of (ii), if t ≥ 0, the Galois representation ρ ⊗ (χ ) corresponds f , p cyc formally to the nearly-overconvergent p-adic modular form θ ( f ),where θ = q is the dq Serre–Tate differential operator on p-adic modular forms. Slightly abusively, we denote such t t a point by θ ( f ),evenif t < 0 (in which case θ ( f ) may not actually exist as a p-adic modular form). Either Theorem 1.2.4 of [17]orthe main theoremof[24], combined with Theorem 0.4 of [35] in the case of equal Hodge–Tate weights, shows that any Q -point ρ of X(ρ) ¯ which is de Rham at p is a nearly-classical point (as predicted by the Fontaine–Mazur conjecture). Proposition 3.7 For any weight k ≥ 2, modular points corresponding to weight k modular forms are dense in X(ρ) ¯ . Proof This is obvious for Spf T (ρ) ¯ ,since T (ρ) ¯ can be written as an inverse limit of locali- sations of Hecke algebras associated to the ﬁnite-level spaces S ( ( p ), O). Since we have k 1 R(ρ) ¯ T (ρ) ¯ by Theorem 3.4, the result follows. Remark 3.8 Note that a crucial step in the proof of Theorem 3.4 is to establish that the set of all modular points (of any weight) is dense in X(ρ) ¯ . However, once this theorem is established, we can obtain the much stronger result of Proposition 3.7 a posteriori. For later constructions we need the fact that there exists a “universal modular form” over X(ρ) ¯ : × × univ (1−k) Deﬁnition 3.9 (i) Let k : Z → R(ρ) ¯ be the character such that det ρ = (χ ) . cyc [ p] (ii) Let G be the formal power series ρ¯ [ p] G = t q ∈ R(ρ) ¯ [[q]], ρ¯ pn where the t are determined by the identity of formal Dirichlet series −1 −s −s univ −1 t n = det 1 − ρ (Frob ) . = p pn [ p] −t The specialisation of G at a nearly-classical point ρ ⊗ (χ ) is precisely the “ p- f , p cyc ρ¯ t [ p] t depletion” θ ( f ) of θ ( f ),where θ is the Serre–Tate differential operator q .If t ≥ 0, dq this p-adic modular form is the image under the unit-root splitting of a classical nearly- holomorphic cuspform, in the sense of Shimura. [ p] Theorem 3.10 (Gouvea) The series G is the q -expansion of a p-adic modular form with ρ¯ coefﬁcients in R(ρ) ¯ , of tame level 1 and weight-character k, which is a normalised eigenform for all Hecke operators. Proof This follows readily from the duality between Hecke algebras and spaces of cusp forms. 123 D. Loefﬂer 3.2 The universal ordinary representation The following deﬁnition is standard: Deﬁnition 3.11 An ordinary reﬁnement of (ρ, ¯ V ) is a choice of 1-dimensional F-subspace + + V ⊆ V stable under ρ( ¯ ), such that the inertia subgroup I acts trivially on V . Q Q p p Let us ﬁx a choice of ordinary reﬁnement V . Then there is a natural deﬁnition of ordinarity for deformations: we say that a deformation ρ of ρ¯ (to some ring A ∈ CNL )isordinaryif ρ| preserves a rank one A-summand lifting V , and the action of I on this summand Q p is trivial. (Note that this summand is unique if it exists, since Hypothesis 3.1(b) implies that + + V /V cannot be isomorphic to V ). Theorem 3.12 Suppose ρ¯ is ordinary. Then there exists a complete local Noetherian O- ord algebra representing the functor of ordinary deformations. We let R (ρ) ¯ be this algebra, ord ord and X (ρ) ¯ = Spf R (ρ) ¯ . ord ord ord ord,+ Remark 3.13 Over R (ρ) ¯ we have a universal triple (ρ , V , V ), but we caution ord ord,+ ord the reader that (V , V ) is not an 0-Panchishkin family over R (ρ) ¯ in the sense of Deﬁnition 2.3, since it interpolates the representations ρ for ordinary modular forms f . f , p These have all their Hodge–Tate weights ≤ 0, and hence cannot satisfy condition (1) in Deﬁnition 2.1. However, we shall build interesting examples of 0-Panchishkin families from ord ord,+ (V , V ) via twists and tensor products. ord On the “modular” side, we can consider the ordinary Hecke algebra T (ρ) ¯ , which is the ord localisation at ρ¯ of the algebra of endomorphisms of e · S(1, Z ) generated by all of the Hecke operators (including U ). There is a natural map ord ord R (ρ) ¯ → T (ρ) ¯ , and by Theorem 3.3 of [38], this map is an isomorphism. (Note that this isomorphism is compatible with the isomorphism R(ρ) ¯ = T (ρ) ¯ of the previous section, via the natural maps ord ord R(ρ) ¯ → R (ρ) ¯ and T (ρ) ¯ → T (ρ) ¯ .) × ord ord Note that the composite Z − → R(ρ) ¯ → R (ρ) ¯ gives R (ρ) ¯ the structure of a - × ord algebra, where = O[[Z ]].Sowehaveamap k : X (ρ) ¯ → X = Spf . cyc Proposition 3.14 (Hida) ord • The ring R (ρ) ¯ is ﬁnite and projective as a -module, and thus has relative dimension 1 over O. × × n • If k ≥ 2 is an integer, and χ : Z → O is a Dirichlet character of conductor p ,then ord the ﬁbre of X (ρ) ¯ at k = k + χ is étale over L = Frac O, and its geometric points biject with the normalised weight k eigenforms of level ( p ) and character χ (if n ≥ 1)or level ( p) (if n = 0) which are ordinary and whose mod p Galois representation is ρ¯. determined (Note that this ﬁbre is empty if k + χ does not lie in the component of X cyc by det ρ¯.) ord Much as above, we can deﬁne a universal ordinary eigenform G with coefﬁcients in ρ¯ [ p] ord ord R (ρ) ¯ (whose p-depletion is the pullback of G along X (ρ) ¯ → X(ρ) ¯ , and whose ρ¯ −1 + U -eigenvalue is the scalar by which Frob acts on V ). However, we shall not use this explicitly here. 123 P-adic L-functions in universal... ord More useful is the following dual construction due to Hida [21]. The ring R (ρ) ¯ ord has ﬁnitely many minimal primes, corresponding to irreducible components of X (ρ) ¯ ord (“branches”). If a is a minimal prime, and we let T be the integral closure of T (ρ) ¯ /a, then we can ﬁnd an invertible ideal I T , and a homomorphism a a ord −1 λ : S (1,) ⊗ T → I , ord a a T (ρ) ¯ a ord characterised by mapping G to 1. ρ¯ 3.3 Nearly ordinary deformations More generally, we can deﬁne a nearly ordinary reﬁnement V of V by dropping the requirement that inertia act trivially on V ; and there is a corresponding deﬁnition of a nearly-ordinary deformation (V , V ) of (V , V ). no Proposition 3.15 There exists a ring R (ρ) ¯ ∈ CNL , and a nearly-ordinary deformation no no,+ (V , V ) of (V , V ) to this ring, which are universal among nearly-ordinary deforma- no tions of (V , V ). Moreover, R (ρ) ¯ is ﬂat over O of relative dimension 2. Proof The representability of this functor follows easily from the ordinary case above. If (V , V ) is nearly-ordinary, we can ﬁnd a unique character χ ¯ : Gal(Q(ζ )/Q) → F such that (V ⊗¯ χ, V ⊗¯ χ) is ordinary. Similarly, the data of a nearly-ordinary deformation of + + (V , V ) is equivalent to the data of an ordinary deformation of (V ⊗¯ χ, V ⊗¯ χ), together with a character χ of Gal(Q(ζ ∞ )/Q) lifting χ ¯ . This shows that the functor of nearly-ordinary no deformations is represented by a ring R (ρ) ¯ , deﬁned as the completed tensor product of ord R (ρ¯ ⊗¯ χ) and the ring parametrising deformations of χ ¯ to a character of Gal(Q(ζ )/Q), ord which is isomorphic to O[[ X ]].Since R (ρ¯ ⊗¯ χ) is ﬂat of dimension 1, we conclude that no R (ρ) ¯ is ﬂat of dimension 2. 3.4 Examples of Panchishkin families The above deformation-theoretic results give rise to the following examples of 0-Panchishkin families, in the sense of Deﬁnition 2.3. Example 3.16 (Nearly-ordinary deformations of modular forms) Suppose V is a modular mod p representation satisfying Hypotheses 3.1, with a nearly-ordinary reﬁnement V .Then no no the universal family V of Galois representations over R (ρ) ¯ , together with its universal no,+ nearly-ordinary reﬁnement V , is an example of a 0-Panchishkin family. In this case, + no Hida theory shows that (V, V ) consists precisely of the Q points of X (ρ) ¯ of the form −s θ ( f ),where f has weight k ≥ 2and 1 ≤ s ≤ k − 1. These are manifestly Zariski-dense. Conjecture 2.8 is known for this family, by work of Mazur and Kitagawa [25] . We are principally interested in examples which (unlike Example 3.16)are not nearly- ordinary. Our ﬁrst examples of such representations come from tensor products: Example 3.17 (Half-ordinary Rankin–Selberg convolutions) Let V and V be two mod p 1 2 representations satisfying Hypotheses 3.1, and suppose V admits a nearly-ordinary reﬁne- ment V . Twisting V by a character and V by the inverse of this character, we can suppose 1 2 See also [19] for a comparison between the Mazur–Kitagawa result and the general conjectures of [18]. 123 D. Loefﬂer that (V , V ) is actually ordinary (not just nearly-so). Then we consider the triple (R, V, V ) given by ord,+ ord ord + ˆ ˆ ˆ R = R (ρ¯ )⊗R(ρ¯ ), V = V ⊗V , V = V ⊗V . 1 2 2 2 ord,+ ord where (V , V ) is the universal ordinary deformation of (V , V ),and V the universal 1 2 1 1 deformation of V (with no ordinarity condition). Note that R has relative dimension 4 over O. + −s The set (V, V ) is the set of points of the form f ,θ (g) ,where f is a classical point −s of weight k ≥ 2, and θ (g) is a nearly-classical point such that g has weight < k and s lies in the range of critical values of the Rankin–Selberg L-function, namely ≤ s ≤ k − 1. This set (V, V ) is Zariski-dense; even the specialisations with (k, , s) = (3, 2, 2) are dense. We shall verify Conjecture 2.8 for this family below. Remark 3.18 A generalisation of the above two examples would be to consider tensor prod- ucts of universal representations over product spaces of the form no X = X (ρ¯ ) × X(ρ¯ ) × ··· × X(ρ¯ ) 1 2 n for general n,where ρ¯ ,..., ρ¯ are irreducible modular representations mod p with ρ¯ nearly 1 n 1 ordinary. This space has dimension 3n −1; but there are n −1 “redundant” dimensions, since the tensor product is not affected by twisting ρ by a character and one of ρ ,...,ρ by the 1 2 n inverse of this character. Quotienting out by this action gives a 0-Panchishkin family over a 2n-dimensional base. Example 3.19 (General tensor products) Let L = Frac O and let V be any L-linear represen- tation of (not necessarily 2-dimensional) which is geometric, satisﬁes the 0-Panchishkin c=1 c=−1 ◦ condition, and has dim V = dim V .Let V be a -stable O-lattice in V (which Q 1 always exists). Then, for any modular mod p representation V , we obtain a 0-Panchishkin family by letting ◦ + ◦ + R = R(ρ¯ ), V = V ⊗ V , V = (V ∩ V ) ⊗ V , 2 2 2 1 1 1 In particular, we can take V to be the Galois representation arising from a cohomological automorphic representation of GSp which is Klingen-ordinary at p. Note that in the last two examples the subspace V will not,ingeneral,extendtoafull ﬂag of -stable subspaces, so V is not nearly ordinary. 3.5 Families of Euler systems For ρ¯ as in Hypotheses 3.1, the canonical 2-dimensional family V over R(ρ) ¯ will not, in general, satisfy the 0-Panchishkin condition. However, it automatically satisﬁes the r- Panchishkin condition for r = 1, as V ={0} satisﬁes the conditions of a 1-Panchishkin + t submodule (with (V, V ) being the set of nearly-classical specialisations θ ( f ) with t ≥ 0). So the more general conjecture sketched in §2.5 predicts that there should exist a family of Euler systems taking values in V (1), interpolating Kato’s Euler systems for each modular form f lifting ρ¯. Such a family of Euler systems has recently been constructed by Nakamura [31]. 123 P-adic L-functions in universal... 4 P-adic L-functions for half-ordinary Rankin convolutions Let us choose two mod p representations ρ¯ , ρ¯ satisfying Hypotheses 3.1, with ρ¯ ordinary 1 2 1 (but no ordinarity assumption on ρ¯ ). ord Choose a branch a of X (ρ¯ ) as before, and let A denote the ring T ⊗ T (ρ¯ ),and 1 a Z 2 X = X × X(ρ) ¯ its formal spectrum. This has relative dimension 4 over Z .Welet V denote a p ord ord ord ∗ + + ∗ + the A-linear representation ρ ⊗ (ρ ) (1),and V = (ρ ) ⊗ (ρ ) (1) where (ρ ) 2 2 1 1 1 ord is the 1-dimensional unramiﬁed subrepresentation of ρ | . Thus V is a rank 4 family of 1 Q -representations over X unramiﬁed outside p,and V a rank 2 local subrepresentation of V. Remark 4.1 This differs from the (V, V ) of Example 3.17 by an automorphism of the base ring R, so Conjecture 2.8 for either one of these examples is equivalent to the other. The present setup is slightly more convenient for the proofs. + t The set (V, V ) contains all points ( f ,θ (g)) where f has weight k ≥ 2, g has weight ≥ 1, and t is an integer with 0 ≤ t ≤ k − − 1. Our goal is to deﬁne a p-adic L-function + + associated to (V, V ), with an interpolating property at the points in (V, V ). × × The ring A is endowed with two canonical characters k , k : Z → A , the former 1 2 [ p] factoring through T and the latter through T (ρ¯ ). We can regard G as a p-adic eigenform a 2 ρ¯ with coefﬁcients in A, of weight k , by base extension. Deﬁnition 4.2 Let denote the p-adic modular form [ p] [ p] ord ord e G · E ∈ S (1, A), ρ¯ k −k 2 1 2 1 [ p] k−1 n where E = ( d )q ∈ S (1,) denotes the p-depleted Eisenstein series of n≥1 k k d|n pn weight k and tame level 1. Let −1 L (ρ¯ , ρ¯ ):=λ () ∈ I ⊗ A. a 1 2 a T a a This is a meromorphic formal-analytic function on the 4-dimensional space X × X(ρ) ¯ , regular along any 3-dimensional slice { f }× X(ρ) ¯ with f classical. We now show that the values of L at points in interpolate values of Rankin L- functions. Let ( f ,θ (g)) be such a point, with f , g newforms of p-power levels, and let k, be the weights of f , g.Let α be the eigenvalue of geometric Frobenius on the unramiﬁed subrepresentation of ρ | ,and let f be the p-stabilisation of f of U -eigenvalue α. f , p α p Remark 4.3 If f has non-trivial level, then f = f ,and α is just the U -eigenvalue of f .If α p 2 k−1 f has level one, then α is the unique unit root of the polynomial X − a ( f ) X + p ,and k−1 f is the level p eigenvector f (τ ) = f (τ ) − f ( pτ). α α ord We deﬁne λ to be the unique linear functional on S (1, L) which factors through f ,α projection to the f eigenspace, and satisﬁes λ ( f ) = 1. By deﬁnition, we have α f ,α α [ p] t t [ p] L (ρ¯ , ρ¯ )( f ,θ (g)) = λ θ (g ) · E . a 1 2 f ,α k− −2t 123 D. Loefﬂer ( p) Deﬁnition 4.4 For f , g newforms as above, we write L ( f × g, s) for the Rankin–Selberg L-function of f and g without its Euler factor at p, ( p) ( p) −s L ( f × g, s) := L (χ χ , 2s + 2 − k − ) a ( f )a (g)n f g n n n≥1 pn −1 −s −1 = det 1 − Frob : V ( f ) ⊗ V (g) , p p = p and let ( p) ( p) ( f ⊗ g, s):= (s) (s − + 1)L ( f ⊗ g, s). C C Theorem 4.5 We have (t +1) t −1 ( p) ∗ p P (g, p α ) ( f , g , + t ) t 1−k t k+ L (ρ¯ , ρ¯ )( f ,θ (g)) = 2 (−1) i λ (g) , a 1 2 ∗ −( +t ) ad P (g , p α) E ( f ) f , f where b is the level at which g is new. Here λ b (g) is the Atkin–Lehner pseudo-eigenvalue of g, P (g, X ) is the polynomial such that −1 r P (g, X ) = a (g) X , p p r ≥0 and k−1 k−2 p p ⎪ 1 − 1 − f crystalline at p, 2 2 α α k−1 ad E ( f ) = − f semistable non-crystalline at p, p 2 ⎪ a k−1 G(χ ) f non-semistable at p, new of level p . 2 f Proof This follows from the Rankin–Selberg integral formula. The computations are virtually identical to the case of ﬁnite-slope forms treated in [27], so we shall not reproduce the computations in detail here. t −1 P (g, p α ) Remark 4.6 Note that the factor can be written as ∗ −( +t ) P (g , p α) −1 −1 −1 + det (1 − ϕ) (1 − p ϕ ) : D (V ) cris + + ∗ + t where V = (ρ ) ⊗ ρ (1 + t ) is the ﬁbre of V at ( f ,θ (g)). On the other hand, the f , p g, p (t +1) factor λ b (g) is essentially the local ε-factor of this representation. 5 Other cases We brieﬂy comment on some other cases which can be treated by the same methods as above. For reasons of space we shall only give a very brief sketch of each construction, and we hope that these sketches will be expanded into a fuller account in future works. 123 P-adic L-functions in universal... 5.1 Relaxing the tame levels Firstly, the assumption in Theorem 4.5 that the tame levels of the Hida families be 1 should not be too difﬁcult to relax. However, handling general tame levels will require much more careful book-keeping about the local Euler factors at the bad primes, as in the construction of p-adic Rankin–Selberg L-functions for the convolution of two ordinary families in [10]. The problem of generalising Theorem 4.5 to non-trivial tame levels will be treated in the forthcoming Warwick PhD thesis of Zeping Hao. 5.2 The case of GSp(4) ×xGL(2) A more ambitious case which can be treated by the same methods is the following. Let be a cohomological automorphic representation of GSp which is globally generic, unram- iﬁed and Klingen-ordinary at p, and contributes to cohomology with coefﬁcients in the algebraic representation of weight (r , r ),for some r ≥ r ≥ 0. (Classically, these corre- 1 2 1 2 spond to holomorphic vector-valued Siegel modular forms taking values in the representation r −r r +3 1 2 2 Sym ⊗ det of GL .) For technical reasons we assume r > 0. 2 2 In [28] we constructed a cyclotomic p-adic L-function interpolating the critical values of L( ⊗ σ, s) where σ is an automorphic representation of GL generated by a holomorphic form of weight ≤ r − r + 1. This is constructed by applying a “push-forward” map to 1 2 [ p] the product of the p-depleted newform g ∈ σ with an auxiliary p-adic Eisenstein series, and pairing this with a coherent H eigenclass coming from . This construction is closely parallel to the construction of the p-adic Rankin–Selberg L-function for GL × GL , and it generalises to universal-deformation families in the same 2 2 way, since the pushforward map of [28] can be applied to any family of p-adic modular forms (over any base). If we assume for simplicity that is unramiﬁed at all ﬁnite places, [ p] and replace g with a universal deformation family G as above, then we obtain an element of ρ¯ R(ρ) ¯ interpolating these p-adic L-functions, with ﬁxed and σ varying through the small- weight specialisations of a 3-dimensional universal-deformation family. We can also add a fourth variable, in which we vary through a 1-dimensional family of Klingen-ordinary representations, with r varying but r ﬁxed. 1 2 5.3 Self-dual triple products If we are given three mod p modular representations ρ ,ρ ,ρ with ρ nearly-ordinary and 1 2 3 1 det(ρ ) · det(ρ ) · det(ρ ) =¯ χ , then the space 1 2 3 cyc no (ρ ,ρ ,ρ ) ∈ X (ρ¯ ) × X(ρ¯ ) × X(ρ¯ ) : det(ρ ) · det(ρ ) · det(ρ ) = χ 1 2 3 1 2 3 1 2 3 cyc carries a natural 8-dimensional 0-Panchishkin family V, given by the tensor product of the three universal deformations V , with the Panchishkin submodule given by V ⊗ V ⊗ V . i 2 3 The base space is apriori 7-dimensional, but it has two “redundant” dimensions (since we can twist either ρ or ρ by a character, and ρ by the inverse of that character, without 2 3 1 changing the tensor product representation), so we obtain an 0-Panchishkin family over a ∗ + 5-dimensional base X, satisfying the self-duality condition V = V (1).The set (V, V ) corresponds to triples of classical modular forms ( f , f , f ) which are “ f -dominant” – 1 2 3 1 i.e. their weights (k , k , k ) satisfy k ≥ k + k . 1 2 3 1 2 3 123 D. Loefﬂer [ p] [ p] Feeding the universal eigenforms G and G into the construction of [14]gives a p- ρ¯ ρ¯ 2 3 adic L-function over this 5-dimensional base space, extending the construction in op.cit. of a p-adic L-function over the 3-dimensional subspace of X where ρ and ρ are nearly-ordinary. 2 3 (Note that this is actually a reﬁnement of Conjecture 2.8, since the resulting p-adic L- function interpolates the square-roots of central L-values.) 5.4 The Bertolini–Darmon–Prasanna case Let ρ¯ be a modular mod p representation of , with universal deformation space X(ρ) ¯ . Q,{ p} We shall suppose that det ρ¯ =¯ χ , and we let X (ρ) ¯ ⊆ X(ρ) ¯ denote the subspace parametris- cyc ing deformations whose determinant is χ ;thisisﬂat over O of relative dimension 2, and cyc is formally smooth if ρ¯ is unobstructed. Meanwhile, we choose an imaginary quadratic ﬁeld K in which p = p p is split, and we 1 2 ac let X = Spf O[[ X ]] be the character space of the anticyclotomic Z -extension of K.Let X ac 0 denote the product X × X (ρ) ¯ .Thisis O-ﬂat of relative dimension 3, and it carries a family of 4-dimensional Galois representations V, given by tensoring the universal deformation ac univ ρ of ρ¯ with the induction to of the universal character over X .Notethat V satisﬁes the “self-duality” condition V (1) V. Locally at p, V is the direct sum of two twists of the universal deformation of ρ¯, corresponding to the two primes above p; and we can deﬁne a 0-Panchishkin submodule V by taking the direct summand corresponding to one of these primes. Note that (V, V ) consists of pairs (ψ, f ) where f is a modular form and ψ an anticyclotomic algebraic Hecke character of weight (n, −n),where n is large compared to the weight of f . [ p] Plugging in the universal family G (more precisely, its pullback to X (ρ) ¯ ) into the ρ¯ ac constructions of [4], we obtain a p-adic analytic function on the 3-dimensional space X × 0 + X (ρ) ¯ interpolating the square-roots of central L-values at specialisations in (V, V ). This reﬁnes the construction due to Castella [9, §2] of a BDP-type L-function over the ac ord 3 2-dimensional space X × X (ρ) ¯ when ρ¯ is ordinary. 5.5 A finite-slope analogue? One can easily formulate a “ﬁnite-slope” analogue of Conjecture 2.8, where the submodule V ⊆ V is replaced by a submodule of the Robba-ring (ϕ, )-module of V| . The analogue ord of Hida’s ordinary deformation space X (ρ) ¯ is now the ρ¯-isotypic component E (ρ) ¯ of the Coleman–Mazur Eigencurve [13]. However, proving a ﬁnite-slope version of the results of Sect. 4, or of the generalisations sketched in the above paragraphs, appears to be much more difﬁcult than the ordinary case. [ p] All of the above constructions rely on the existence of the universal eigenform G as a ρ¯ family of p-adic modular forms over X(ρ) ¯ . However, in the ﬁnite-slope case, we need to pay attention to overconvergence conditions, since the ﬁnite-slope analogue of the projectors λ [ p] are only deﬁned on overconvergent spaces. Clearly G is not overconvergent (as a family), ρ¯ since it has specialisations which are nearly-classical rather than classical. So we need to work in an appropriate theory of nearly-overconvergent families. Such a theory has recently been introduced by Andreatta and Iovita [1]. We might make the following optimistic conjecture: 3 ord 0 This is slightly imprecise since X (ρ) ¯ is not contained in X (ρ) ¯ ; more precisely, the correspondence ord no 0 between the two constructions is given by identifying X (ρ) ¯ with X (ρ) ¯ ∩ X (ρ) ¯ , via twisting by a suitable ab character of = Z . Q,{ p} 123 P-adic L-functions in universal... Conjecture 5.1 Let f be a nearly-classical point of X(ρ) ¯ , corresponding to a modular form an f of prime-to- p level. Then there is an afﬁnoid neighbourhood X = Max A of f in X(ρ) ¯ f f [ p] over which the universal eigenform G is a family of nearly-overconvergent forms in the ρ¯ sense of [1]. If this conjecture holds, one might realistically hope to deﬁne (for instance) a p-adic Rankin–Selberg L-function over neighbourhoods of crystalline classical points in E (ρ¯ ) × an X(ρ¯ ) . 6 Conjectures on P-nearly-ordinary families In this section, we’ll use Galois deformation theory to deﬁne universal parameter spaces for Galois representations valued in reductive groups, which satisfy a Panchishkin-type condition relative to a parabolic subgroup; and we formulate a “parabolic R = T ” conjecture, pre- dicting that these should have an alternative, purely automorphic description. We expect that these parameter spaces should be the natural base spaces for families of p-adic L-functions, and of Euler systems. 6.1 Nearly-ordinary Galois deformations Let G be a reductive group scheme over O and P a parabolic subgroup. In [6, §7], Böckle deﬁnes a homomorphism ρ : → G( A),for A ∈ CNL ,tobe P-nearly ordinary if Q,S ρ| lands in a conjugate of P( A). Theorem 7.6 of op.cit. shows that under some mild hypotheses, the functor of P-nearly-ordinary deformations of a given P-nearly-ordinary residual representation is representable. Remark 6.1 This extends the deﬁnition of near-ordinarity described in Remark 2.4 above, which corresponds to taking G = GL and P the Borel subgroup. On the other hand, if (V, V ) is an r-Panchishkin family in the sense of Deﬁnition 2.3,thenitis P-nearly-ordinary where P is the corresponds to taking G = GL and P to be the parabolic subgroup of block- c=1 c=−1 upper-triangular matrices with blocks of sizes dim V −r and dim V +r. So the notion of P-near-ordinarity gives a framework covering both of these classes of representations. The reason why we consider general reductive groups G, rather than just GL ,isthat the geometry of deformation spaces for GL -valued global Galois representations is rather mysterious when n > 2; in particular, it is not expected that these spaces will always have a Zariski-dense set of specialisations which are de Rham. However, the geometry of defor- mation spaces is much simpler and better-understood for Galois representations arising from Shimura varieties (or, more generally, from automorphic representations that are discrete- series at ∞). 6.2 Nearly-ordinary automorphic representations We now introduce the corresponding condition on the automorphic side. We let G be a reductive group over Q; for simplicity, we assume here G is split. We also suppose G has a “twisting element” in the sense of [8], and ﬁx a choice of such an element . Then Conjecture 4 ∨ Alternatively, one could replace G by the identity component of the “C-group” of op.cit., which the quotient of G × G by a central element of order 2. We can also allow non-split G, by considering representations into a larger, non-connected quotient of the C-group. 123 D. Loefﬂer 5.3.4 of op.cit. predicts that cohomological cuspidal automorphic representations of G ∨ ∨ give rise to Galois representations ρ : → G (Q ),where G is the Langlands dual , p Q p of G. Let P be a parabolic in G, with Levi decomposition P = MN. We say that a cohomo- logical, cuspidal automorphic representation is P-nearly-ordinary if the Hecke operators at p associated to cocharacters valued in the centre Z ( M ) of M act on with unit eigen- values. For example, if P is the parabolic in GSp consisting of matrices whose lower-left 2n n × n block is zero, then the condition of P-near-ordinarity is that the double coset of diag( p,..., p, 1,..., 1) should act as a unit. See also [22] for a discussion of near-ordinarity for general parabolics in the case G = SL . The link between the “Galois” and “automorphic” notions of near-ordinarity is the fol- lowing. Since the root datum of G is the dual of that of G, there is a canonical bijection ∨ ∨ P ↔ P between conjugacy classes of parabolics in P and parabolics in G ; and one expects that if is nearly-ordinary for P,then ρ should be a P -nearly-ordinary rep- , p resentation. (This is known in many cases; see e.g. [36] for the group GSp .) In particular, families of P-nearly-ordinary cohomological automorphic representations of G should give ∨ ∨ rise to families of P -nearly-ordinary Galois representations into G . If we also choose a linear representation ξ : G → GL , then for suitably chosen P and r, the resulting families of n-dimensional Galois representations will be r-Panchishkin families. The example of §4 is of this type, taking G = GL × GL ,and P = B × GL 2 2 2 2 where B is the Borel subgroup of GL .Herewetake ξ to be the 4-dimensional tensor 2 2 product representation of G G, which maps B × GL to the parabolic in GL with 2 2 4 block sizes (2, 2). Similarly, the self-dual triple-product setting of Sect. 5.3 corresponds to taking G to be the group (GL × GL × GL )/ GL ,and P the image of B × GL × GL .Then G is a 2 2 2 1 2 2 2 subgroup of GL × GL × GL , and the 8-dimensional tensor-product representation of G 2 2 2 sends the dual of P to the parabolic in GL with blocks (4, 4). 6.3 Big and small Galois eigenvarieties In the setting of the previous paragraph, we deﬁne the big P-nearly-ordinary Galois eigenva- ∨ ∨ riety for G to be the following space. Suppose G and P have smooth models over O,and ∨ ∨ ﬁx some choice of ρ¯ : → G (F) which is P -nearly-ordinary. Then – assuming the Q,S hypotheses of Böckle’s construction are satisﬁed – we obtain a universal deformation ring P −no ∨ R (ρ) ¯ for for P -nearly-ordinary liftings of ρ¯.Wedeﬁne thebig P-nearly-ordinary P −no Galois eigenvariety X (ρ) ¯ to be the formal spectrum of this ring R (ρ) ¯ . The methods of [6] give a formula for the dimension of this space. Suppose ρ¯ satisﬁes the Ad ρ( ¯ c)=1 “oddness” condition that dim g = dim(G/ B ),where g is the Lie algebra of G/F, G F c is complex conjugation and B is a Borel subgroup of G. (This condition is expected to hold P −no for representations arising from Shimura varieties; see [11, Introduction].) Then R (ρ) ¯ has a presentation as a quotient of a power series ring in d variables by an ideal with d 1 2 generators, where d − d = dim P − dim(G/ B ) = dim B , 1 2 G M where M is the Levi factor of P and B ⊆ M is a Borel subgroup of M. It seems reasonable to conjecture that X (ρ) ¯ is in fact ﬂat over O, and its relative dimension is dim B . P M The term big is intended to contrast with the following alternative construction (which is perhaps less immediately natural; we introduce it because it is the Galois counterpart 123 P-adic L-functions in universal... of an existing construction on the automorphic side, as we shall recall below). Let M = ∨ ∨ ∨ ∨ ∨ der M / Z ( M ),where Z ( M ) is the centre of M ; hence M is the Langlands dual of M . We ﬁx a Hodge type v and an inertial type τ for M -valued representations of ,inthe sense of [3]. Then we say a lifting ρ of ρ¯ to Q is P -nearly-ordinary of type (τ, v) if it ∨ ∨ is P -nearly-ordinary, and the composition − → P (Q ) → M (Q ) has the given p p p Hodge and inertial types. We deﬁne the small P-nearly-ordinary Galois eigenvariety to be the universal deformation space X (ρ¯; τ, v) for deformations that are P -nearly-ordinary of the speciﬁed type. Using the formulae of [3] applied to M to compute the dimension of the local lifting rings, and assuming that ρ¯ is odd and v is sufﬁciently regular, we compute that the expected dimension of X (ρ¯; τ, v) is now given by dim Z ( M ) = dim Z ( M ). Remark 6.2 Note that the big and small Galois eigenvarieties coincide if P is a Borel sub- group; but the dimension of the big eigenvariety grows with P, while the dimension of the small eigenvariety shrinks as P grows. For instance, if G = GL and P = G,then X (ρ) ¯ is 2 P just the unrestricted deformation space, which is 3-dimensional over O as we have seen; but X (ρ¯; τ, v) has dimension 1, since for any (τ, v) there are only ﬁnitely many deformations of that type, so X (ρ¯; τ, v) has only ﬁnitely many points up to twisting by characters. 6.4 Big and small automorphic eigenvarieties We can now ask if the above Galois-theoretic spaces have automorphic counterparts. 6.4.1 The big eigenvariety Seeking an automorphic counterpart of the big Galois eigenvariety leads to the following question: Question:If G is reductive over Q,and P is a parabolic in G/Q as above, is there a natural purely automorphic construction of a parameter space E for systems of Hecke eigenvalues arising from cohomological automorphic representations for G that are nearly ordinary for the parabolic P? We call this conjectural object E the big P-nearly-ordinary automorphic Eigenvariety. We expect its dimension to be the same as its Galois analogue; in particular, if G has discrete series its dimension should be dim B ,where B is a Borel subgroup of the Levi of P as M M before. The case when P = B is a Borel subgroup is relatively well-understood; this is the setting of Hida theory. However, the case of non-Borel parabolics is much more mysterious. In this case, one can give a candidate for this space E as follows. For any open compact K ⊂ G(A ), we can form the H (K , O) of Betti cohomology of the symmetric space for G of level K , which is a ﬁnitely-generated graded O-module. This has an action of Hecke operators, and the subalgebra of its endomorphisms generated by Hecke operators at primes where K is unramiﬁed, the spherical Hecke algebra of level K,is commutative. We ﬁx an open compact subgroup K ⊂ G(A ),and let K ={g ∈ G(Z ) : g mod n, p p n n ∗ p ∈ N (Z/ p )},where N is the unipotent radical of P. Then, for any n ≥ 1, H (K , O) P P has a canonical idempotent endomorphism e (the Hida ordinary projector associated to r ! P), deﬁned by lim U where U is a suitable Hecke operator; this commutes with the r →∞ P spherical Hecke algebra. 123 D. Loefﬂer P−no p Deﬁnition 6.3 With the above notations, let T (K ) be the quotient of the spher- ∗ p P−no p ical Hecke algebra acting faithfully on e H (K K , O); and deﬁne T (K ) = P p,n P−no p lim T (K ). ← − P−no p We conjecture that the formal spectrum of T (K ) should be the big P-nearly- ordinary eigenvariety. However, from this deﬁnition alone it is rather difﬁcult to obtain much information about the properties of the resulting space (for instance, it is not clear whether P−no p T (K ) is Noetherian). As far as the author is aware, the only non-Borel cases where this construction is well-understood are the following: • G = GL and P = G, as in Theorem 3.4. • G = Res + (U ),where U is a totally deﬁnite unitary group for some CM extension F /Q + + F / F , with p split in F and F / F unramiﬁed at all ﬁnite places; and P is a parabolic [ F :Q ] ∼ p subgroup of G(Q ) GL (Q ) whose Levi subgroup is a product of copies of p n p GL and GL . This case has been studied extensively by Yiwen Ding [16]. 1 2 P−no p In the deﬁnite unitary case, Ding proves that the localisation of T (K ) at the maximal ideal corresponding to an irreducible ρ¯ is a quotient of the global Galois deformation ring P −no R (ρ) ¯ , and is therefore Noetherian; and he gives a lower bound for the relative dimension P−no p of T (K ) over O (localised at the maximal ideal corresponding to some ρ¯). This lower bound is exactly dim B , the dimension conjectured for the Galois eigenvariety above. Remark 6.4 Note that Ding’s construction uses the p-adic local Langlands correspondence for GL (Q ) in an essential way, so this approach will be much harder to generalise to cases 2 p where the Levi of P is not a product of tori and copies of GL (Q ). 2 p 6.4.2 The small eigenvariety In contrast to the rather disappointing situation described above, there does seem to be a well-established theory for the “little brother” of this space—the small P-nearly-ordinary automorphic eigenvariety. This would be a parameter space for P-nearly-ordinary cohomo- logical automorphic representations satisfying two additional conditions: • the highest weight λ of the algebraic representation of G to whose cohomology con- der tributes should lie in a ﬁxed equivalence class modulo characters of M / M ; no • the ordinary part J ( ) of J ( ), which is an irreducible smooth representation of P p P p no M (Q ), should satisfy e · J ( ) = 0where e is some ﬁxed idempotent in the Hecke p P p der algebra of M (Q ). Note that both conditions are vacuous if P is a Borel. These conditions are the automorphic counterparts of the ﬁxed Hodge and inertial types up to twisting used to deﬁne the small P- nearly-ordinary Galois eigenvariety. See e.g. Mauger [30] for the construction of the small P-nearly-ordinary automorphic eigenvariety, and [23]for a“ P-ﬁnite-slope” analogue. Remark 6.5 The most obvious choice of e would be the idempotent projecting to the invariants der for some choice of open compact subgroup of M (Q ). For instance, Mauger’s theory no der applies to such that J ( ) has non-zero invariants under M (Z ), although it can P p p be extended without difﬁculty to allow other more general idempotents. However, a craftier choice would be to take e to be a special idempotent in the sense of [7], corresponding to der a choice of Bernstein component for M (Q ); these Bernstein components are expected to biject with inertial types on the Galois side (the inertial local Langlands correspondence 123 P-adic L-functions in universal... der for M (Q )), while the highest weights λ biject with Hodge types, so we obtain a natural dictionary between the deﬁning data at p for the Galois and automorphic versions of the small P-nearly-ordinary eigenvariety. 6.4.3 R = T theorems Both big and small automorphic eigenvarieties should, clearly, decompose into disjoint unions of pieces indexed by mod p Hecke eigenvalue systems. We can then formulate the (extremely speculative) “parabolic R = T ” conjecture that each of these pieces should correspond to one of the big or small Galois eigenvarieties of the previous section, for a mod p Galois representation ρ¯ determined by the mod p Hecke eigensystem. In the case when G is a deﬁnite unitary group, results of this kind have been proven by Geraghty [20]when P is a Borel subgroup; and when the Levi of P is a product of GL ’s P −no and GL ’s,Dingprovesin[16] the slightly weaker result that the map from R (ρ) ¯ to P−no p the ρ¯-localisation of T (K ) is surjective with nilpotent kernel, after possibly extending + red red the totally real ﬁeld F (an “ R = T ” theorem). 6.5 Miscellaneous remarks Remark 6.6 The 4-dimensional parameter space for GSp × GL mentioned at the end of §5.2 is a slightly artiﬁcial hybrid: the it is the product of the big automorphic (or Galois) eigenvariety for P = G = GL with the small automorphic eigenvariety for the Klingen parabolic of GSp . Of course, we expect that the “correct” parameter space for this construc- tion is the product of the big eigenvarieties for the two groups, which would have dimension 7 (or 6 if we factor out a redundant twist, which corresponds to working with the group GSp × GL ). However, we do not know how to construct p-adic L-functions on this GL 2 4 1 eigenvariety at present. Remark 6.7 The small P-nearly-ordinary eigenvariety is ﬁnite over the “weight space” der parametrising characters of ( M / M )(Z ). Moreover, in Shimura-variety settings it is ﬂat over this space (up to a minor grain of salt if the centre Z (G) has inﬁnite arithmetic sub- groups). It is natural to ask if there is an analogous, purely locally deﬁned “big P-weight space” over which the big eigenvariety E is ﬁnite; the results of [16] suggest that a candidate could be a universal deformation space for p-adic Banach representations of M (Q ) on the automorphic side, or M -valued representations of on the Galois side. However, these spaces will in general have much larger dimension than the eigenvariety, so there does not seem to be a natural choice of local parameter space over which E is ﬁnite and ﬂat. Acknowledgements It is a pleasure to dedicate this article to Bernadette Perrin-Riou, in honour of her immense and varied contributions to number theory in general, and to p-adic L-functions in particular, which have been an inspiration to me throughout my career. I would also like to thank Daniel Barrera Salazar, Yiwen Ding, and Chris Williams for informative discussions in connection with this paper; Sarah Zerbes for her feedback on an earlier draft (and her assistance translating the abstract into French); and the anonymous referee for numerous valuable comments and corrections. 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. 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Abstract

We construct examples of p-adic L-functions over universal deformation spaces for GL .We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions and Euler systems are not the usual eigenvarieties (parametrising nearly-ordinary families of automorphic representations), but other, larger spaces depending on a choice of a parabolic subgroup, which we call ‘big parabolic eigenvarieties’. Résumé Nous construisons des exemples de fonctions Lp-adiques déﬁnies sur les espaces des défor- mations universels de GL . Nous formulons une conjecture qui prédit que les espaces (2) naturels des paramétres pour les fonctions Lp-adiques et les systémes d’Euler ne sont pas les variétés de Hecke usuelles, mais d’autres espaces plus grands, qui dépendent d’un choix de sous-groupe parabolique. Mathematics Subject Classiﬁcation 11F67 · 11F85 1 Introduction It is well known that many interesting automorphic L-functions L(π, s) have p-adic coun- terparts; and that these can often be extended to multi-variable p-adic L-functions, in which the automorphic representation π itself also varies in a p-adic family of some kind. In the literature so far, the p-adic families considered have been Hida families, or more generally Coleman families—families of automorphic representations which are principal series at p, together with the additional data of a “ p-reﬁnement” (a choice of one among the Weyl-group orbit of characters from which π is induced). In Galois-theoretic terms, this corresponds to a full ﬂag of subspaces in the local Galois representation at p (or in its (ϕ, )-module, for Coleman families). The parameter spaces for these families are known as eigenvarieties. The aim of this note is to give an example of a p-adic L-function varying in a family of a rather different type: it arises from a family of automorphic representations of GL × GL ,but 2 2 the parameter space for this family (arising from Galois deformation theory) has strictly bigger Supported by Royal Society University Research Fellowship “L-functions and Iwasawa theory”. B David Loefﬂer d.a.loefﬂer@warwick.ac.uk Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK 123 D. Loefﬂer dimension than the eigenvariety for this group—it has dimension 4, while the eigenvariety in this case has dimension 3. We also sketch some generalisations of the result which can be proved by the same methods. This corresponds to the fact that a p-reﬁnement is a little more data than is actually needed to deﬁne a p-adic L-function: rather than a full ﬂag, it sufﬁces to have a single local subrepresentation of a speciﬁc dimension (a Panchishkin subrepresentation), which is a weaker condition and hence permits variation over a larger parameter space. This weaker condition is also sufﬁcient to interpolate Selmer groups in families, and hence to formulate an Iwasawa main conjecture. We conclude with some speculative conjectures whose aim is to identify the largest param- eter spaces on which p-adic L-functions and Euler systems can make sense. We conjecture that, given a reductive group G and parabolic subgroup P (and appropriate auxiliary data), there should be two natural p-adic formal schemes, the big and small P-nearly-ordinary eigenvarieties. These coincide if P is a Borel subgroup, but not otherwise; if G = GL and P is the whole of G, then the big eigenvariety is the 3-dimensional Galois deformation space of a modular mod p representation (with no local conditions at p). In general, we expect that the “natural home” of p-adic L-functions – and also of Euler systems – should be a big ordinary eigenvariety for an appropriate parabolic subgroup. 2 Families of Galois representations 2.1 The Panchishkin condition Let L be a ﬁnite extension of Q and let V be a ﬁnite-dimensional L-vector space with a continuous linear action of = Gal(Q/Q). Recall that V is said to be geometric if it is unramiﬁed at all but ﬁnitely many primes and de Rham at p; in particular it is Hodge–Tate at p, so we may consider its Hodge–Tate weights. (In this paper, we adopt the common, but not entirely universal, convention that the cyclotomic character has Hodge–Tate weight +1.) We recall the following condition introduced in [29, Deﬁnition 7.2] (generalising the more familiar case r = 0, which is the condition originally studied by Panchishkin in [33, Deﬁnition 5.5]). Deﬁnition 2.1 Let r ≥ 0 be an integer. We say V satisﬁes the r-Panchishkin condition if it is geometric, and the following conditions hold: (1) We have (c=+1) (number of Hodge–Tate weights ≥ 1ofV ) = dim V − r , where c ∈ is (any) complex conjugation. + + (2) There exists a subspace V ⊆ V stable under such that V has all Hodge–Tate weights ≥ 1, and V /V has all Hodge–Tate weights ≤ 0. Remark 2.2 (i) Note that V is unique if exists; we call it the Panchishkin subrepresentation of V at p. (ii) Condition (1) is equivalent to requiring that the Tate dual V (1) be “r-critical”, in the sense of [29, Deﬁnition 6.4]. This generalises the notion of “critical values” of an L- function due to Deligne: if V is the p-adic realisation of a motive M, then condition (1) for r = 0 is equivalent to requiring that L( M , 0) is a critical value of the L-function L( M , s) in the sense of [15]. (Note that this use of the word “critical” is unrelated to the usage in the theory of eigenvarieties, as in [2] for instance.) 123 P-adic L-functions in universal... (iii) The Panchishkin condition is closely related to the concept of near-ordinarity:arep- resentation V is said to be nearly-ordinary if it is geometric with distinct Hodge–Tate weights, and there exists a full ﬂag of subspaces of V such that the Hodge–Tate weights of the graded pieces are in strictly increasing order. Clearly, if V is nearly-ordinary and r-critical, then it is also r-Panchishkin; but the condition of near-ordinarity is much more restrictive, and we want to emphasise here that near-ordinarity is an unnecessarily strong hypothesis for the study of p-adic L-functions. 2.2 Panchishkin families By a “Panchishkin family”, we mean a family of p-adic Galois representations equipped with a family of Panchishkin subrepresentations, in the sense of the following deﬁnition. For simplicity, we shall suppose throughout this paper that p > 2, so that we can diagonalise the action of complex conjugation without introducing denominators. Let O be the ring of integers of L,and F its residue ﬁeld. We let CNL be the category of complete Noetherian local O-algebras with residue ﬁeld F. Deﬁnition 2.3 Let R be an object of CNL ,and r ≥ 1aninteger.An r-Panchishkin family of Galois representations (V, V ) over R consists of the following data: • a ﬁnite free R-module V with an R-linear continuous action of , unramiﬁed at almost all primes. + c=1 • an R-direct-summand V ⊆ V stable under ,of R-rank equal to that of V . These are required to satisfy the following condition: • The set (V, V ) of maximal ideals x of R[1/ p] such that V satisﬁes the r-Panchishkin condition and V is its Panchishkin subrepresentation is dense in Spec R[1/ p]. + + + We call V an r-Panchishkin submodule of V,and (V, V ) the interpolation region of V . It is important to note that V is not uniquely determined by V (orevenbythe pair (V, r )), in contrast to the case of Q -linear representations, as the examples of the next section will show. Remark 2.4 The natural notion of a nearly-ordinary family in this case would be a ﬁnite free R-module V with -action as above, together with a ﬁltration by -stable submodules Q Q 0 1 V = F V ⊃ F V ⊃ ... with graded pieces free of rank 1 over R. Our main focus in the present work will be in examples of r-Panchishkin families which do not admit a nearly- ordinary ﬁltration. 2.3 Examples from character twists We brieﬂy illustrate the above deﬁnitions using families of representations arising by twisting a ﬁxed, geometric representation by a family of characters. Example 2.5 (Cyclotomic twists) The original examples of Panchishkin families are those of the following form. Let V be an L-linear representation of satisfying the r-Panchishkin ◦ + condition, and V a O-lattice in V stable under .Let V be the Panchishkin subrepre- ◦+ + ◦ sentation, and V = V ∩ V . × × × We let denote the Iwasawa algebra O[[Z ]],and j the canonical character Z → . p p c=1 c=−1 If dim V = dim V , then we can take R to be the localisation of at any of its ( p −1) 123 D. Loefﬂer × × maximal ideals, corresponding to characters Z → F ; otherwise, we need to assume our maximal ideal corresponds to a character trivial on −1. We can then let V = V ⊗ R(χ ), O cyc × × where χ : → R denotes the composite of j and the p-adic cyclotomic cyc Q + ◦+ ◦+ + ◦ + character. If we let V = V ⊗ R(χ ),where V = V ∩ V ,then V is an R-direct- O cyc summand of V stable under . By construction, the set (V, V ) contains all points of Spec R[1/ p] corresponding to characters of the form j + χ,where χ is of ﬁnite order and j is an integer in some interval + + containing 0 (depending on the gap between the Hodge–Tate weights of V and V /V ). In particular, it is Zariski-dense, as required; so (V, V ) is an r-Panchishkin family over R. Example 2.6 (Varying r) More generally, if V is a geometric L-linear -representation and ◦ c=+1 V a lattice in V,wecan deﬁne R(V ) to be the set of integers 0 ≤ r ≤ dim V such + c=+1 that V posesses a -stable subrepresentation V of dimension dim V − r with all p r + + Hodge–Tate weights of V strictly larger than those of V /V . Note that R(V ) is never r r c=+1 + empty, since it always contains dim V (with V ={0}); and if V is nearly ordinary, c=+1 then R(V ) is the whole interval [0, dim V ]. For r ∈ R(V ), deﬁning R and V as in the previous example, we can deﬁne V = ◦+ + V ⊗ R(χ ).Then (V, V ) is an r-Panchishkin family, with interpolation region O cyc r r (V, V ) consisting of all j + χ with j in some nonempty interval of integers depend- ing on V and r. Example 2.7 (Twisting by imaginary quadratic Grössencharacters) As a ﬁnal “character- twist” example, we consider the following setting: let K be an imaginary quadratic ﬁeld with p = p p split in K,and let V be a geometric representation of ,ofdimension d.We 1 2 K let K be the unique Z -extension of K , with Galois group G ,and R = O[[G ]].We ∞ K K consider the rank 2d family over R deﬁned by V = Ind V ⊗ R(χ ) O univ ◦ × for some lattice V ⊂ V , with χ the composite map G → R (the “universal univ K K character” valued in R). We now equip this with Panchishkin submodules. Choosing decomposition groups at the p , we obtain an isomorphism Suppose u , u are integers ≥ 0 with u + u ≤ d. We assume that for each i = 1, 2, the 1 2 1 2 representation V has subrepresentations V stable under the decomposition group at p ,of + + dimension u , with each V having strictly larger Hodge–Tate weights than V /V .Ifwe i p p i i + ◦+ deﬁne V to be the image of V ⊗ R(χ )| under the isomorphism (†),then O univ i i Kp + + V is an r-Panchishkin submodule of V where r = d − u − u ;the set (V, V ) consists of 1 2 all locally-algebraic characters of G whose Hodge–Tate weights lie in a certain non-empty region of Z , depending on V and the u . See [29, §10] for an example of this kind with d = 2; in this case, we obtain 6 different Panchishkin submodules, one with r = 2, two with r = 1 and three with r = 0. The interpo- lation regions for these submodules are illustrated in Figure 1 of op.cit.. (More generally, if 1 j We use additive notation for characters, so j + χ is a shorthand for the character z → z χ(z). 123 P-adic L-functions in universal... V is nearly-ordinary of any dimension d ≥ 1, this construction gives rise to (d + 1)(d + 2) different Panchishkin submodules, d + 1 of which have r = 0.) 2.4 Conjectures on p-adic L-functions The following conjecture is due to Coates–Perrin-Riou [12] and Panchishkin [33]inthe case of cyclotomic twists of a ﬁxed representation. The generalisation to families as above is “folklore”; we have been unable to locate its ﬁrst appearance, but it is a special case of more general conjectures of Fukaya and Kato [18] (who have also investigated the case of non-commutative base rings R, which we shall not attempt to consider here). Conjecture 2.8 Suppose (V, V ) is an 0-Panchishkin family. There exists an element + + L(V, V ) ∈ Frac R such that for all x ∈ (V, V ) we have L( M , 0) L(V, V )(x ) = (Euler factor) · , (period) where M is the (conjectural) motive whose realisation is V . x x If V is semistable at p, the expected form of the Euler factor is −1 −1 + + det (1 − p ϕ ) : D (V ) · det (1 − ϕ) : D (V /V ) . cris cris We refer to [18] for more details of the interpolation factors involved. 2.5 Euler systems In [29], Zerbes and the present author conjectured that when V is the family of cyclotomic twists of a ﬁxed representation, the r-Panchishkin condition was the “correct” condition for a family of Euler systems of rank r to exist, taking values in the r-th wedge power of the Galois ∗ + cohomology of the Tate dual V (1), and satisfying a local condition at p determined by V . This extends the conjectures formulated by Perrin-Riou in [34], which correspond to taking c=1 r to be the maximal value dim V (in which case the r-Panchishkin condition is automatic, as we have seen). It is also consistent with the above conjectures of Coates–Perrin-Riou and Panchishkin for r = 0, if we understand a “rank 0 Euler system” to be a p-adic L-function. It seems natural to expect that an analogue of Conjecture 2.8 should hold for arbitrary r-Panchishkin families, predicting the existence of families of rank r Euler systems over R; and, as in the rank 0 case, one can show that this would follow as a consequence of the very general conjectures of [18]. Remark 2.9 There are a number of (unconditional) results concerning the variation of Euler systems in Hida families of automorphic representations, which are examples of nearly- ordinary families; see e.g. [32] for Kato’s Euler system, and [26]for theGL × GL 2 2 Beilinson–Flach Euler system. However, the above conjecture predicts that Euler systems should vary in more general families, which are not nearly-ordinary but are still r-Panchishkin. Some examples of cyclo- tomic twist type for r = 1 are discussed in [29]. A much more sophisticated example due to Nakamura, in which R is the universal deformation space of a 2-dimensional modular Galois representation, is discussed in §3.5 below. 123 D. Loefﬂer 3 Examples from GL Notation: Suppose f is a modular cusp form which is a normalised eigenform for the Hecke operators, with a chosen embedding of its coefﬁcient ﬁeld Q( f ) into L.Wedenoteby ρ the unique Galois representation → GL (L) characterised as follows: we have f , p Q 2 −1 tr ρ (Frob ) = a ( f ) for almost all primes , where Frob denotes an arithmetic Frobe- f , p 1−k nius at . Thus det ρ is χ up to ﬁnite-order characters, where k is the weight of f .(We f , p cyc warn the reader that some references use the notation ρ for the dual of this representation.) f , p 3.1 The universal deformation ring Let ρ¯ : → GL(V ) GL (F) be a 2-dimensional, odd, irreducible (hence, by Khare– Q 2 Wintenberger, modular) representation. Hypotheses 3.1 We shall assume ρ¯ satisﬁes the following: (a) ρ¯| is irreducible, where K = Q(ζ ) (Taylor–Wiles condition). (b) if ρ¯| is not absolutely irreducible, with semisimpliﬁcation χ ⊕ χ (after possibly 1, p 2, p Q p ±1 extending F), then we have χ /χ ∈{ / 1,ε } where ε is the mod p cyclotomic 1, p 2, p p character. (c) ρ¯ is unramiﬁed away from p (i.e. its tame level is 1). Remark 3.2 Note that the ﬁrst two assumptions are essential to our method, because they are hypotheses for major theorems which we need to quote. On the other hand, the third is much less fundamental and is imposed chieﬂy in order to simplify the calculations in Sect. 4 below. See Sect. 5.1 below for further discussion. Deﬁnition 3.3 Let R(ρ) ¯ ∈ CNL be the universal deformation ring over O parametrising deformations of ρ¯ as a -representation, and ρ : → GL (R(ρ) ¯ ) the universal Q,{ p} Q,{ p} 2 deformation. Let X(ρ) ¯ = Spf R(ρ) ¯ . Theorem 3.4 (Böckle, Emerton) • The ring R(ρ) ¯ is a reduced complete intersection ring, and is ﬂat over O of relative dimension 3. • We have a canonical isomorphism R(ρ) ¯ = T (ρ) ¯ ,where T (ρ) ¯ is the localisation at the maximal ideal corresponding to ρ¯ of the prime-to- p Hecke algebra acting on the space S(1, O) of cuspidal p-adic modular forms of tame level 1. Proof This is proved in [5] assuming that ρ¯| has a twist which is either ordinary, or Q p irreducible and ﬂat. This was extended to the setting described above (allowing irreducible but non-ﬂat ρ¯) by Emerton, see [17, Theorem 1.2.3]. Remark 3.5 If ρ¯ is unobstructed in the sense that H , Ad(ρ) ¯ = 0, then R(ρ) ¯ is Q,{ p} isomorphic to a power-series ring in 3 variables over O. It is shown in [37]thatif f is a ﬁxed newform of weight ≥ 3, then for all but ﬁnitely many primes p of the coefﬁcient ﬁeld Q( f ), the mod p representation ρ¯ is unobstructed. f ,p Deﬁnition 3.6 (i) If f is a classical modular newform of p-power level (and any weight) such that ρ¯ =¯ ρ,then ρ is a deformation of ρ¯ and hence determines a Q -point of f , p f , p p X(ρ) ¯ . We shall call these points classical. 123 P-adic L-functions in universal... (ii) More generally, a Q -point of X(ρ) ¯ will be called nearly classical if the corresponding −t Galois representation ρ has the form ρ ⊗ (χ ) , for some (necessarily unique) f , p cyc newform f and t ∈ Z. −t In the setting of (ii), if t ≥ 0, the Galois representation ρ ⊗ (χ ) corresponds f , p cyc formally to the nearly-overconvergent p-adic modular form θ ( f ),where θ = q is the dq Serre–Tate differential operator on p-adic modular forms. Slightly abusively, we denote such t t a point by θ ( f ),evenif t < 0 (in which case θ ( f ) may not actually exist as a p-adic modular form). Either Theorem 1.2.4 of [17]orthe main theoremof[24], combined with Theorem 0.4 of [35] in the case of equal Hodge–Tate weights, shows that any Q -point ρ of X(ρ) ¯ which is de Rham at p is a nearly-classical point (as predicted by the Fontaine–Mazur conjecture). Proposition 3.7 For any weight k ≥ 2, modular points corresponding to weight k modular forms are dense in X(ρ) ¯ . Proof This is obvious for Spf T (ρ) ¯ ,since T (ρ) ¯ can be written as an inverse limit of locali- sations of Hecke algebras associated to the ﬁnite-level spaces S ( ( p ), O). Since we have k 1 R(ρ) ¯ T (ρ) ¯ by Theorem 3.4, the result follows. Remark 3.8 Note that a crucial step in the proof of Theorem 3.4 is to establish that the set of all modular points (of any weight) is dense in X(ρ) ¯ . However, once this theorem is established, we can obtain the much stronger result of Proposition 3.7 a posteriori. For later constructions we need the fact that there exists a “universal modular form” over X(ρ) ¯ : × × univ (1−k) Deﬁnition 3.9 (i) Let k : Z → R(ρ) ¯ be the character such that det ρ = (χ ) . cyc [ p] (ii) Let G be the formal power series ρ¯ [ p] G = t q ∈ R(ρ) ¯ [[q]], ρ¯ pn where the t are determined by the identity of formal Dirichlet series −1 −s −s univ −1 t n = det 1 − ρ (Frob ) . = p pn [ p] −t The specialisation of G at a nearly-classical point ρ ⊗ (χ ) is precisely the “ p- f , p cyc ρ¯ t [ p] t depletion” θ ( f ) of θ ( f ),where θ is the Serre–Tate differential operator q .If t ≥ 0, dq this p-adic modular form is the image under the unit-root splitting of a classical nearly- holomorphic cuspform, in the sense of Shimura. [ p] Theorem 3.10 (Gouvea) The series G is the q -expansion of a p-adic modular form with ρ¯ coefﬁcients in R(ρ) ¯ , of tame level 1 and weight-character k, which is a normalised eigenform for all Hecke operators. Proof This follows readily from the duality between Hecke algebras and spaces of cusp forms. 123 D. Loefﬂer 3.2 The universal ordinary representation The following deﬁnition is standard: Deﬁnition 3.11 An ordinary reﬁnement of (ρ, ¯ V ) is a choice of 1-dimensional F-subspace + + V ⊆ V stable under ρ( ¯ ), such that the inertia subgroup I acts trivially on V . Q Q p p Let us ﬁx a choice of ordinary reﬁnement V . Then there is a natural deﬁnition of ordinarity for deformations: we say that a deformation ρ of ρ¯ (to some ring A ∈ CNL )isordinaryif ρ| preserves a rank one A-summand lifting V , and the action of I on this summand Q p is trivial. (Note that this summand is unique if it exists, since Hypothesis 3.1(b) implies that + + V /V cannot be isomorphic to V ). Theorem 3.12 Suppose ρ¯ is ordinary. Then there exists a complete local Noetherian O- ord algebra representing the functor of ordinary deformations. We let R (ρ) ¯ be this algebra, ord ord and X (ρ) ¯ = Spf R (ρ) ¯ . ord ord ord ord,+ Remark 3.13 Over R (ρ) ¯ we have a universal triple (ρ , V , V ), but we caution ord ord,+ ord the reader that (V , V ) is not an 0-Panchishkin family over R (ρ) ¯ in the sense of Deﬁnition 2.3, since it interpolates the representations ρ for ordinary modular forms f . f , p These have all their Hodge–Tate weights ≤ 0, and hence cannot satisfy condition (1) in Deﬁnition 2.1. However, we shall build interesting examples of 0-Panchishkin families from ord ord,+ (V , V ) via twists and tensor products. ord On the “modular” side, we can consider the ordinary Hecke algebra T (ρ) ¯ , which is the ord localisation at ρ¯ of the algebra of endomorphisms of e · S(1, Z ) generated by all of the Hecke operators (including U ). There is a natural map ord ord R (ρ) ¯ → T (ρ) ¯ , and by Theorem 3.3 of [38], this map is an isomorphism. (Note that this isomorphism is compatible with the isomorphism R(ρ) ¯ = T (ρ) ¯ of the previous section, via the natural maps ord ord R(ρ) ¯ → R (ρ) ¯ and T (ρ) ¯ → T (ρ) ¯ .) × ord ord Note that the composite Z − → R(ρ) ¯ → R (ρ) ¯ gives R (ρ) ¯ the structure of a - × ord algebra, where = O[[Z ]].Sowehaveamap k : X (ρ) ¯ → X = Spf . cyc Proposition 3.14 (Hida) ord • The ring R (ρ) ¯ is ﬁnite and projective as a -module, and thus has relative dimension 1 over O. × × n • If k ≥ 2 is an integer, and χ : Z → O is a Dirichlet character of conductor p ,then ord the ﬁbre of X (ρ) ¯ at k = k + χ is étale over L = Frac O, and its geometric points biject with the normalised weight k eigenforms of level ( p ) and character χ (if n ≥ 1)or level ( p) (if n = 0) which are ordinary and whose mod p Galois representation is ρ¯. determined (Note that this ﬁbre is empty if k + χ does not lie in the component of X cyc by det ρ¯.) ord Much as above, we can deﬁne a universal ordinary eigenform G with coefﬁcients in ρ¯ [ p] ord ord R (ρ) ¯ (whose p-depletion is the pullback of G along X (ρ) ¯ → X(ρ) ¯ , and whose ρ¯ −1 + U -eigenvalue is the scalar by which Frob acts on V ). However, we shall not use this explicitly here. 123 P-adic L-functions in universal... ord More useful is the following dual construction due to Hida [21]. The ring R (ρ) ¯ ord has ﬁnitely many minimal primes, corresponding to irreducible components of X (ρ) ¯ ord (“branches”). If a is a minimal prime, and we let T be the integral closure of T (ρ) ¯ /a, then we can ﬁnd an invertible ideal I T , and a homomorphism a a ord −1 λ : S (1,) ⊗ T → I , ord a a T (ρ) ¯ a ord characterised by mapping G to 1. ρ¯ 3.3 Nearly ordinary deformations More generally, we can deﬁne a nearly ordinary reﬁnement V of V by dropping the requirement that inertia act trivially on V ; and there is a corresponding deﬁnition of a nearly-ordinary deformation (V , V ) of (V , V ). no Proposition 3.15 There exists a ring R (ρ) ¯ ∈ CNL , and a nearly-ordinary deformation no no,+ (V , V ) of (V , V ) to this ring, which are universal among nearly-ordinary deforma- no tions of (V , V ). Moreover, R (ρ) ¯ is ﬂat over O of relative dimension 2. Proof The representability of this functor follows easily from the ordinary case above. If (V , V ) is nearly-ordinary, we can ﬁnd a unique character χ ¯ : Gal(Q(ζ )/Q) → F such that (V ⊗¯ χ, V ⊗¯ χ) is ordinary. Similarly, the data of a nearly-ordinary deformation of + + (V , V ) is equivalent to the data of an ordinary deformation of (V ⊗¯ χ, V ⊗¯ χ), together with a character χ of Gal(Q(ζ ∞ )/Q) lifting χ ¯ . This shows that the functor of nearly-ordinary no deformations is represented by a ring R (ρ) ¯ , deﬁned as the completed tensor product of ord R (ρ¯ ⊗¯ χ) and the ring parametrising deformations of χ ¯ to a character of Gal(Q(ζ )/Q), ord which is isomorphic to O[[ X ]].Since R (ρ¯ ⊗¯ χ) is ﬂat of dimension 1, we conclude that no R (ρ) ¯ is ﬂat of dimension 2. 3.4 Examples of Panchishkin families The above deformation-theoretic results give rise to the following examples of 0-Panchishkin families, in the sense of Deﬁnition 2.3. Example 3.16 (Nearly-ordinary deformations of modular forms) Suppose V is a modular mod p representation satisfying Hypotheses 3.1, with a nearly-ordinary reﬁnement V .Then no no the universal family V of Galois representations over R (ρ) ¯ , together with its universal no,+ nearly-ordinary reﬁnement V , is an example of a 0-Panchishkin family. In this case, + no Hida theory shows that (V, V ) consists precisely of the Q points of X (ρ) ¯ of the form −s θ ( f ),where f has weight k ≥ 2and 1 ≤ s ≤ k − 1. These are manifestly Zariski-dense. Conjecture 2.8 is known for this family, by work of Mazur and Kitagawa [25] . We are principally interested in examples which (unlike Example 3.16)are not nearly- ordinary. Our ﬁrst examples of such representations come from tensor products: Example 3.17 (Half-ordinary Rankin–Selberg convolutions) Let V and V be two mod p 1 2 representations satisfying Hypotheses 3.1, and suppose V admits a nearly-ordinary reﬁne- ment V . Twisting V by a character and V by the inverse of this character, we can suppose 1 2 See also [19] for a comparison between the Mazur–Kitagawa result and the general conjectures of [18]. 123 D. Loefﬂer that (V , V ) is actually ordinary (not just nearly-so). Then we consider the triple (R, V, V ) given by ord,+ ord ord + ˆ ˆ ˆ R = R (ρ¯ )⊗R(ρ¯ ), V = V ⊗V , V = V ⊗V . 1 2 2 2 ord,+ ord where (V , V ) is the universal ordinary deformation of (V , V ),and V the universal 1 2 1 1 deformation of V (with no ordinarity condition). Note that R has relative dimension 4 over O. + −s The set (V, V ) is the set of points of the form f ,θ (g) ,where f is a classical point −s of weight k ≥ 2, and θ (g) is a nearly-classical point such that g has weight < k and s lies in the range of critical values of the Rankin–Selberg L-function, namely ≤ s ≤ k − 1. This set (V, V ) is Zariski-dense; even the specialisations with (k, , s) = (3, 2, 2) are dense. We shall verify Conjecture 2.8 for this family below. Remark 3.18 A generalisation of the above two examples would be to consider tensor prod- ucts of universal representations over product spaces of the form no X = X (ρ¯ ) × X(ρ¯ ) × ··· × X(ρ¯ ) 1 2 n for general n,where ρ¯ ,..., ρ¯ are irreducible modular representations mod p with ρ¯ nearly 1 n 1 ordinary. This space has dimension 3n −1; but there are n −1 “redundant” dimensions, since the tensor product is not affected by twisting ρ by a character and one of ρ ,...,ρ by the 1 2 n inverse of this character. Quotienting out by this action gives a 0-Panchishkin family over a 2n-dimensional base. Example 3.19 (General tensor products) Let L = Frac O and let V be any L-linear represen- tation of (not necessarily 2-dimensional) which is geometric, satisﬁes the 0-Panchishkin c=1 c=−1 ◦ condition, and has dim V = dim V .Let V be a -stable O-lattice in V (which Q 1 always exists). Then, for any modular mod p representation V , we obtain a 0-Panchishkin family by letting ◦ + ◦ + R = R(ρ¯ ), V = V ⊗ V , V = (V ∩ V ) ⊗ V , 2 2 2 1 1 1 In particular, we can take V to be the Galois representation arising from a cohomological automorphic representation of GSp which is Klingen-ordinary at p. Note that in the last two examples the subspace V will not,ingeneral,extendtoafull ﬂag of -stable subspaces, so V is not nearly ordinary. 3.5 Families of Euler systems For ρ¯ as in Hypotheses 3.1, the canonical 2-dimensional family V over R(ρ) ¯ will not, in general, satisfy the 0-Panchishkin condition. However, it automatically satisﬁes the r- Panchishkin condition for r = 1, as V ={0} satisﬁes the conditions of a 1-Panchishkin + t submodule (with (V, V ) being the set of nearly-classical specialisations θ ( f ) with t ≥ 0). So the more general conjecture sketched in §2.5 predicts that there should exist a family of Euler systems taking values in V (1), interpolating Kato’s Euler systems for each modular form f lifting ρ¯. Such a family of Euler systems has recently been constructed by Nakamura [31]. 123 P-adic L-functions in universal... 4 P-adic L-functions for half-ordinary Rankin convolutions Let us choose two mod p representations ρ¯ , ρ¯ satisfying Hypotheses 3.1, with ρ¯ ordinary 1 2 1 (but no ordinarity assumption on ρ¯ ). ord Choose a branch a of X (ρ¯ ) as before, and let A denote the ring T ⊗ T (ρ¯ ),and 1 a Z 2 X = X × X(ρ) ¯ its formal spectrum. This has relative dimension 4 over Z .Welet V denote a p ord ord ord ∗ + + ∗ + the A-linear representation ρ ⊗ (ρ ) (1),and V = (ρ ) ⊗ (ρ ) (1) where (ρ ) 2 2 1 1 1 ord is the 1-dimensional unramiﬁed subrepresentation of ρ | . Thus V is a rank 4 family of 1 Q -representations over X unramiﬁed outside p,and V a rank 2 local subrepresentation of V. Remark 4.1 This differs from the (V, V ) of Example 3.17 by an automorphism of the base ring R, so Conjecture 2.8 for either one of these examples is equivalent to the other. The present setup is slightly more convenient for the proofs. + t The set (V, V ) contains all points ( f ,θ (g)) where f has weight k ≥ 2, g has weight ≥ 1, and t is an integer with 0 ≤ t ≤ k − − 1. Our goal is to deﬁne a p-adic L-function + + associated to (V, V ), with an interpolating property at the points in (V, V ). × × The ring A is endowed with two canonical characters k , k : Z → A , the former 1 2 [ p] factoring through T and the latter through T (ρ¯ ). We can regard G as a p-adic eigenform a 2 ρ¯ with coefﬁcients in A, of weight k , by base extension. Deﬁnition 4.2 Let denote the p-adic modular form [ p] [ p] ord ord e G · E ∈ S (1, A), ρ¯ k −k 2 1 2 1 [ p] k−1 n where E = ( d )q ∈ S (1,) denotes the p-depleted Eisenstein series of n≥1 k k d|n pn weight k and tame level 1. Let −1 L (ρ¯ , ρ¯ ):=λ () ∈ I ⊗ A. a 1 2 a T a a This is a meromorphic formal-analytic function on the 4-dimensional space X × X(ρ) ¯ , regular along any 3-dimensional slice { f }× X(ρ) ¯ with f classical. We now show that the values of L at points in interpolate values of Rankin L- functions. Let ( f ,θ (g)) be such a point, with f , g newforms of p-power levels, and let k, be the weights of f , g.Let α be the eigenvalue of geometric Frobenius on the unramiﬁed subrepresentation of ρ | ,and let f be the p-stabilisation of f of U -eigenvalue α. f , p α p Remark 4.3 If f has non-trivial level, then f = f ,and α is just the U -eigenvalue of f .If α p 2 k−1 f has level one, then α is the unique unit root of the polynomial X − a ( f ) X + p ,and k−1 f is the level p eigenvector f (τ ) = f (τ ) − f ( pτ). α α ord We deﬁne λ to be the unique linear functional on S (1, L) which factors through f ,α projection to the f eigenspace, and satisﬁes λ ( f ) = 1. By deﬁnition, we have α f ,α α [ p] t t [ p] L (ρ¯ , ρ¯ )( f ,θ (g)) = λ θ (g ) · E . a 1 2 f ,α k− −2t 123 D. Loefﬂer ( p) Deﬁnition 4.4 For f , g newforms as above, we write L ( f × g, s) for the Rankin–Selberg L-function of f and g without its Euler factor at p, ( p) ( p) −s L ( f × g, s) := L (χ χ , 2s + 2 − k − ) a ( f )a (g)n f g n n n≥1 pn −1 −s −1 = det 1 − Frob : V ( f ) ⊗ V (g) , p p = p and let ( p) ( p) ( f ⊗ g, s):= (s) (s − + 1)L ( f ⊗ g, s). C C Theorem 4.5 We have (t +1) t −1 ( p) ∗ p P (g, p α ) ( f , g , + t ) t 1−k t k+ L (ρ¯ , ρ¯ )( f ,θ (g)) = 2 (−1) i λ (g) , a 1 2 ∗ −( +t ) ad P (g , p α) E ( f ) f , f where b is the level at which g is new. Here λ b (g) is the Atkin–Lehner pseudo-eigenvalue of g, P (g, X ) is the polynomial such that −1 r P (g, X ) = a (g) X , p p r ≥0 and k−1 k−2 p p ⎪ 1 − 1 − f crystalline at p, 2 2 α α k−1 ad E ( f ) = − f semistable non-crystalline at p, p 2 ⎪ a k−1 G(χ ) f non-semistable at p, new of level p . 2 f Proof This follows from the Rankin–Selberg integral formula. The computations are virtually identical to the case of ﬁnite-slope forms treated in [27], so we shall not reproduce the computations in detail here. t −1 P (g, p α ) Remark 4.6 Note that the factor can be written as ∗ −( +t ) P (g , p α) −1 −1 −1 + det (1 − ϕ) (1 − p ϕ ) : D (V ) cris + + ∗ + t where V = (ρ ) ⊗ ρ (1 + t ) is the ﬁbre of V at ( f ,θ (g)). On the other hand, the f , p g, p (t +1) factor λ b (g) is essentially the local ε-factor of this representation. 5 Other cases We brieﬂy comment on some other cases which can be treated by the same methods as above. For reasons of space we shall only give a very brief sketch of each construction, and we hope that these sketches will be expanded into a fuller account in future works. 123 P-adic L-functions in universal... 5.1 Relaxing the tame levels Firstly, the assumption in Theorem 4.5 that the tame levels of the Hida families be 1 should not be too difﬁcult to relax. However, handling general tame levels will require much more careful book-keeping about the local Euler factors at the bad primes, as in the construction of p-adic Rankin–Selberg L-functions for the convolution of two ordinary families in [10]. The problem of generalising Theorem 4.5 to non-trivial tame levels will be treated in the forthcoming Warwick PhD thesis of Zeping Hao. 5.2 The case of GSp(4) ×xGL(2) A more ambitious case which can be treated by the same methods is the following. Let be a cohomological automorphic representation of GSp which is globally generic, unram- iﬁed and Klingen-ordinary at p, and contributes to cohomology with coefﬁcients in the algebraic representation of weight (r , r ),for some r ≥ r ≥ 0. (Classically, these corre- 1 2 1 2 spond to holomorphic vector-valued Siegel modular forms taking values in the representation r −r r +3 1 2 2 Sym ⊗ det of GL .) For technical reasons we assume r > 0. 2 2 In [28] we constructed a cyclotomic p-adic L-function interpolating the critical values of L( ⊗ σ, s) where σ is an automorphic representation of GL generated by a holomorphic form of weight ≤ r − r + 1. This is constructed by applying a “push-forward” map to 1 2 [ p] the product of the p-depleted newform g ∈ σ with an auxiliary p-adic Eisenstein series, and pairing this with a coherent H eigenclass coming from . This construction is closely parallel to the construction of the p-adic Rankin–Selberg L-function for GL × GL , and it generalises to universal-deformation families in the same 2 2 way, since the pushforward map of [28] can be applied to any family of p-adic modular forms (over any base). If we assume for simplicity that is unramiﬁed at all ﬁnite places, [ p] and replace g with a universal deformation family G as above, then we obtain an element of ρ¯ R(ρ) ¯ interpolating these p-adic L-functions, with ﬁxed and σ varying through the small- weight specialisations of a 3-dimensional universal-deformation family. We can also add a fourth variable, in which we vary through a 1-dimensional family of Klingen-ordinary representations, with r varying but r ﬁxed. 1 2 5.3 Self-dual triple products If we are given three mod p modular representations ρ ,ρ ,ρ with ρ nearly-ordinary and 1 2 3 1 det(ρ ) · det(ρ ) · det(ρ ) =¯ χ , then the space 1 2 3 cyc no (ρ ,ρ ,ρ ) ∈ X (ρ¯ ) × X(ρ¯ ) × X(ρ¯ ) : det(ρ ) · det(ρ ) · det(ρ ) = χ 1 2 3 1 2 3 1 2 3 cyc carries a natural 8-dimensional 0-Panchishkin family V, given by the tensor product of the three universal deformations V , with the Panchishkin submodule given by V ⊗ V ⊗ V . i 2 3 The base space is apriori 7-dimensional, but it has two “redundant” dimensions (since we can twist either ρ or ρ by a character, and ρ by the inverse of that character, without 2 3 1 changing the tensor product representation), so we obtain an 0-Panchishkin family over a ∗ + 5-dimensional base X, satisfying the self-duality condition V = V (1).The set (V, V ) corresponds to triples of classical modular forms ( f , f , f ) which are “ f -dominant” – 1 2 3 1 i.e. their weights (k , k , k ) satisfy k ≥ k + k . 1 2 3 1 2 3 123 D. Loefﬂer [ p] [ p] Feeding the universal eigenforms G and G into the construction of [14]gives a p- ρ¯ ρ¯ 2 3 adic L-function over this 5-dimensional base space, extending the construction in op.cit. of a p-adic L-function over the 3-dimensional subspace of X where ρ and ρ are nearly-ordinary. 2 3 (Note that this is actually a reﬁnement of Conjecture 2.8, since the resulting p-adic L- function interpolates the square-roots of central L-values.) 5.4 The Bertolini–Darmon–Prasanna case Let ρ¯ be a modular mod p representation of , with universal deformation space X(ρ) ¯ . Q,{ p} We shall suppose that det ρ¯ =¯ χ , and we let X (ρ) ¯ ⊆ X(ρ) ¯ denote the subspace parametris- cyc ing deformations whose determinant is χ ;thisisﬂat over O of relative dimension 2, and cyc is formally smooth if ρ¯ is unobstructed. Meanwhile, we choose an imaginary quadratic ﬁeld K in which p = p p is split, and we 1 2 ac let X = Spf O[[ X ]] be the character space of the anticyclotomic Z -extension of K.Let X ac 0 denote the product X × X (ρ) ¯ .Thisis O-ﬂat of relative dimension 3, and it carries a family of 4-dimensional Galois representations V, given by tensoring the universal deformation ac univ ρ of ρ¯ with the induction to of the universal character over X .Notethat V satisﬁes the “self-duality” condition V (1) V. Locally at p, V is the direct sum of two twists of the universal deformation of ρ¯, corresponding to the two primes above p; and we can deﬁne a 0-Panchishkin submodule V by taking the direct summand corresponding to one of these primes. Note that (V, V ) consists of pairs (ψ, f ) where f is a modular form and ψ an anticyclotomic algebraic Hecke character of weight (n, −n),where n is large compared to the weight of f . [ p] Plugging in the universal family G (more precisely, its pullback to X (ρ) ¯ ) into the ρ¯ ac constructions of [4], we obtain a p-adic analytic function on the 3-dimensional space X × 0 + X (ρ) ¯ interpolating the square-roots of central L-values at specialisations in (V, V ). This reﬁnes the construction due to Castella [9, §2] of a BDP-type L-function over the ac ord 3 2-dimensional space X × X (ρ) ¯ when ρ¯ is ordinary. 5.5 A finite-slope analogue? One can easily formulate a “ﬁnite-slope” analogue of Conjecture 2.8, where the submodule V ⊆ V is replaced by a submodule of the Robba-ring (ϕ, )-module of V| . The analogue ord of Hida’s ordinary deformation space X (ρ) ¯ is now the ρ¯-isotypic component E (ρ) ¯ of the Coleman–Mazur Eigencurve [13]. However, proving a ﬁnite-slope version of the results of Sect. 4, or of the generalisations sketched in the above paragraphs, appears to be much more difﬁcult than the ordinary case. [ p] All of the above constructions rely on the existence of the universal eigenform G as a ρ¯ family of p-adic modular forms over X(ρ) ¯ . However, in the ﬁnite-slope case, we need to pay attention to overconvergence conditions, since the ﬁnite-slope analogue of the projectors λ [ p] are only deﬁned on overconvergent spaces. Clearly G is not overconvergent (as a family), ρ¯ since it has specialisations which are nearly-classical rather than classical. So we need to work in an appropriate theory of nearly-overconvergent families. Such a theory has recently been introduced by Andreatta and Iovita [1]. We might make the following optimistic conjecture: 3 ord 0 This is slightly imprecise since X (ρ) ¯ is not contained in X (ρ) ¯ ; more precisely, the correspondence ord no 0 between the two constructions is given by identifying X (ρ) ¯ with X (ρ) ¯ ∩ X (ρ) ¯ , via twisting by a suitable ab character of = Z . Q,{ p} 123 P-adic L-functions in universal... Conjecture 5.1 Let f be a nearly-classical point of X(ρ) ¯ , corresponding to a modular form an f of prime-to- p level. Then there is an afﬁnoid neighbourhood X = Max A of f in X(ρ) ¯ f f [ p] over which the universal eigenform G is a family of nearly-overconvergent forms in the ρ¯ sense of [1]. If this conjecture holds, one might realistically hope to deﬁne (for instance) a p-adic Rankin–Selberg L-function over neighbourhoods of crystalline classical points in E (ρ¯ ) × an X(ρ¯ ) . 6 Conjectures on P-nearly-ordinary families In this section, we’ll use Galois deformation theory to deﬁne universal parameter spaces for Galois representations valued in reductive groups, which satisfy a Panchishkin-type condition relative to a parabolic subgroup; and we formulate a “parabolic R = T ” conjecture, pre- dicting that these should have an alternative, purely automorphic description. We expect that these parameter spaces should be the natural base spaces for families of p-adic L-functions, and of Euler systems. 6.1 Nearly-ordinary Galois deformations Let G be a reductive group scheme over O and P a parabolic subgroup. In [6, §7], Böckle deﬁnes a homomorphism ρ : → G( A),for A ∈ CNL ,tobe P-nearly ordinary if Q,S ρ| lands in a conjugate of P( A). Theorem 7.6 of op.cit. shows that under some mild hypotheses, the functor of P-nearly-ordinary deformations of a given P-nearly-ordinary residual representation is representable. Remark 6.1 This extends the deﬁnition of near-ordinarity described in Remark 2.4 above, which corresponds to taking G = GL and P the Borel subgroup. On the other hand, if (V, V ) is an r-Panchishkin family in the sense of Deﬁnition 2.3,thenitis P-nearly-ordinary where P is the corresponds to taking G = GL and P to be the parabolic subgroup of block- c=1 c=−1 upper-triangular matrices with blocks of sizes dim V −r and dim V +r. So the notion of P-near-ordinarity gives a framework covering both of these classes of representations. The reason why we consider general reductive groups G, rather than just GL ,isthat the geometry of deformation spaces for GL -valued global Galois representations is rather mysterious when n > 2; in particular, it is not expected that these spaces will always have a Zariski-dense set of specialisations which are de Rham. However, the geometry of defor- mation spaces is much simpler and better-understood for Galois representations arising from Shimura varieties (or, more generally, from automorphic representations that are discrete- series at ∞). 6.2 Nearly-ordinary automorphic representations We now introduce the corresponding condition on the automorphic side. We let G be a reductive group over Q; for simplicity, we assume here G is split. We also suppose G has a “twisting element” in the sense of [8], and ﬁx a choice of such an element . Then Conjecture 4 ∨ Alternatively, one could replace G by the identity component of the “C-group” of op.cit., which the quotient of G × G by a central element of order 2. We can also allow non-split G, by considering representations into a larger, non-connected quotient of the C-group. 123 D. Loefﬂer 5.3.4 of op.cit. predicts that cohomological cuspidal automorphic representations of G ∨ ∨ give rise to Galois representations ρ : → G (Q ),where G is the Langlands dual , p Q p of G. Let P be a parabolic in G, with Levi decomposition P = MN. We say that a cohomo- logical, cuspidal automorphic representation is P-nearly-ordinary if the Hecke operators at p associated to cocharacters valued in the centre Z ( M ) of M act on with unit eigen- values. For example, if P is the parabolic in GSp consisting of matrices whose lower-left 2n n × n block is zero, then the condition of P-near-ordinarity is that the double coset of diag( p,..., p, 1,..., 1) should act as a unit. See also [22] for a discussion of near-ordinarity for general parabolics in the case G = SL . The link between the “Galois” and “automorphic” notions of near-ordinarity is the fol- lowing. Since the root datum of G is the dual of that of G, there is a canonical bijection ∨ ∨ P ↔ P between conjugacy classes of parabolics in P and parabolics in G ; and one expects that if is nearly-ordinary for P,then ρ should be a P -nearly-ordinary rep- , p resentation. (This is known in many cases; see e.g. [36] for the group GSp .) In particular, families of P-nearly-ordinary cohomological automorphic representations of G should give ∨ ∨ rise to families of P -nearly-ordinary Galois representations into G . If we also choose a linear representation ξ : G → GL , then for suitably chosen P and r, the resulting families of n-dimensional Galois representations will be r-Panchishkin families. The example of §4 is of this type, taking G = GL × GL ,and P = B × GL 2 2 2 2 where B is the Borel subgroup of GL .Herewetake ξ to be the 4-dimensional tensor 2 2 product representation of G G, which maps B × GL to the parabolic in GL with 2 2 4 block sizes (2, 2). Similarly, the self-dual triple-product setting of Sect. 5.3 corresponds to taking G to be the group (GL × GL × GL )/ GL ,and P the image of B × GL × GL .Then G is a 2 2 2 1 2 2 2 subgroup of GL × GL × GL , and the 8-dimensional tensor-product representation of G 2 2 2 sends the dual of P to the parabolic in GL with blocks (4, 4). 6.3 Big and small Galois eigenvarieties In the setting of the previous paragraph, we deﬁne the big P-nearly-ordinary Galois eigenva- ∨ ∨ riety for G to be the following space. Suppose G and P have smooth models over O,and ∨ ∨ ﬁx some choice of ρ¯ : → G (F) which is P -nearly-ordinary. Then – assuming the Q,S hypotheses of Böckle’s construction are satisﬁed – we obtain a universal deformation ring P −no ∨ R (ρ) ¯ for for P -nearly-ordinary liftings of ρ¯.Wedeﬁne thebig P-nearly-ordinary P −no Galois eigenvariety X (ρ) ¯ to be the formal spectrum of this ring R (ρ) ¯ . The methods of [6] give a formula for the dimension of this space. Suppose ρ¯ satisﬁes the Ad ρ( ¯ c)=1 “oddness” condition that dim g = dim(G/ B ),where g is the Lie algebra of G/F, G F c is complex conjugation and B is a Borel subgroup of G. (This condition is expected to hold P −no for representations arising from Shimura varieties; see [11, Introduction].) Then R (ρ) ¯ has a presentation as a quotient of a power series ring in d variables by an ideal with d 1 2 generators, where d − d = dim P − dim(G/ B ) = dim B , 1 2 G M where M is the Levi factor of P and B ⊆ M is a Borel subgroup of M. It seems reasonable to conjecture that X (ρ) ¯ is in fact ﬂat over O, and its relative dimension is dim B . P M The term big is intended to contrast with the following alternative construction (which is perhaps less immediately natural; we introduce it because it is the Galois counterpart 123 P-adic L-functions in universal... of an existing construction on the automorphic side, as we shall recall below). Let M = ∨ ∨ ∨ ∨ ∨ der M / Z ( M ),where Z ( M ) is the centre of M ; hence M is the Langlands dual of M . We ﬁx a Hodge type v and an inertial type τ for M -valued representations of ,inthe sense of [3]. Then we say a lifting ρ of ρ¯ to Q is P -nearly-ordinary of type (τ, v) if it ∨ ∨ is P -nearly-ordinary, and the composition − → P (Q ) → M (Q ) has the given p p p Hodge and inertial types. We deﬁne the small P-nearly-ordinary Galois eigenvariety to be the universal deformation space X (ρ¯; τ, v) for deformations that are P -nearly-ordinary of the speciﬁed type. Using the formulae of [3] applied to M to compute the dimension of the local lifting rings, and assuming that ρ¯ is odd and v is sufﬁciently regular, we compute that the expected dimension of X (ρ¯; τ, v) is now given by dim Z ( M ) = dim Z ( M ). Remark 6.2 Note that the big and small Galois eigenvarieties coincide if P is a Borel sub- group; but the dimension of the big eigenvariety grows with P, while the dimension of the small eigenvariety shrinks as P grows. For instance, if G = GL and P = G,then X (ρ) ¯ is 2 P just the unrestricted deformation space, which is 3-dimensional over O as we have seen; but X (ρ¯; τ, v) has dimension 1, since for any (τ, v) there are only ﬁnitely many deformations of that type, so X (ρ¯; τ, v) has only ﬁnitely many points up to twisting by characters. 6.4 Big and small automorphic eigenvarieties We can now ask if the above Galois-theoretic spaces have automorphic counterparts. 6.4.1 The big eigenvariety Seeking an automorphic counterpart of the big Galois eigenvariety leads to the following question: Question:If G is reductive over Q,and P is a parabolic in G/Q as above, is there a natural purely automorphic construction of a parameter space E for systems of Hecke eigenvalues arising from cohomological automorphic representations for G that are nearly ordinary for the parabolic P? We call this conjectural object E the big P-nearly-ordinary automorphic Eigenvariety. We expect its dimension to be the same as its Galois analogue; in particular, if G has discrete series its dimension should be dim B ,where B is a Borel subgroup of the Levi of P as M M before. The case when P = B is a Borel subgroup is relatively well-understood; this is the setting of Hida theory. However, the case of non-Borel parabolics is much more mysterious. In this case, one can give a candidate for this space E as follows. For any open compact K ⊂ G(A ), we can form the H (K , O) of Betti cohomology of the symmetric space for G of level K , which is a ﬁnitely-generated graded O-module. This has an action of Hecke operators, and the subalgebra of its endomorphisms generated by Hecke operators at primes where K is unramiﬁed, the spherical Hecke algebra of level K,is commutative. We ﬁx an open compact subgroup K ⊂ G(A ),and let K ={g ∈ G(Z ) : g mod n, p p n n ∗ p ∈ N (Z/ p )},where N is the unipotent radical of P. Then, for any n ≥ 1, H (K , O) P P has a canonical idempotent endomorphism e (the Hida ordinary projector associated to r ! P), deﬁned by lim U where U is a suitable Hecke operator; this commutes with the r →∞ P spherical Hecke algebra. 123 D. Loefﬂer P−no p Deﬁnition 6.3 With the above notations, let T (K ) be the quotient of the spher- ∗ p P−no p ical Hecke algebra acting faithfully on e H (K K , O); and deﬁne T (K ) = P p,n P−no p lim T (K ). ← − P−no p We conjecture that the formal spectrum of T (K ) should be the big P-nearly- ordinary eigenvariety. However, from this deﬁnition alone it is rather difﬁcult to obtain much information about the properties of the resulting space (for instance, it is not clear whether P−no p T (K ) is Noetherian). As far as the author is aware, the only non-Borel cases where this construction is well-understood are the following: • G = GL and P = G, as in Theorem 3.4. • G = Res + (U ),where U is a totally deﬁnite unitary group for some CM extension F /Q + + F / F , with p split in F and F / F unramiﬁed at all ﬁnite places; and P is a parabolic [ F :Q ] ∼ p subgroup of G(Q ) GL (Q ) whose Levi subgroup is a product of copies of p n p GL and GL . This case has been studied extensively by Yiwen Ding [16]. 1 2 P−no p In the deﬁnite unitary case, Ding proves that the localisation of T (K ) at the maximal ideal corresponding to an irreducible ρ¯ is a quotient of the global Galois deformation ring P −no R (ρ) ¯ , and is therefore Noetherian; and he gives a lower bound for the relative dimension P−no p of T (K ) over O (localised at the maximal ideal corresponding to some ρ¯). This lower bound is exactly dim B , the dimension conjectured for the Galois eigenvariety above. Remark 6.4 Note that Ding’s construction uses the p-adic local Langlands correspondence for GL (Q ) in an essential way, so this approach will be much harder to generalise to cases 2 p where the Levi of P is not a product of tori and copies of GL (Q ). 2 p 6.4.2 The small eigenvariety In contrast to the rather disappointing situation described above, there does seem to be a well-established theory for the “little brother” of this space—the small P-nearly-ordinary automorphic eigenvariety. This would be a parameter space for P-nearly-ordinary cohomo- logical automorphic representations satisfying two additional conditions: • the highest weight λ of the algebraic representation of G to whose cohomology con- der tributes should lie in a ﬁxed equivalence class modulo characters of M / M ; no • the ordinary part J ( ) of J ( ), which is an irreducible smooth representation of P p P p no M (Q ), should satisfy e · J ( ) = 0where e is some ﬁxed idempotent in the Hecke p P p der algebra of M (Q ). Note that both conditions are vacuous if P is a Borel. These conditions are the automorphic counterparts of the ﬁxed Hodge and inertial types up to twisting used to deﬁne the small P- nearly-ordinary Galois eigenvariety. See e.g. Mauger [30] for the construction of the small P-nearly-ordinary automorphic eigenvariety, and [23]for a“ P-ﬁnite-slope” analogue. Remark 6.5 The most obvious choice of e would be the idempotent projecting to the invariants der for some choice of open compact subgroup of M (Q ). For instance, Mauger’s theory no der applies to such that J ( ) has non-zero invariants under M (Z ), although it can P p p be extended without difﬁculty to allow other more general idempotents. However, a craftier choice would be to take e to be a special idempotent in the sense of [7], corresponding to der a choice of Bernstein component for M (Q ); these Bernstein components are expected to biject with inertial types on the Galois side (the inertial local Langlands correspondence 123 P-adic L-functions in universal... der for M (Q )), while the highest weights λ biject with Hodge types, so we obtain a natural dictionary between the deﬁning data at p for the Galois and automorphic versions of the small P-nearly-ordinary eigenvariety. 6.4.3 R = T theorems Both big and small automorphic eigenvarieties should, clearly, decompose into disjoint unions of pieces indexed by mod p Hecke eigenvalue systems. We can then formulate the (extremely speculative) “parabolic R = T ” conjecture that each of these pieces should correspond to one of the big or small Galois eigenvarieties of the previous section, for a mod p Galois representation ρ¯ determined by the mod p Hecke eigensystem. In the case when G is a deﬁnite unitary group, results of this kind have been proven by Geraghty [20]when P is a Borel subgroup; and when the Levi of P is a product of GL ’s P −no and GL ’s,Dingprovesin[16] the slightly weaker result that the map from R (ρ) ¯ to P−no p the ρ¯-localisation of T (K ) is surjective with nilpotent kernel, after possibly extending + red red the totally real ﬁeld F (an “ R = T ” theorem). 6.5 Miscellaneous remarks Remark 6.6 The 4-dimensional parameter space for GSp × GL mentioned at the end of §5.2 is a slightly artiﬁcial hybrid: the it is the product of the big automorphic (or Galois) eigenvariety for P = G = GL with the small automorphic eigenvariety for the Klingen parabolic of GSp . Of course, we expect that the “correct” parameter space for this construc- tion is the product of the big eigenvarieties for the two groups, which would have dimension 7 (or 6 if we factor out a redundant twist, which corresponds to working with the group GSp × GL ). However, we do not know how to construct p-adic L-functions on this GL 2 4 1 eigenvariety at present. Remark 6.7 The small P-nearly-ordinary eigenvariety is ﬁnite over the “weight space” der parametrising characters of ( M / M )(Z ). Moreover, in Shimura-variety settings it is ﬂat over this space (up to a minor grain of salt if the centre Z (G) has inﬁnite arithmetic sub- groups). It is natural to ask if there is an analogous, purely locally deﬁned “big P-weight space” over which the big eigenvariety E is ﬁnite; the results of [16] suggest that a candidate could be a universal deformation space for p-adic Banach representations of M (Q ) on the automorphic side, or M -valued representations of on the Galois side. However, these spaces will in general have much larger dimension than the eigenvariety, so there does not seem to be a natural choice of local parameter space over which E is ﬁnite and ﬂat. Acknowledgements It is a pleasure to dedicate this article to Bernadette Perrin-Riou, in honour of her immense and varied contributions to number theory in general, and to p-adic L-functions in particular, which have been an inspiration to me throughout my career. I would also like to thank Daniel Barrera Salazar, Yiwen Ding, and Chris Williams for informative discussions in connection with this paper; Sarah Zerbes for her feedback on an earlier draft (and her assistance translating the abstract into French); and the anonymous referee for numerous valuable comments and corrections. 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. 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P-adic L-functions in universal deformation families

P-adic L-functions in universal deformation families

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P-adic L-functions in universal deformation families

P-adic L-functions in universal deformation families

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