On exceptional zeros of Garrett–Hida p-adic L-functions

This article proves a case of the p-adic Birch and Swinnerton-Dyer conjecture for Garrett p-adic L-functions of [6], in the exceptional zero setting of extended analytic rank 2.


Introduction
Let A be an elliptic curve defined over Q, having ordinary reduction at a rational prime p > 3. Let 1 and 2 be odd, irreducible, two-dimensional Artin representations of the absolute Galois group of Q, which are unramified at p and satisfy the self-duality condition det( 1 ) = det( 2 ) −1 .
By modularity, the triple (A, 1 , 2 ) arises from a triple ( f , g, h) of cuspidal p-ordinary newforms of weights w o = (2, 1, 1). Let f α be the ordinary p-stabilisation of f , and fix p-stabilisations g α and h α of g and h respectively. Set = 1 ⊗ 2 . In the recent paper [6] we proposed a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for the leading term at w o of the 3-variable Garrett-Hida p-adic L-function L αα p (A, ) = L p ( f , g α , h α ) associated with the triple ( f , g α , h α ) of Hida families specialising to ( f α , g α , h α ) at w o . In this article we verify our conjecture in the analytic rank-zero exceptional cases, viz. when the complex Garrett L-function L(A, , s) = L( f ⊗ g ⊗ h, s) does not vanish at s = 1 and L αα p (A, ) has an exceptional zero at w o in the sense of Mazur-Tate-Teitelbaum (cf. Theorem 2.1 and Sect. 2.1 below). Moreover, when L(A, , 1) = 0 and L αα p (A, ) has an exceptional zero, we propose a conjecture relating the value at w o of the fourth partial derivative of L αα p (A, ) along the f -direction to the p-adic logarithms of two global points on A rational over the number field cut out by (cf. Conjecture 2.3).

Setting and notations
Fix algebraic closuresQ andQ p of Q and Q p respectively, and field embeddings i p :Q −→Q p and i ∞ :Q −→ C. With the notations of the Introduction, let ξ = n≥1 a n (ξ ) · q n ∈ S u (N ξ , χ ξ )Q denote one of the cuspidal newforms f , g and h. Here u and N ξ are the weight and the conductor of ξ respectively, and S u (N ξ , χ ξ ) F is the space of cuspidal modular forms of level 1 (N ξ ), weight u, character χ ξ and Fourier coefficients in the subfield F ofQ p . Fix a number field Q( ) containing for any ξ the Fourier coefficients a n (ξ ), as well as the roots α ξ and β ξ of the pth Hecke polynomials P ξ, p = X 2 − a p (ξ ) · X + χ ξ ( p) · p. Let V i be a two-dimensional Q( )-vector space affording the representation i , and let K be a Galois number field such that i factors through Gal(K /Q). Set where V p (A) = H 1 ét (AQ, Q p (1)) is the p-adic Tate module of A with Q p -coefficients. Throughout this note we make the following Assumption 1.1 1. (Self-duality) The characters χ g and χ h are inverse to each other. 2. (Local signs) The conductors N g and N h are coprime to p · N f . 3. (Étaleness) The forms g and h are cuspidal, p-regular and do not have RM by a real quadratic field in which p splits.
The first condition is a reformulation of the self-duality condition mentioned in the Introduction, namely det( 1 ) = det( 2 ) −1 . Recall that the form ξ is p-regular if P ξ, p has distinct roots. Moreover, one says that a weight-one eigenform has RM (real multiplication) if it is the theta series associated with a ray class character of a real quadratic field. Assumption 1.1.3 is equivalent to require that V i is irreducible, not isomorphic to Ind Q K χ for a finite order character χ : G K −→ Q( ) * of a real quadratic field K in which p splits, and that an arithmetic Frobenius at p acts on V i with distinct eigenvalues. For ξ = g, h, this assumption guarantees that the p-adic Coleman-Mazur-Buzzard eigencurve of tame level N ξ is étale over the weight space at the points corresponding to the p-stabilisations of ξ (cf. [2]). It is used in [6] to construct the Garrett-Nekovář height ⟪·, ·⟫ f g α h α which appears in the main result of this note. To explain the relevance of Assumptions 1.1.1 and 1.1.2, let α f be the unit root of P f , p and fix roots α g and α h of P g, p and P h, p respectively. Fix a finite extension L of Q p containing Q( ) and the roots of unity of order lcm(N f , N g , N h ). Let ξ be one of f , g and h, and let u o be the weight of ξ . According to the results of [2,10,18], there exists a unique Hida family which specialises at u o to the p-stabilised newform Here , and O ξ is the ring of bounded analytic functions on a (sufficiently small) connected open disc U ξ in the p-adic weight space over L. For each classical weight u in U ξ ∩ Z ≥3 , the weight-u specialisation ξ α,u = n≥1 a n (ξ α )(u) · q n ∈ L[[q]] of ξ α is the q-expansion of the ordinary p-stabilisation of a newform ξ u in S u (M ξ , χ ξ ) L . Since f has a unique p-ordinary p-stabilisation f α , we simply write f for f α . Assumption 1.1.1 guarantees that for each classical triple w = (k, l, m) in the set the complex Garrett L-function L( f k ⊗ g l ⊗ h m , s) admits an analytic continuation to all of C and satisfies a functional equation relating its values at s and k + l + m − 2 − s, with root number ε(w) = ≤∞ ε (w) equal to +1 or to −1. Assumption 1.1.2 implies that all the local signs ε (w) are equal to +1 for every w in the f -unbalanced region f = {w = (k, l, m) ∈ : k ≥ l + m} (cf. [11]). Under these assumptions, [12] associates with ( f , g α , h α ) an analytic function satisfies the following interpolation property. For each w = (k, l, m) in f , the value of L αα p (A, ) at w is an explicit non-zero complex multiple of (1) Here c w = k+l+m−2 2 , and for ξ = f , g α , h α one denotes by α u the unit root of P ξ u , p and sets β u · α u = χ ξ ( p) · p u−1 , where χ ξ is the prime-top part of χ ξ (so that χ ξ = χ ξ for ξ = g, h, and χ f is the trivial character modulo M f ). We refer to Theorem A of loc. cit. for the precise interpolation formula. We call L αα p (A, ) = L p ( f , g α , h α ) the Garrett-Hida p-adic L-function associated with (A, ) (or with ( f , g α , h α )).

Exceptional zero formulae
The p-adic variant of the Birch and Swinnerton-Dyer conjecture formulated in [6] predicts that the leading term of L αα p (A, ) at w o = (2, 1, 1) is encoded by the discriminant of the Garrett-Nekovář height pairing constructed in Section 2 of loco citato, where I is the ideal of functions in O f gh which vanish at w o and the p-extended Mordell-Weil group A † (K ) is defined as follows. When A has good reduction at p, one sets is a 1-dimensional Q p -vector space on which an arithmetic Frobenius acts as multiplication by α f . Let q A in pZ p be the p-adic Tate period of the base change A Q p of A to Q p (cf. Chapter V of [15]), and let Q p 2 be the quadratic unramified extension of Q p . The Tate uniformisation yields a rigid analytic morphism with kernel q Z A and unique up to sign. Set is any compatible system of p n -th roots of q A , and define to be the direct sum of A(K ) and the Q( )-submodule The Garrett-Nekovář height ⟪·, ·⟫ f g α h α depends on the choice of suitably normalised G Q -equivariant embeddings where is the weight-one specialisation of the big Galois representation V (ξ α ) associated with ξ α . (We refer to Sect. 3.1 below for precise definitions.) More precisely, denote by V ( f ) the f α -isotypic component of the cohomology Section 2 of [6] constructs a canonical Garrett-Nekovář p-adic height pairing on the naive extended Selmer group of V ( f , g, h) over Q, defined as the direct sum of the Bloch-Kato Selmer group Sel(Q, , is independent of the choice of ℘ ∞ , γ g and γ h . We refer to [6] for more details. If ξ denotes either g or h, then the restriction to G Q p of the Artin representation V (ξ ) is the direct sum of the submodules V (ξ ) α and V (ξ ) β on which an arithmetic Frobenius acts as multiplication by α ξ and β ξ respectively (cf. Assumption 1.1.3). The G Q p -representation V ( f , g, h) − then decomposes as the direct sum of the subspaces is a pair of elements of {α, β}. If ξ denotes either g or h, Sect. 3.1.1 below recalls the definition of canonical weight-one differentials where Q nr p is the maximal unramified extension of Q p . If A is multiplicative at p, set in which case it has dimension 2 and one says that (A, ) is exceptional at p. More precisely, note that α g = β g by Assumptions 1.1.3, hence only one of the previous identities can be satisfied. Moreover Equation (1) shows that the value of the square-root Garrett- . The previous discussion then shows that (A, ) is exceptional at p precisely if one of the Euler factors which appear in the previous expression is zero, id est if L αα p (A, ) (or L αα p (A, )) has an exceptional zero in the sense of Mazur-Tate-Teitelbaum [13]. In this case Lemma 9.8 of [7] proves that the restriction L αα to the improving line L defined by the equations m = 1 and k = l + 1 admits the factorisation in the ring O(L) of analytic functions on L, where Moreover, the value at w o of the improved p-adic L-function L αα p (A, ) is an explicit algebraic number in Q( ), equal to zero precisely if L(A, , s) vanishes at s = 1. We refer to the proof of Proposition 8.3 of [12] for details.
The following is the main result of this note.
Then the following equality holds in I /I 2 up to sign.
Theorem 2.1 is proved in Sect. 4 below. More precisely, Sects. 3.3 and 3.4 below prove that the following equality holds in I /I 2 up to sign: is the value at the centre u o of U ξ of the logarithmic derivative of the p-th Fourier coefficient of the Hida family ξ = f , g α , h α . In Sect. 4 we then deduce Theorem 2.1 from Eq. (6) and the study carried out in [7, Section 9] of the linear term of L αα p (A, ) at w o in the exceptional case.
It should be possible to extend Theorem 2.1 (and Conjecture 2.3 below) to the case of p-new eigenforms of even weight k ≥ 2 and trivial character (cf. Section 1.1 of [6]). We have not checked the details.

The rank-zero exceptional case of [6, Conjecture 1.1]
Assume in this section that (A, ) is exceptional at p, and that the Garrett complex L-function According to the main result of [8] (see also Theorem B of [3]), one has (3)). The normalisation imposed on the embeddings γ g and γ h (and described in Sect. 3.1.1 below) implies that the matrix M in GL 2 (L) defined by the identity (q q ) · M = (γ gh (q 1 ) γ gh (q 2 )) has determinant in Q( ) * . In light of the above discussion, Theorem 2.1 then proves the following corollary, which together with Eq. (6)

Exceptional zeros and rational points (cf. [14])
Assume in this section that (A, ) is exceptional at p, and that the Garrett complex L-function L(A, , s) vanishes at the central critical point s = 1: The p-adic L-function L αα p (A, ) belongs to I 2 (cf. Theorem 2.1) and Conjecture 2.3 of [6] predicts that its image in ( for two rational points P and Q in A(K ) . (Recall that the p-adic height ⟪·, ·⟫ f g α h α is skew-symmetric, hence the previous expression is a square root of its discriminant on the Q( )-submodule of A † (K ) generated by q , q , P and Q.) One has ⟪q , q ⟫ f g α h α (k, 1, 1) = 0 by Eq. (6). Moreover, Sect. 3.5 below proves that Eq. (11)). We are then led to the following holds in L up to multiplication by a non-zero scalar in Q( ) * .
As explained in [5], the main result of [1] can be used to prove cases of Conjecture 2.3 when g and h are theta series associated with certain ray class characters of the same imaginary quadratic field in which p is inert (and P and Q are Heegner points). By combining this with an extension of the height computations carried out in [16,17], the article [4] proves instances of Conjecture 1.1 of [6] in this setting.

Remark 2.4
In light of the aforementioned results of [5], Rivero when A has good reduction at p. The previous discussion places Rivero's conjecture within a conceptual framework and sheds some light on this question.

Height computations
Throughout the rest of this note we assume that (A, ) is exceptional at p. In particular A has multiplicative reduction at p, id est p divides exactly N f .

Setting and notations
This subsection briefly recalls the needed definitions and notations from our previous articles [6,7].

Galois representations
Set N = lcm (N f , N g , N h ) and let G Q,N be the Galois group of the maximal extension of Q contained inQ and unramified outside N ∞. If ξ denotes one of f , g α and h α , let V (ξ ) be the big Galois representation associated with ξ (cf. Section 5 of [7]). It is a free O ξ -module of rank two, equipped with a continuous linear action G Q,N . For each u in U ξ ∩ Z ≥2 the base change V (ξ ) ⊗ u L of V (ξ ) along evaluation at u on O ξ is canonically isomorphic to the homological p-adic Deligne representation of ξ u with coefficients in L (cf. loco citato for more details). In particular if ξ = f and u = 2 there is a natural specialisation isomorphism Sect. 1). It is a two-dimensional L-vector space affording the dual of the p-adic Deligne-Serre representation of ξ = g, h with coefficients in L. In order to have a uniform notation, in this case one defines ρ 1 : The restriction of V (ξ ) to G Q p (via the embedding i p fixed at the outset) fits into a short More precisely, let χ cyc : G Q −→ Z * p be the p-adic cyclotomic character, and letǎ p (ξ ) : G Q p −→ O * ξ be the unramified character sending an arithmetic Frobenius to the p-th Fourier coefficients a p (ξ ) of ξ . Then where χ u−1 cyc : (The freeness of V (ξ ) ± is guaranteed by Assumption 1.1.3, cf. Section 5 of [7].) If ξ = f and u = 2 the specialisation isomorphism ρ 2 β is the submodule of V (ξ ) on which an arithmetic Frobenius Frob p acts as multiplication by γ ξ = α ξ , β ξ (cf. Assumption 1.
Denote by f gh = χ If · denotes one of the symbols ∅, + and −, define is a free O f gh -module of rank 8, resp. 4, equipped with a continuous action of G Q,N , resp. G Q p . As χ g ·χ h = 1 (cf. Assumption 1.1), the product of the perfect dualities π ξ , for ξ = f , g α , h α , yields a perfect skew-symmetric Kummer duality π : (1), inducing a perfect local Kummer duality π : (1). After setting (where · ⊗ w o L denotes the base change along evaluation at w o on O f gh ). Let be the specialisation of π via ρ w o , and define π : V ± ⊗ L V ∓ −→ L(1) similarly.

Weight one differentials
nr p is the p-adic completion of the maximal unramified extension of Q p (and as usual ξ denotes one of f , g α and h α ). For each u in U ξ ∩ Z ≥2 there is a natural comparison isomorphism between D(ξ ) − ⊗ u L and the ξ u -isotypic component of the space of cuspidal modular forms of weight u, level 1 (N ξ p) and Fourier coefficients in L. Assumption 1.1.3 guarantees that D(ξ ) − is free (of rank one) over O ξ , and admits a basis ω ξ whose image in D(ξ ) − ⊗ u L corresponds to ξ u under the aforementioned comparison isomorphism, for each u in U ξ ∩ Z ≥2 . (We refer to Section 3.1 of [6] and the references therein for more details.) For ξ = g α , h α , the holomorphic weight-one differential Let c be the common conductor of χ g and χ h , and identify (L(χ ξ ) ⊗ Q p Q nr p ) G Q p with L via the Gauß sum G(χ ξ ) = (−c) i ξ a∈(Z/cZ) * χ ξ (a) −1 ⊗ e 2πia/c , where i g = 0 and i h = 1 (so that G(χ g ) · G(χ h ) = 1 by Assumption 1.1.1). The pairing π ξ then induces a perfect duality ·, · ξ : One defines the antiholomorphic weight-one differential (cf. Eq. (5)) to be the dual of ω ξ α under ·, · ξ , viz. the element satisfying ω ξ α , η ξ α ξ = 1.
The embeddings g and h With the notations of Sect. 1, set V g = V 1 and V h = V 2 . Let ξ denote either g or h. As recalled above, the Artin representation V (ξ ) = V (ξ ) ⊗ 1 L affords the dual of the p-adic Deligne representation of ξ with coefficients in L, id est is isomorphic to (3)) by requiring that the composition π ξ • (γ ξ ⊗ γ ξ ) takes values in the number field Q( ) (via the embedding i p :Q −→Q p fixed at the outset).

Selmer complexes
Let R˜ f (Q, V ) be the Nekovář Selmer complex associated with (V , V + ) (cf. Section 2.2 of [6]). It is an element of the derived category D b ft (L) of cohomologically bounded complexes of L-modules with cohomology of finite type over L, sitting is an exact triangle where R cont (G, ·) is the complex of continuous non-homogeneous cochains of G with values in ·, res p is the restriction map (induced by the embedding i p :Q −→Q p fixed at the outset) and p − is the map induced by the projection V −→ V − . Denote bỹ the cohomology of R˜ (Q, V ), let Sel(Q, V ) be the Bloch-Kato Selmer group of V over Q, and let i + : V + −→ V be the natural inclusion. Then there is a commutative and exact diagram of L-vector spaces (cf. loc. cit.) where the first line arises from the exact triangle (12). In addition there is a unique section ı ur : Sel(Q, V ) −→H 1 f (Q, V ) of the above projection such that ı ur (x) + belongs to the Bloch-Kato finite subspace H 1 fin (Q p , V + ) for each x in Sel(Q, V ). We often use j and ı ur to identify Nekovář's extended Selmer groupH 1 f (Q, V ) with the naive extended Selmer group One similarly associates with (V , V + ) a Selmer complex sitting in an exact triangle analogous to (12). (We refer to loc. cit. for more details.)

Preliminary lemmas
This section gives a concrete description of the functionals ⟪q, ·⟫ f g α h α :

Bockstein maps
Let (C, C) denote one of the pairs where R p (·) and R (·) are shorthands for R cont (Q p , ·) = R cont (G Q p , ·) and R cont (G Q,N , ·) respectively (cf. Sect. 3.1.2). The specialisation maps ρ w o (cf. Eq. (10)) induce isomorphisms Applying C ⊗ L O f gh · to the exact triangle (arising from evaluation on w o ) then yields a derived Bockstein map which in turn induces in cohomology a Bockstein map If no risk of confusion arises, we simply write β for β C/C . Let be the maps arising from the exact triangle (12).

Lemma 3.1 The following diagram commutes.
For M = V , V one has an exact triangle (cf. Equation (12)) . Eq. (14)). It follows from the definition of the derived Bockstein maps β − and β on R cont (Q p , V − ) and R˜ (Q, V ) respectively that j V ⊗ I /I 2 [1] • β − is equal to β • j V . Since by definition the maps j are the ones induced in cohomology by j V , the lemma follows.
The following lemma gives a concrete description of β C/C . Lemma 3.2 Let (C, C) be as above, let z be a 1-cocycle in C, let Z be a 1-cochain in C, and let Z k , Z l and Z m be 2-cochains in C such that Then z · = ρ w o (Z · ) is a 2-cocycle for · = k, l, m, and one has the equality Proof The proof is very similar to that of [16,Lemma 5.5]. We omit it.

Local and global duality
Nekovář's generalised Poitou-Tate duality associates with the perfect duality π f gh introduced in Eq. (11) a global cup-product pairing (cf. Section 2.4 of [6]) The pairing π f gh induces a Kummer duality V − ⊗ L V + −→ L(1) and we denote by the induced local Tate duality pairing. Recall finally the map (13).

Lemma 3.3 For each ζ in H
Proof This is proved as in [16,Lemma 5.7].

The Garrett-Nekovář p-adic height pairing
After identifyingH 1 for each x and y inH 1 f (Q, V ), where we write again ·, · Nek for the I /I 2 -base change of Nekovář's cup-product (15). Lemmas 3.1 and 3.3 give the following

Lemma 3.4 For each q in H
and we write again ·, · Tate for the I /I 2 -base change of the local Tate pairing (16)).

Computation of ⟪qˇˇ, q˛˛⟫ f g˛hĄ
ssume in this subsection α f = α g · α h , so that H 0 (Q p , V − ) is generated over L by the periods Recall that χ cyc : G Q −→ Z * p denotes the p-adic cyclotomic character. Fix a lift q ββ in where d denotes the differentials of the complex R cont (Q p , V − ) and The assumption α f = α g · α h implies that takes value in I , and that its composition with the projection I −→ I /I 2 is of the form with the Q p -vector space Hom(Q * p , Q p ) of continuous morphisms of groups from Q * p to Q p via the local reciprocity map rec p : Q * p −→ G ab Q p , normalised by requiring rec p ( p −1 ) to be an arithmetic Frobenius. By local class field theory, for each p-adic unit u one has where · : Z * p −→ 1 + pZ p denotes the projection to principal units, and Eq. (7)). As a consequence −2 · ϕ k is equal to . Similarly one shows that 2 · ϕ l and 2 · ϕ m are equal to the logarithms log g α = log p −L an g α · ord p and log h α = log p −L an g α · ord p . It then follows from Eq. (17) and Lemma 3.2 that in H 1 (Q p , V − ) ⊗ L I /I 2 , where (with the notations introduced in Sect. 3.2.1) one writes β − f g α h α for the Bockstein map β C/C associated with C = R p (V − ). Note that is an L[G Q p ]-direct summand of V − on which G Q p acts trivially, so that log ξ ⊗q ββ (for ξ = f , g α , h α ) belongs to the direct summand Similarly is a direct summand of H 1 (Q p , V + ). The local Tate pairing ·, · Tate introduced in Sect. 3.2.2 induces a perfect duality (denoted by the same symbol) between via the local Kummer map, local class field theory gives Here for each z inH 1 f (Q, V ), where ξ = f , g α , h α , u o = 2, 1, 1 is the centre of U ξ , and is defined as follows. Let pr αα denote the projection onto the direct summand H 1 (Q p , V ( f ) + αα ) of the local cohomology group H 1 (Q p , V + ), and let q * ββ be the generator of V ( f ) + αα (−1) dual to q ββ under π f gh (−1), namely satisfying Then z + αα is defined (via the natural isomorphism (19)) by the identity We now determine z + αα for z = j (q αα ). By definition j (q αα ) is represented by whereq αα in V is a lift of q αα under the the projection V −→ V − , and where dq αα : G Q p −→ V + is its image under the differential in R cont (Q p , V ). By construction dq αα represents the class q + αα = j (q αα ) + in H 1 (Q p , V + ). Since V (ξ ) is the direct sum of V (ξ ) α and V (ξ ) β for ξ = g, h, we can (and will) chooseq αα of the form arising from the short exact sequence of G Q p -modules where a(ζ p ∞ ) = (ζ p n · q Z A ) n≥1 for each compatible system ζ p ∞ = (ζ p n ) n≥1 of p n -th roots of unity, and b is the Q p -linear extension of the inverse limit of (canonical) maps is the image of a compatible system p ∞ √ q A of p n -th roots of the Tate period q A under the composition of ℘ Tate and the inverse of the isomorphism ℘ ∞ : V ( f ) V p (A) induced by the fixed modular parametrisation ℘ ∞ : X 1 (N f ) −→ A. As a consequence 1 in Q p maps to q A⊗ 1 under the connecting map Q p −→ H 1 (Q p , Q p (1)) = Q * p⊗ Q p associated with the short exact sequence (23), hence is the map induced in cohomology by the composition of ℘ −1 ∞ and ℘ + Tate = ℘ Tate • a. If A denotes either A or E, denote by is its dual basis, and set . (1) and z in Q p (−1), the functoriality of the Poincaré duality under finite morphisms yields then (by the definition of the weight-one differentials η ξ α , cf. Sect. 3.1.1) Together with Eq. (24) this gives id est j (q αα ) + αα = m p deg(℘ ∞ ) · q A⊗ 1.