Heegner Points and Exceptional Zeros of Garrett 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 (Bertolini et al. in On p -adic analogues of the Birch and Swinnerton–Dyer conjecture for Garrett L -functions, 2021), in the imaginary dihedral exceptional zero setting of extended analytic rank 4.

Fix an algebraic closureQ p of Q p , an embedding i p :Q −→Q p , and a finite extension L of Q p containing (the images under i p of) the values of ν ξ and α ξ , for ξ = g, h. Denote by ξ α in S 1 (pN ξ , χ ξ ) the p-stabilisation of ξ with U p -eigenvalue α ξ . According to [1,7], there exist unique Hida families f = n 1 a n (f ) · q n ∈ O f [[q]] and ξ α = n 1 a n (ξ α ) · q n ∈ O ξ α specialising to f = f 2 and ξ α = ξ α,1 in weights two and one respectively. Here O f is the ring of bounded analytic functions on a (small) connected open disc U f centred at 2 in the weight space W = Hom cont (Z * p , C * p ) over Q p . For each k in U f ∩ Z 4 , the weight-k specialisation f k of f is the ordinary p-stabilisation of a p-ordinary newform f k of weight k and level Γ 0 (N f /p). Similarly O ξ α is the ring of bounded analytic functions on a connected open disc U ξ α centred at 1 in W L = W ⊗ Q p L, and ξ α,u is the p-stabilisation of a newform ξ u of weight u and level Γ 1 (N ξ ) for each l in U ξ α ∩ Z 1 , with ξ 1 = ξ. In order to lighten the notation, we write U ξ = U ξ α and O ξ = O ξ α . Set = g ⊗ h and O f gh = O f⊗Q p O g⊗L O h . Under Assumption 1.1, Theorem A of [8] associates with (f , g α , h α ) a Garrett-Hida square root p-adic L-function (denoted L f F in loc. cit., where F = (f , g α , h α )), whose square interpolates the central critical values L(f k ⊗g l ⊗h m , (k+l+m−2)/2) of the Garrett L-functions attached to (f k , g l , h m ) for classical triples (k, l, m) in the f -unbalanced region, viz. triples (k, l, m) in U f × U g × U h ∩ Z 3 1 satisfying k l + m. The first equality in (1) implies that L αα p (A, ) has an exceptional zero in the sense of [9] at the "Birch and Swinnerton-Dyer point" w o = (2, 1, 1) (cf. [5,Section 1.2]).
Fix a number field Q( ) containing the values of ν g and ν h , and for ξ = g, h fix a Q( )[G Q ]-module V ξ , two-dimensional over Q( ), affording the Artin representation where Q p (A, ) is a two-dimensional Q( )-vector space depending only on the base change of A to Q p and on the restriction of V gh to G Q p (cf. Sect. 2.1.3 below). Moreover, Section 2 of [4] constructs a Garrett-Nekovář height-pairing where I is the kernel of evaluation at w o on O f gh . It is a skew-symmetric bilinear form, arising from an application of Nekovář's theory of Selmer complexes to the big self-dual Galois representation associated with (f , g α , h α ). After setting Conjecture 1.1 of [4] predicts that L αα p (A, ) belongs to I r † − I r † +1 , and that its image in ( The following theorem is the main result of this note.  In the present setting, the Garrett L-function L(f ⊗ g ⊗ h, s) factors as the product of the Rankin-Selberg L-functions L(A/K, ϕ, s) and L(A/K, ψ, s), where ϕ = ν g · ν h and ψ = ν g · ν c h , and ν c h is the conjugate of ν h by the nontrivial element of Gal(K/Q). Note that ϕ and ψ are dihedral by Assumption 1.1. (2), and that both L(A/K, ϕ, s) and L(A/K, ψ, s) have sign −1 in their functional equation by Assumption 1.1.(1). In particular the assumptions of the Theorem imply that L(A/K, χ, s) has a simple zero at s = 1 for χ = ϕ and χ = ψ, hence A(K ) is two-dimensional over Q( ) and generated by Heegner points by the Kolyvagin-Gross-Zagier-Zhang theorem. If Under the assumptions of the Theorem, the results of [2,5] imply that L αα p (A, ) belongs to I 4 − I 5 . The actual contribution of this note is the proof of the identity L αα p (A, ) (mod I 5 ) = R αα p (A, ), which grounds on the results of loc. cit. and an extension of the techniques of [10][11][12].
M. Bertolini et al.

Preliminaries 2.1.1. Galois Representations.
To lighten the notation, set (g, h) = (g α , h α ). For ξ = f , g, h let V (ξ) be the big Galois representation attached to ξ (cf. Section 5 of [5]). Under the current assumptions, it is a free O ξ -module of rank two, equipped with a continuous O ξ -linear action of G Q . For each u in U ξ ∩ Z 2 , evaluation at u on U ξ induces a natural specialisation isomorphism where · ⊗ u E denotes the base change along evaluation at u on O ξ , and where V (ξ u ) is the homological p-adic Deligne representation of ξ u with coefficients in E (cf. Section 2.4 of [5]).
When ξ = f and u = 2, the representation which induces an isomorphism of Q p [G Q ]-modules between V (f ) and the p-adic affords the dual of the Deligne-Serre representation of ξ, id est the induced from G K to G Q of the character ν ξ with coefficients in L. (Recall that ξ 1 = ξ α . Here we favour the lighter notation V (ξ) for V (ξ) ⊗ 1 L over the more consistent one V (ξ α ).) There exists a perfect G Q -equivariant and skew-symmetric pairing (With the notations of [5,Section 5], the pairing π ξ is the composition of the twist by [5,Equation (114)].) Up to sign, the pairing π f : V (f ) ⊗ Q p V (f ) −→ Q p (1) arising from the base change of π f along evaluation at k = 2 on O f and the specialisation isomorphism ρ 2 is the one induced on the f -isotypic components by the Poincaré duality on H 1 et (X 1 (N f )Q, Q p (1)). If ξ = g, h, the weight-one specialisation of π ξ yields a perfect skew-symmetric duality Identify G Q p with a subgroup of G Q via the embedding i p :Q −→Q p fixed at the outset, and letǎ p (ξ) : G Q p −→ O * ξ be the unramified character sending an arithmetic Frobenius to the p-th Fourier coefficient a p (ξ) of ξ. In the present setting there is a natural short exact where V (ξ) + and V (ξ) − are free O ξ -modules of rank one and G Q p acts on them via the characters χ ξ ·χ u −1 cyc ·ǎ p (ξ) −1 andǎ p (ξ) respectively (cf. Section 5 of [5]). If ξ = f , the specialisation isomorphism Evaluation at w o = (2, 1, 1) on O f gh induces a specialisation isomorphism The product of the pairing π ξ for ξ = f , g, h yields a perfect, G Q -equivariant and skew-symmetric duality (cf. Assumption 1.1. (2)) (1), whose base change along evaluation at w o on O f gh recasts (via ρ w o ) the perfect duality defined by the product of the perfect pairings π ξ for ξ = f, g, h. For which are maximal isotropic with respect to the skew-symmetric duality π f gh . After setting with i F and i f the natural inclusions and p − the natural projection. Note that p − • i F and i f have the same image, hence the morphism p f is defined by the commutativity of the diagram. One defines the balanced local subspace to be the image of the morphism induced in cohomology by i F . This morphism is injective (cf. Section 7.2 of [5]), hence gives a natural identification In particular the base change of the commutative diagram (2) along evaluation at w o on O f gh is equal to with i F and i f the natural inclusions and p − the natural projection.
The Bloch-Kato finite subspace of H 1 (Q p , V ) is equal to the kernel of the map [5]. (With a slight abuse of notation, we denote by the same symbol a morphism of G Q p -modules and the maps it induces in cohomology.) By construction (cf. Eqs. (2) and (5)), the specialisation (3)) belongs to the kernel of the specialisation map ρ w o : . We have proved the following

p-Adic
Periods. LetQ nr p be the p-adic completion of the maximal unramified extension of Q p , let c = c(χ g ) be the conductor of χ g , and for ξ = g, h define As explained in Section 3.1 of [4], p is free of rank one over O ξ , and its base change u corresponds to ξ u under the aforementioned isomorphism for each u in U ξ ∩Z 2 . (We refer to loc. cit. and the references therein for the details.) The weight-two specialisation of ω f equals the de Rham class For ξ = g, h, the weight-one specialisation of ω ξ yields a class (with ϕ the crystalline Frobenius). The pairing π ξ = π ξ ⊗ 1 L induces a perfect duality ·, · ξ : D cris (V (ξ)) ⊗ L D cris (V (ξ)) −→ D cris (χ ξ ) and one defines η ξ α in D cris (V (ξ) β ) = D cris (V (ξ)) ϕ=β −1 ξ by the identity Along with ω f , it is important to consider another p-adic period arising from the Tate uniformisation of A Q p , cf. Section 2 of [3]. Let K p be the completion of K at p (namely the quadratic unramified extension of Q p ). Tate's theory gives a rigid analytic uniformisation ℘ Tate : G rig m,K p −→ A K p , unique up to sign, with kernel the lattice generated by the Tate period q A in pZ p of A Q p . One sets where p ∞ √ q A is any compatible system of p n th roots of q A , ℘ ∞ : As in loc. cit., define the generators

The Garrett-Nekovář p-Adic
and there exists a unique section ı ur : Sel(Q, V ) −→H 1 f (Q, V ) of π such that the composition ı ur (·) + takes values in the finite subspace H 1 fin (Q p , V + ) of H 1 (Q p , V + ) (cf. Section 2.3 of [4]). As in loc. cit. we use the maps j and ı ur to identify Nekovář's extended Selmer groupH 1 f (Q, V ) with the naive extended Selmer group Enlarging L if necessary, for ξ = g, h fix an isomorphism of L[G Q ]-modules for each x and y in V ξ (cf. Eq. (4) of [4]). Set (cf. Eq. (6)) The modular parametrisation ℘ ∞ : X 1 (N f ) −→ A fixed in Sect. 2.1.1, the global Kummer map on A(K )⊗Q p and the isomorphisms γ g and γ h induce an embedding to be the restriction of the canonical height ⟪·, ·⟫ f gh onH 1 f (Q, V ) along γ gh . Note that the discriminant R αα p (A, ) of ⟪·, ·⟫ f gh on A † (K ) (cf. Sect. 1) is independent of the choice of the modular parametrisation ℘ ∞ and the isomorphisms γ g and γ h .
The Bloch-Kato exponential map exp p gives an isomorphism between the tangent space V dR /Fil 0 of V and the finite (viz. crystalline) subspace H 1 fin (Q p , V ) of H 1 (Q p , V ). Denote by log p the inverse of exp p and define the αα-logarithm to be the composition of log p with evaluation at ω f ⊗ η g α ⊗ η h α in Fil 0 V dR under the perfect duality ·, · fgh . Similarly define the ββ-logarithm (Note that log ii factors through the projection H 1 where K χ is the ring class field of K cut-out by χ and A(K χ ) is viewed as a subgroup of A(K p ) via the embedding i p :Q −→Q p fixed at the outset. (Recall that p is inert in K and that χ is dihedral, hence pO K splits completely in K χ .)

Big Logarithms and Diagonal Classes Let
be the big logarithm map constructed in Proposition 7.3 of [5] using the work of Coleman, Perrin-Riou et alii. (Note that the tame character χ f of f is trivial in the present setting and that the logarithm L f takes values in I by the exceptional zero condition α f = α g · α h .) With a slight abuse of notation denote by Eq. (3)).
Let H 1 bal (Q, V ) be the group of global classes in H 1 (Q, V ) whose restriction at p belongs to the balanced local condition H 1 bal (Q p , V ). According to Theorem A of [5] (cf. [2, Section 2.1]) there exists a canonical big diagonal class κ(f , g, h))) = L αα p (A, ).
Define the diagonal class to be the image in H 1 (Q, V ) of κ (f , g, h) under the map induced in cohomology by the specialisation isomorphism ρ w o : V ⊗ w o L V . Since by assumption the complex Garrett L-function L(A, , s) = L(f ⊗ g ⊗ h, s) vanishes at s = 1, Theorem B of [5] implies that κ(f, g α , h α ) is crystalline at p, hence a Selmer class: Identify  κ(f , g, h) Equation (11) gives The following key lemma, proved in Part 1 of Proposition 9.3 of [5], gives an explicit description of the linear term of L f (Y u ) at w o . Identify the p-adic completion of the Galois group of the maximal abelian extension of Q p with that of Q * p via the local Artin map, normalised in such a way that p −1 corresponds to the arithmetic Frobenius. This identifies H 1 (Q p , Q p ) with Hom cont (Q * p , Q p ), hence (recalling that G Q p acts trivially on V − ββ , cf. Eq. (4)) and the Bloch-Kato dual exponential exp * ββ ) (with ϕ and v as above) and q in Q * p⊗ Q p , set If (ξ, u) denotes one of the pairs (f , k), (g, l) and (h, m), definẽ Heegner Points and Exceptional to be the linear map which on Y in H 1 (Q p , V f ) takes the valuẽ Here . We can finally state the aforementioned key lemma.

Lemma 2.2. For each local class
For each pair (u, v) of distinct elements of {k, l, m}, define (cf. Eq. (13)) Equation (14) and Lemma 2.2 give the following lemma (which implies that the derivativesD · (κ(f , g, h)) are independent of the choice of the classes Y u satisfying (13)).

An Exceptional Zero Formulaà la Rubin-Perrin-Riou
For a positive integer n and each 2n-tuple y = (y 1 , . . . , y 2n ) of elements ofH 1 f (Q, V ) denote by R αα p (y) = Pf ⟪y i , y j ⟫ f gh 1 i,j 2n ∈ I n /I n+1 the Pfaffian of the skew-symmetric 2n × 2n matrix whose ij-entry is ⟪y i , y j ⟫ f gh , and define the extended Garrett-Nekovář p-adic height pairing to be the bilinear form which on y ⊗ y in Sel(Q, V ) ⊗2 takes the valuẽ h αα p (y ⊗ y ) = R αα p (q αα , q ββ , y, y ).
The aim of this section is to prove the following proposition. We divide the proof of Proposition 2.4 in a series of lemmas. Define Proof. See Equations (17) and (27) of [3]. (Note that the p-adic logarithm denoted by log αα in [3] is equal to log p , q ββ fgh = −c p (f ) · log αα .) the local Tate pairing arising from the perfect duality π fgh : V ⊗ L V −→ L (1). (7). Lemma 2.6. There exist 1-cochains X k , X l and X m in C 1 cont (Q p , V − ) such that , g, h))), and Proof. This follows from Equations (30)-(37) in Section 3.4 of [4]. (The paragraphs containing the aforementioned equations do not use the non-exceptionality assumption [4,Equation (26)] imposed in [4,Section 3]

.)
Fix in what follows 1-cochains X k , X l and X m satisfying the conclusions of Lemma 2.6. For i = αα, ββ let pr i : for each σ and σ in G Q . For ξ = g, h, the Q( )[G Q ]-module Ind Q K ν ξ affords the representation ξ . With the notations of Sect. 1 we can then take where ϕ = ν g · ν h and ψ = ν g · ν c h are dihedral characters of K (cf. Sect. 1). The Artin formalism then yields the factorisation where L(A/K, χ, s) = L(f ⊗ ϑ χ , s) is the Hasse-Weil L-function of the base change of A to K twisted by χ = ϕ, ψ (viz. the Rankin-Selberg convolution of f and the weight-one theta series ϑ χ associated with χ). Let χ denote either ϕ or ψ, let K χ be the ring class field of K cut out by χ, and let A(K χ ) χ be the submodule of A(K χ ) ⊗ Z Q( ) on which Gal(K χ /K) acts via χ. Fix a primitive Heegner point P in A(K χ ) and set Equations (25) and (27) and Assumption 1.1.(1) imply that L(A/K, χ, s) has a simple zero at s = 1, hence the Gross-Zagier-Kolyvagin-Zhang theorem yields where Sel(K χ , V p (A)) is the Bloch-Kato Selmer group of the restriction of V p (A) to G K χ , one denotes by Sel(K χ , V p (A)) χ the submodule of Sel(K χ , V p (A)) ⊗ Q p L on which the Galois group of K χ /K acts via the character χ, and one considers A(K χ ) χ as a submodule of Sel(K χ , V p (A)) χ via the K χ -rational Kummer map. Let σ p in G Q − G K be an arithmetic Frobenius at p. For ξ = g, h and each pair (a, b) of elements of Q( ), denote by [a, b] ξ in V ξ the Q( )-valued function on G Q sending the identity to a and σ p to b. Then G K acts on the line L · [1, 0] ξ via ν ξ , and on the line L · [0, 1] ξ via the conjugate ν c ξ of ν ξ by the nontrivial element c = σ p | K of Gal(K/Q). Moreover, since ν ξ (σ 2 p ) = ν cen (recall that β ξ = −α ξ ), and for each pair (i, j) of elements of {α, β} set A direct computation shows that the vectors of V gh are qual to 4 · [1, 0] g ⊗ [1, 0] h and 4α ξ · [1, 0] g ⊗ [0, 1] h respectively, hence G K acts on them via ϕ = ν g · ν h and ψ = ν g · ν c h respectively. For χ = ϕ, ψ define P (χ) = γ gh P χ ⊗ σ p (v χ ) + σ p (P χ ) ⊗ v χ in Sel(Q, V ) to be image of P χ ⊗σ p (v χ )+σ p (P χ )⊗v χ in A(K ) under the embedding γ gh introduced in Eq. (10), so that (cf. Eqs. (26) and (28)) Sel(Q, V ) = L · P (ϕ) ⊕ L · P (ψ). (29) Write ε = α f and for χ equal to ϕ or ψ define P ε χ = P χ + ε · σ p (P χ ). The point P ε χ is non-zero. This follows from Eq. (28) if χ is not quadratic. When χ is quadratic, one has σ p (P χ ) = χ 1 (p) · P χ , hence P ε χ is non-zero by Eq. (28) and Assumption 1.2. In order to lighten the notation, set κ αα = κ(f, g α , h α ). The main result Theorem A of [2] proves the identity log ββ (res p (κ(f, g α , h α ))) = log ω f (P ε ϕ ) · log ω f (P ε ψ ) ∈ L * /Q( ) * .
Here log ω f : A(K χ ) ⊗ Z L −→ L ⊗ Q p K p denotes the L-linear extension of the logarithm log ω f on A(K χ ) introduced in Sect. 2.1.4. (Note that the right hand side of the previous identity is an element of L ⊗ Q p K p fixed by the action of σ p , id est of L.) Recall that the roots α ξ and β ξ = −α ξ of the pth Hecke polynomial of ξ = g, h are distinct, and that α g · α h = α f = β g · β h (cf. Eq. (1)). We can then replace in the above constructions the Hida family ξ = ξ α with the one ξ β specialising to the p-stabilisation ξ β (q) = ξ(q) − α ξ · ξ(q p ) at weight one, for ξ = g, h. This produces a diagonal class κ(f, g β , h β ) in the Selmer group Sel(Q, W ) of the p-adic Since by the definition of the balanced local condition (cf. Sect. 2.1.1) one has log αα (res p (κ αα )) = log ββ (res p (κ ββ )) = 0, it follows that the diagonal classes κ αα and κ ββ are linearly independent, hence Sel(Q, V ) = L · κ αα ⊕ L · κ ββ .