* 2021 The Author(s)*c **Milan Journal of Mathematics**

**Heegner Points and Exceptional Zeros** **of Garrett** **p-Adic L-Functions**

**p-Adic L-Functions**

### Massimo Bertolini, Marco Adamo Seveso, and Rodolfo Venerucci

**Abstract. 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.

**1. Statement of the Main Result**

*Let A be an elliptic curve deﬁned over the ﬁeld* **Q of rational numbers, having**
*multiplicative reduction at a rational prime p > 3. Let K be a quadratic imaginary*
*ﬁeld of discriminant d**K* *coprime to the conductor N**A* *of A, and let*

*ν**g* *: G**K* *−→ ¯***Q*** ^{∗}* and

*ν*

*h*

*: G*

*K*

*−→ ¯*

**Q**

^{∗}*be ﬁnite order characters of the absolute Galois group G** _{K}* = Gal( ¯

**Q/K) of K, where***¯*

**Q is the ﬁeld of algebraic complex numbers. Write N***A*

*= N*

_{A}^{+}

*·N*

*A*

*, where each prime*

^{−}*divisor of N*

_{A}^{+}

*(resp., N*

_{A}

^{−}*) splits (resp., is inert) in K. We make the following*

**Assumption 1.1.**

*1. (Heegner assumption) The prime p is inert in K (id est divides*

*N*_{A}^{−}*) and N*_{A}^{−}*is a square-free product of an even number of primes.*

*2. (Self-duality) The central characters of ν**g* *and ν**h* are inverse to each other.

*3. (Cuspidality) The characters ν**g* *and ν**h*are not induced by Dirichlet characters.

*4. (Local signs) The conductors of ν*_{g}*and ν*_{h}*are coprime to d*_{K}*· N**A*.
*Let f =*

*n1**a**n**(f )· q*^{n}*in S*_{2}(Γ_{0}*(N**f**)) be the newform of conductor N**f* *= N**A*

*attached to A by the modularity theorem. For ν**ξ**= ν**g**, ν**h**, let **ξ**: G*_{Q}*−→ GL*2(**C)**
be the odd irreducible (cf. Assumption 1.1.(3)) Artin representation of G** _{Q}** induced

*by ν*

*ξ*, corresponding by modularity to the cuspidal weight one theta series

*ξ =*

(a,f*ξ*)=O*K*

*ν**ξ*(*a) · q*^{Na}*∈ S*1*(N**ξ**, χ**ξ**).*

Here *a runs the set of non-zero ideals of O**K* coprime to the conductor f*ξ* *of ν**ξ*,
**Na = |O***K**/a|, N**ξ* *= d**K* **· Nf***ξ* *and χ**ξ* *= ε**K* *· ν**ξ*^{cen}*, where ε**K* : (**Z/d***K***Z)**^{∗}*−→ μ*2 is

0123456789().: V,-vol

*the quadratic character of K and ν*_{ξ}^{cen} *: G*_{Q}*−→ ¯***Q**^{∗}*is the central character of ν**ξ*.
*Since p is inert in K by Assumption*1.1.(1), the p-th Hecke polynomial of ξ equals
*X*^{2}*+ χ**ξ**(p) (id est the p-th Fourier coeﬃcient of ξ is equal to zero). In addition*
*χ**ξ**(p) is non-zero by Assumption* 1.1.(4), hence X^{2}*+ χ**ξ**(p) = (X− α**ξ*)*· (X − β**ξ*)
*has distinct roots α**ξ* *and β**ξ* = *−α**ξ*. According to Assumption 1.1.(2) one has
*α**g* *· α**h* *= β**g* *· β**h* = *±1 and α**g* *· β**h* *= β**g* *· α**h* = *−α**g* *· α**h*, hence we can, and will,
assume

*α**f* *= α**g**· α**h**= β**g**· β**h*and *− α**f* *= β**g**· α**h**= α**g**· β**h* (1)
*by reordering the roots (α**ξ**, β**ξ**) of X*^{2}*+ χ**ξ**(p) if necessary, where α**f* *= a**p**(f ) =±1.*

Fix an algebraic closure ¯**Q***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 =**

*n1*

*a*_{n}**(f )**· q^{n}*∈ O***f***[[q]]* and **ξ*** _{α}*=

*n1*

*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*

*is*

**f***the ring of bounded analytic functions on a (small) connected open disc U*

*centred at 2 in the weight space*

**f***W = Hom*cont(

**Z**

^{∗}*p*

*,*

**C**

^{∗}*p*) over

**Q**

*p*

*. For each k in U*

**f***4,*

**∩ Z**

**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***is the ring of bounded*

_{α}*analytic functions on a connected open disc U*

**ξ***centred at 1 in*

_{α}*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}*1*

**∩ Z***, 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***⊗*ˆ

**Q**

*p*

*O*

**g***⊗*ˆ

*L*

*O*

*. Under Assumption1.1, Theorem A of [8] associates with (f , g*

**h**

_{α}

**, h***α*

*) a Garrett–Hida square root p-adic L-function*

*L**p*^{αα}*(A, ) =L**p***(f , g**_{α}**, h***α*)*∈ O***f gh**

(denoted*L*^{f}**F****in loc. cit., where F = (f , g**_{α}**, h***α*)), whose square
*L*^{αα}_{p}*(A, ) = L**p***(f , g**_{α}**, h***α*) =*L**p***(f , g**_{α}**, h***α*)^{2}

*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**

_{ξ}*. Deﬁne A(K** _{}*)

^{}*= H*

^{0}

*(Gal(K*

_{}*/*)

**Q), A(K***⊗*

_{Z}*V*

_{gh}*), where V*

_{gh}*= V*

_{g}*⊗*

_{Q()}*V*

*and*

_{h}*K*

*is the number ﬁeld cut-out by =*

*g*

*⊗*

*h*. Following [9] one exploits Tate’s

*p-adic uniformisation to deﬁne an extended Mordell–Weil group*

*A*^{†}*(K*)^{}*= A(K*)^{}*⊕ Q**p**(A, ),*

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}*(cf. Sect. 2.1.3 below).*

_{p}Moreover, Section 2 of [4] constructs a Garrett–Nekov´*aˇr height-pairing*

*⟪·, ·⟫***f g**_{α}**h**_{α}*: A*^{†}*(K** _{}*)

^{}*⊗*

_{Q()}*A*

^{†}*(K*

*)*

_{}

^{}*−→ I /I*

^{2}

*,*

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 Nekov´aˇr’s theory of Selmer complexes to the

**big self-dual Galois representation associated with (f , g**

_{α}

**, h***α*). After setting

*r** ^{†}*= dim

_{Q()}*A(K*)

^{}*,*

Conjecture 1.1 of [4] predicts that L^{αα}_{p}*(A, ) belongs toI*^{r}^{†}*− I*^{r}^{†}^{+1}, and that its
image in (*I*^{r}^{†}*/I*^{r}^{†}^{+1}*)/ Q()*

*is equal to the discriminant*

^{∗2}*R*^{αα}_{p}*(A, ) = det*

*⟪P**i**, P**j*⟫_{f g}_{α}_{h}_{α}

1i,jr^{†}

*of the p-adic height* *⟪·, ·⟫*_{f g}_{α}_{h}_{α}*, where P*_{1}*, . . . , P*_{r}*†* is any**Q()-basis of A**^{†}*(K*)* ^{}*.
The following theorem is the main result of this note.

* Theorem. Assume that Assumptions* 1.1

*and*1.2

*(stated below) are satisfied. If the*

*complex L-function L(f⊗ g ⊗ h, s) has order of vanishing 2 at s = 1, then*

dim_{Q()}*A*^{†}*(K*)^{}*= 4,* *L*^{αα}_{p}*(A, )∈ I*^{4}*− I*^{5}
*and the equality*

*L*^{αα}_{p}*(A, ) (mod* *I*^{5}*) = R*^{αα}_{p}*(A, )*

*holds in the quotient of* *I*^{4}*/I*^{5} *by the multiplicative action of* **Q()**^{∗2}*.*

*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*_{h}^{c}*, and ν*_{h}^{c}*is the conjugate of ν** _{h}* by the nontrivial element

*of Gal(K/*1.1.(2), and that both

**Q). Note that ϕ and ψ are dihedral by Assumption***L(A/K, ϕ, s) and L(A/K, ψ, s) have sign−1 in their functional equation by Assump-*tion 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*

^{}*theorem.*

**Q() and generated by Heegner points by the Kolyvagin–Gross–Zagier–Zhang***If χ = ϕ, ψ is quadratic, ¯***Q*** ^{ker(χ)}* =

**Q(**

*√*

*cd*

_{1}

*,√*

*cd*_{2}*), where c, d*_{1} *and d*_{2} are
*fundamental discriminants such that d**K* *= d*_{1}*· d*2. (We consider 1 as a fundamental
*discriminant). In this case L(A/K, χ, s) further factors as the product of the Hasse-*
*Weil L-functions L(A/ Q, χ*1

*, s) and L(A/*2

**Q, χ***, s) of the twists of A by the quadratic*

*characters χ*

*i*of

**Q(**

*√*

*cd**i*). By Assumptions1.1.(1) and1.1.(4), we can order χ_{1}and
*χ*_{2} *in such a way that sign(A, χ*_{1}) =*−1 and sign(A, χ*2*) = +1, where sign(A, χ** _{i}*) is

*the sign in the functional equation satisﬁed by L(A/*

**Q, χ***i*

*, s).*

**Assumption 1.2. If χ = ϕ or χ = ψ is quadratic, then χ**_{1}*(p) = α** _{f}*.

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 [10–12].

**2. Proof of the Main Result**

**2.1. 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*

*-module of rank two, equipped with a continuous*

**ξ***O*

**ξ***-linear action of G*

_{Q}*. For each u in U*

**ξ***2*

**∩ Z***, evaluation at u*

*on U*

*induces a natural specialisation isomorphism*

**ξ***ρ**u***: V (ξ)**⊗*u**E* ** V (ξ***u**),*

*where E =* **Q***p* **if ξ = f and E = L if ξ = g, h, 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 V (f ) = V (f**_{2}*) is equal to the f -*
*isotypic component of the cohomology group H*_{´}_{et}^{1}*(X*_{1}*(N**f*)**Q**¯*,***Q***p**(1)), where X*_{1}*(N**f*)**Q**¯

is the base change to ¯* Q of the compact modular curve X*1

*(N*

*) of level Γ*

_{f}_{1}

*(N*

*) deﬁned over*

_{f}**Q. Fix a modular parametrisation (viz. a non-constant morphism of**

**Q-curves)**

*℘*_{∞}*: X*_{1}*(N**f*)*−→ A,*

which induces an isomorphism of **Q***p**[G*_{Q}*]-modules between V (f ) and the p-adic*
*Tate module V**p**(A) = H*_{´}_{et}^{1}*(A***Q**¯*,***Q***p**(1)) of A with* **Q***p*-coeﬃcients.

**When ξ = g, h and u = 1, the L[G**** _{Q}**]-module

*1*

**V (ξ) = V (ξ)**⊗*L*

*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

*π*

**ξ**

**: V (ξ)**⊗

_{O}

**ξ**

**V (ξ)**−→ O

**ξ***(χ*

*ξ*

*· χ*

^{u−1}_{cyc}

*),*

*where χ*_{cyc}*: G*_{Q}**−→ Z**^{∗}*p* *is the p-adic cyclotomic character and χ*^{u−1}_{cyc} *: G*_{Q}*−→ O***ξ**^{∗}

*satisﬁes χ*^{u−1}_{cyc} *(σ)(u) = χ*_{cyc}*(σ)*^{u−1}*for each σ in G*_{Q}*and each u in U***ξ*** ∩ Z. (With*
the notations of [5, Section 5], the pairing π

*is the composition of the twist by*

**ξ***χ*

*ξ*

*· χ*

^{u−1}_{cyc}of the

*O*

*-adic Poincar´e duality*

**ξ***·, ·*

**f**

**: V (ξ)**⊗

_{O}

**ξ***V*

^{∗}

**(ξ)***−→ O*

*deﬁned in [5, Equation (103)] with id*

**ξ**

_{V (ξ)}*⊗ w*

^{−1}

_{N}

_{ξ}

_{p}*, where w*

*N*

_{ξ}*p*

*: V*

^{∗}

**(ξ)(χ***ξ*

*· χ*

^{u−1}_{cyc})

*the*

**V (ξ) is***O*

*-adic Atkin–Lehner isomorphism deﬁned in [5, Equation (114)].) Up to sign,*

**ξ***the pairing π*

_{f}*: V (f )⊗*

_{Q}*p*

*V (f )*

**−→ Q***p*

*(1) arising from the base change of π*

*along*

_{f}*evaluation at k = 2 onO*

**f***and the specialisation isomorphism ρ*

_{2}is the one induced

*on the f -isotypic components by the Poincar´e duality on H*

_{´}

_{et}

^{1}

*(X*

_{1}

*(N*

*f*)

**Q**¯

*,*

**Q**

*p*(1)). If

**ξ = g, h, the weight-one specialisation of π***yields a perfect skew-symmetric duality*

**ξ***π**ξ* *: V (ξ)⊗**L**V (ξ)−→ L(χ**ξ**).*

*Identify G*_{Q}_{p}*with a subgroup of G*_{Q}*via the embedding i**p* : ¯**Q −→ ¯****Q***p* ﬁxed
at the outset, and let ˇ*a**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 sequence of

*O*

**ξ***[G*

_{Q}*]-modules*

_{p}**V (ξ)**^{+}* −→ V (ξ) − V (ξ)*

^{−}*,*

**where V (ξ)**^{+} **and V (ξ)*** ^{−}*are free

*O*

**ξ***-modules of rank one and G*

_{Q}*acts on them via*

_{p}*the characters χ*

*ξ*

*·χ*

^{u−1}_{cyc}

*·ˇa*

*p*

**(ξ)***and ˇ*

^{−1}*a*

*p*

**(ξ) respectively (cf. Section 5 of [5]). If ξ = f ,***the specialisation isomorphism ρ*

_{2}

*2*

**: V (f )**⊗**Q**

*p*

**V (f) identiﬁes V (f)**

^{−}*⊗*2

**Q**

*p*with

**the maximal p-unramiﬁed quotient of V (f ) and V (ξ)**^{+}

*⊗*2

**Q**

*p*

*with the kernel V (f )*

^{+}

*of the projection V (f )−→ V (f)*

^{−}

**. If ξ = g, h deﬁne***V (ξ)**α***= V (ξ)**^{−}*⊗*1*L* and *V (ξ)**β* **= V (ξ)**^{+}*⊗*1*L,*

*so that V (ξ)**γ* *(for γ = α, β) is the submodule of V (ξ) on which an arithmetic*
*Frobenius in G*_{Q}_{p}*acts as multiplication by γ**ξ**, and (as L[G*_{Q}* _{p}*]-modules)

*V (ξ) = V (ξ)**α**⊕ V (ξ)**β**.*
Deﬁne

**V = V (f, g, h) = V (f) ˆ**⊗**Q***p***V (g) ˆ**⊗*L***V (h)(Ξ****f gh***),*
where Ξ**f gh***= χ*(4−k−l−m )/2

cyc *: G*_{Q}*−→ O***f gh*** ^{∗}* satisﬁes Ξ

**f gh***(σ)(w) = χ*

_{cyc}

*(σ)*

^{4−k−l−m}^{2}

*for each σ in G*

_{Q}*and each w = (k, l, m) in U*

**f***× U*

**g***× U*

**h**

**∩ Z**^{3}, and

*V = V (f, g, h) = V (f )⊗***Q***p**V (g)⊗**L**V (h).*

*Evaluation at w*_{o}*= (2, 1, 1) on* *O** f gh* induces a specialisation isomorphism

*ρ*

*w*

_{o}

**: V***⊗*

*w*

_{o}*L V.*

*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))

*π***f gh****: V***⊗*_{O}**f g h****V −→ O****f gh***(1),*

*whose base change along evaluation at w**o* on *O***f gh***recasts (via ρ**w**o*) the perfect
duality

*π**fgh**: V* *⊗**L**V* *−→ L(1)*

*deﬁned by the product of the perfect pairings π**ξ* *for ξ = f, g, h.*

**For ξ = f , g, h let***F*^{•}**V (ξ) be the decreasing ﬁltration on the***O***f gh***[G*_{Q}* _{p}*]-

**module V (ξ) deﬁned by***F*

^{1}

**V (ξ) = V (ξ)**^{+},

*F*

^{i}*0 and*

**V (ξ) = V (ξ) for each i***F*

^{i}

**V (ξ) = 0 for each i**2. Deﬁne the balanced submodule F^{2}

**V of V by***F*^{2}**V =**

*a+b+c=2*

*F*^{a}**V (f ) ˆ**⊗**Q***p**F*^{b}**V (g) ˆ**⊗*L**F*^{c}**V (h)**

*⊗*_{O}* f g h* Ξ

**f gh***,*

**and the f -unbalanced submodule V**^{+} **of V by**

**V**^{+} **= V (f )**^{+}*⊗*ˆ**Q***p***V (g) ˆ**⊗*L***V (h)**⊗_{O}* f g h* Ξ

**f gh***.*

*These are G*_{Q}* _{p}*-invariant free

*O*

**f gh**

**-submodules of V of rank 4 =**^{1}

_{2}rank

_{O}

_{f g h}

**V , which***are maximal isotropic with respect to the skew-symmetric duality π*

*. After set- ting*

**f gh****V**^{−}**= V /V**^{+} and **V***f* **= V (f )**^{−}*⊗*ˆ**Q***p***V (g)**^{+}*⊗*ˆ*L***V (h)**^{+}*⊗*_{O}* f g h* Ξ

**f gh***,*one has a commutative diagram of

*O*

**f gh***[G*

_{Q}*]-modules*

_{p}*F*^{2}**V**^{ } ^{i}^{F}^{//}

*p**f*

**V**

*p*^{−}

**V***f* ^{i}* ^{f}* //

**V**

^{−}(2)

*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 H*

_{bal}

^{1}(

**Q**

*p*

**, V )***of H*

^{1}(

**Q**

*p*

**, V ) to be the image of the morphism induced in cohomology by i***. This morphism is injective (cf. Section 7.2 of [5]), hence gives a natural identiﬁcation*

_{F}*H*_{bal}^{1} (**Q***p***, V ) = H**^{1}(**Q***p**,F*^{2}* V )* (3)

*Set V*

^{±}*= V (f )*

^{±}*⊗*

**Q**

*p*

*V (g)⊗*

*L*

*V (h). For each pair (i, j) of elements of*

*{α, β}*

*deﬁne V*_{ij}^{·}*= V (f )*^{·}*⊗***Q***p**V (g)**i**⊗**L**V (h)**j*, where*· is one of symbols ∅, + and −. Then*
*V*^{·}*= V*_{αα}^{·}*⊕ V**αβ*^{·}*⊕ V**βα*^{·}*⊕ V**ββ*^{·}

*as L[G*_{Q}* _{p}*]-modules, and Eq. (1) implies

*H*^{0}(**Q***p**, V*^{−}*) = V*_{αα}^{−}*⊕ V*_{ββ}* ^{−}* and

*H*

^{0}(

**Q**

*p*

*, V*

^{+}(

*−1)) = V*

*αα*

^{+}(

*−1) ⊕ V*

_{ββ}^{+}(

*−1). (4)*

*The specialisation isomorphism ρ*

_{w}

_{o}

**identiﬁes V**

^{±}*⊗*

*w*

*o*

*L,F*

^{2}

**V ⊗***w*

*o*

**L and V**

_{f}*⊗*

*w*

*o*

*L*

*with V*

*,*

^{±}*F*

^{2}

*V = V*

*ββ*

*+ V*

_{αβ}^{+}

*+ V*

_{βα}^{+}

*and V*

_{ββ}*respectively. In particular the base change of the commutative diagram (2) along evaluation at w*

^{−}*o*on

*O*

*is equal to*

**f gh***F*^{2}*V* ^{ } ^{i}^{F}^{//}

*p*_{f}

*V*

*p*^{−}

*V*_{ββ}^{−}^{ } ^{i}^{f}^{//} *V*^{−}

(5)

*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*
*p*^{−}*: H*^{1}(**Q***p**, V )* *−→ H*^{1}(**Q***p**, V** ^{−}*), cf. Section 9.1 of [5]. (With a slight abuse of

*notation, we denote by the same symbol a morphism of G*

_{Q}*-modules and the maps it induces in cohomology.) By construction (cf. Eqs. (2) and (5)), the specialisation*

_{p}*κ = ρ*

*w*

_{o}

**(κ) in H**^{1}(

**Q**

*p*

*, V ) at w*

*o*

**of a local balanced class κ in H**_{bal}

^{1}(

**Q**

*p*

**, V ) belongs***to the kernel of the map H*

^{1}(

**Q**

*p*

*, V )*

*−→ H*

^{1}(

**Q**

*p*

*, V*

_{ij}

^{−}*) for ij = αα, αβ, βα. Then κ*

*is crystalline precisely if it belongs to the kernel of H*

^{1}(

**Q**

*p*

*, V )−→ H*

^{1}(

**Q**

*p*

*, V*

_{ββ}*), id*

^{−}*est if p*

*f*

**(κ) in H**^{1}(

**Q**

*p*

**, V***f*) (cf. Eq. (3)) belongs to the kernel of the specialisation

*map ρ*

*w*

*o*

*: H*

^{1}(

**Q**

*p*

**, V***f*)

*−→ H*

^{1}(

**Q**

*p*

*, V*

_{ββ}*). Since the ideal*

^{−}*I of O*

*is generated*

**f gh***by a regular sequence and H*^{2}(**Q***p**, V*_{ββ}^{−}*) = 0, the specialisation map ρ**w** _{o}* induces an

*isomorphism H*

^{1}(Q

*p*

**, V***f*)

*⊗*

*w*

*o*

*L H*

^{1}(Q

*p*

*, V*

_{ββ}*). We have proved the following*

^{−}

**Lemma 2.1. Let κ be a local balanced class in H**_{bal}

^{1}(

**Q**

*p*

**, V ) and set κ = ρ***w*

_{o}

**(κ) in***H*

^{1}(

**Q**

*p*

*, V ). Then κ is crystalline if and only if p*

*f*

**(κ) belongs to***I*

*· H*

^{1}(

**Q**

*p*

**, V***f*

*).*

**2.1.2.** **p-Adic Periods. Let ˆ****Q**^{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*

*G(χ**ξ*) = (*−c)*^{i}^{ξ}*·*

**a∈(Z/cZ)**^{∗}

*χ**ξ**(a)*^{−1}*⊗ e*^{2πia/c}*∈ D*cris*(χ**ξ**),*

*where i**g* *= 0, i**h*=*−1 and D*cris*(χ**ξ**) is a shorthand for H*^{0}(**Q***p**, L(χ**ξ*)*⊗***Q***p***Q**ˆ^{nr}*p* ).

As explained in Section 3.1 of [4], for ξ = f , g, h the module D(ξ)^{−}*of G*_{Q}* _{p}*-

**invariants of V (ξ)**

^{−}*⊗*ˆ

**Q**

*p*

**Q**ˆ

^{nr}

*p*is free of rank one over

*O*

*, and its base change*

**ξ****D(ξ)**^{−}_{u}**= D(ξ)**^{−}*⊗**u**L*

*along evaluation at a classical weight u in U***ξ*** ∩ Z*2on

*O*

*is canonically isomorphic*

**ξ**

**to the ξ**

_{u}*-isotypic component L*

**· ξ***u*

*of S*

_{u}*(pN*

_{ξ}*, χ*

*)*

_{ξ}*. Moreover there exists an*

_{L}*O*

*- basis*

**ξ***ω*_{ξ}**∈ D(ξ)**^{−}

*whose image ω***ξ**_{u}**in D(ξ)**^{−}_{u}**corresponds to ξ*** _{u}*under the aforementioned isomorphism

*for each u in U*

**ξ***2. (We refer to loc. cit. and the references therein for the details.)*

**∩Z***The weight-two specialisation of ω*

*equals the de Rham class*

_{f}*ω**f* *∈ D*cris*(V (f )** ^{−}*)

*Fil*

^{0}

*D*

_{dR}

*(V (f ))*

*associated with f under the Faltings–Tsuji comparison isomorphism between the*

´

*etale and de Rham cohomology of X*_{1}*(N**f*)_{Q}* _{p}*. (The isomorphism in the previous

*equation arises from the projection V (f )−→ V (f)*

*.) Denote by*

^{−}* ·, ·**f* *: D*_{dR}*(V (f ))⊗**L**D*_{dR}*(V (f ))−→ L*

*the perfect duality induced by π**f**, and deﬁne η**f* *in D*dR*(V (f ))/Fil*^{0}by the identity
* η**f**, ω**f**f* *= 1.*

**For ξ = g, h, the weight-one specialisation of ω*** ξ* yields a class

*ω*

*ξ*

*α*

*∈ D*cris

*(V (ξ)*

*α*

*) = D*cris

*(V (ξ))*

^{ϕ=α}

^{−1}

^{ξ}*(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}^{ξ}* ^{−1}* by the identity

* η**ξ**α**, ω*_{ξ}_{α}*ξ* *= G(χ*_{ξ}*).*

*Along with ω**f**, it is important to consider another p-adic period*
*q(f )∈ D*cris*(V (f )** ^{−}*) = Fil

^{0}

*D*

_{dR}

*(V (f ))*

*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*

*, unique up to*

_{p}*sign, with kernel the lattice generated by the Tate period q*

*A*

*in p*

**Z**

*p*

*of A*

_{Q}*. One sets*

_{p}*q(A) = p*^{−}

*℘*_{Tate}(^{p∞}*√*
*q** _{A}*)

*∈ V**p**(A)** ^{−}* and

*q(f ) =√m*

_{p}*· ℘*

^{−1}

_{∞}*(q(A)),*(6) where

^{p∞}*√q*

*A*

*is any compatible system of p*

^{n}*th roots of q*

*A*

*, ℘*

_{∞}*: V (f )*

^{−}*V*

*p*

*(A)*

^{−}*is the isomorphism arising from the ﬁxed modular parametrisation ℘*

_{∞}*, m*

*p*= 1 if

*α*

_{f}*= 1 and m*

_{p}*= d*

_{K}*if α*

*=*

_{f}*−1. As in loc. cit., deﬁne the generators*

*q**αα**= q(f )⊗ ω**g*_{α}*⊗ ω**h** _{α}* and

*q*

*ββ*

*= q(f )⊗ η*

*g*

_{α}*⊗ η*

*h*

_{α}*of the subspaces V*_{αα}^{−}*and V*_{ββ}^{−}*respectively of H*^{0}(**Q***p**, V*^{−}*) = D*_{cris}*(V** ^{−}*)

*.*

^{ϕ=1}**2.1.3. The Garrett–Nekov´aˇr** **p-Adic Height Pairing. Section 2 of [4] constructs a***canonical skew-symmetric p-adic height pairing*

*⟪·, ·⟫**_{f gh}* : ˜

*H*

_{f}^{1}(

**Q, V ) ⊗***L*

*H*˜

_{f}^{1}(

**Q, V ) −→ I /I**^{2}

on the extended Selmer group ˜*H*_{f}^{1}(**Q, V ) associated with the Greenberg local condi-***tion at p arising from the inclusion i*^{+}*: V*^{+}* −→ V . Let Sel(Q, V ) denote the Bloch–*

*Kato Selmer group of V , which is equal to the kernel of H*^{1}(**Q, V ) −→ H**^{1}(**Q***p**, V** ^{−}*)
in the present setting (cf. [5, Section 9.1]). One has a commutative exact diagram

0 ^{//}*H*^{0}(**Q***p**, V** ^{−}*)

^{j}^{//}

*H*˜

_{f}^{1}(

**Q, V )***·*^{+}

*π* //Sel(**Q, V )**

res*p*

//0

*H*^{1}(**Q***p**, V*^{+}) ^{i}^{+} ^{//}*H*^{1}(**Q***p**, V )*

(7)

*and there exists a unique section ı*_{ur} : Sel(**Q, V ) −→ ˜**H_{f}^{1}(**Q, V ) of π such that the***composition ı*_{ur}(*·)*^{+} *takes values in the ﬁnite subspace H*_{fin}^{1} (**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 Nekov´aˇr’s
extended Selmer group ˜*H*_{f}^{1}(**Q, V ) with the naive extended Selmer group**

Sel* ^{†}*(

**Q, V ) = H**^{0}(

**Q**

*p*

*, V*

*)*

^{−}

**⊕ Sel(Q, V ).***Enlarging L if necessary, for ξ = g, h ﬁx an isomorphism of L[G*** _{Q}**]-modules

*γ*

*ξ*

*: V*

*ξ*

*⊗*

_{Q()}*L V (ξ) such that π*

*ξ*

*(γ*

*ξ*

*(x)⊗ γ*

*ξ*

*(y))*

**∈ Q()(χ***ξ*) (8)

*for each x and y in V*

*ξ*(cf. Eq. (4) of [4]). Set (cf. Eq. (6))

*Q**p**(A, ) = H*^{0}(**Q***p**, Q() · q(A) ⊗*

**Q()***V*

*gh*

*).*(9)

*The modular parametrisation ℘*

_{∞}*: X*

_{1}

*(N*

*f*)

*−→ A ﬁxed in Sect.*2.1.1, the global

*Kummer map on A(K*) ˆ

**⊗Q***p*

*and the isomorphisms γ*

*g*

*and γ*

*h*induce an embedding

*γ**gh**: A*^{†}*(K*)^{}* −→ Sel** ^{†}*(

**Q, V ) = ˜**H

_{f}^{1}(

*(10) and one deﬁnes the Garrett–Nekov´*

**Q, V ),***aˇr p-adic pairing (cf. Sect.*1)

*⟪·, ·⟫***f gh***: A*^{†}*(K*)^{}*⊗***Q()***A*^{†}*(K*)^{}*−→ I /I*^{2}

to be the restriction of the canonical height *⟪·, ·⟫** f gh* on ˜

*H*

_{f}^{1}(

**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*.

**2.1.4. Logarithms. Let V**_{dR}

*= D*

_{dR}

*(V ) be the de Rham module of V = V (f, g, h).*

*The duality π**fgh* *: V* *⊗**L**V* *−→ L(1) induces a perfect pairing*
* ·, ·**fgh* *: V*_{dR}*⊗**L**V*_{dR}*−→ L.*

*After identifying V*_{dR} *with D*_{dR}*(V (f ))⊗***Q***p* *D*_{cris}*(V (g))⊗**L**D*_{cris}*(V (h)) and L with*
*D*_{cris}*(χ**g*)*⊗**L**D*_{cris}*(χ**h*) under the natural isomorphisms (cf. Assumption1.1.(2)), the
pairing * ·, ·**fgh* agrees with the product of the pairings* ·, ·**ξ* *for ξ = f, g, h.*

The Bloch–Kato exponential map exp* _{p}*gives an isomorphism between the tan-

*gent space V*

_{dR}

*/Fil*

^{0}

*of V and the ﬁnite (viz. crystalline) subspace H*

_{fin}

^{1}(

**Q**

*p*

*, V ) of*

*H*

^{1}(

**Q**

*p*

*, V ). Denote by log*

*the inverse of exp*

_{p}

_{p}*and deﬁne the αα-logarithm*

log* _{αα}*=

*log*

*p*

*, ω*

*f*

*⊗ η*

*g*

_{α}*⊗ η*

*h*

_{α}*fgh*

*: H*

_{fin}

^{1}(

**Q**

*p*

*, V )−→ L*

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*

log* _{ββ}* =

*log*

*p*

*, ω*

_{f}*⊗ ω*

*g*

*α*

*⊗ ω*

*h*

*α*

*: H*

_{fin}

^{1}(

**Q**

*p*

*, V )−→ L.*

(Note that log_{ii}*factors through the projection H*_{fin}^{1} (**Q***p**, V )−→ H*^{1}(**Q***p**, V**ii*).)
Set tg_{dR,K}_{p}*(f ) = H*^{0}*(K**p**, V (f )⊗***Q***p* *B*_{dR}*)/Fil*^{0} and consider the composition

log_{A,p}*: A(K**p*) ˆ**⊗Q***p* * H*fin^{1} *(K**p**, V**p**(A)) H*fin^{1} *(K**p**, V (f )) tg**dR,K**p**(f ),*
where the ﬁrst isomorphism is the local Kummer map, the second is induced by the
*ﬁxed modular parametrisation ℘*_{∞}*: X*_{1}*(N**f*) *−→ A (cf. Sect.* 2.1.1), and the third
*is the inverse of the Bloch–Kato exponential map. For χ = ϕ, ψ (cf. Sect.* 1) deﬁne

log_{ω}* _{f}* =

*log*

*A,p*

*, ω*

*f*

*f*

*: A(K*

*χ*)

*−→ K*

*p*

*,*

*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*.)

**2.2. Big Logarithms and Diagonal Classes**
Let

*L***f***: H*^{1}(**Q***p***, V***f*)*−→ I*

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*

*takes values in*

**f***I by the exceptional*

*zero condition α*

*f*

*= α*

*g*

*· α*

*h*.) With a slight abuse of notation denote by

*L***f***: H*_{bal}^{1} (**Q***p** , V )−→ I*
also the composition

*L*

**f***◦ p*

*f*(cf. Eq. (3)).

*Let H*_{bal}^{1} (**Q, V ) be the group of global classes in H**^{1}(**Q, V ) whose restriction***at p belongs to the balanced local condition H*_{bal}^{1} (**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) = κ(f , g**_{α}**, h***α*)*∈ H*bal^{1} (* Q, V )*
such that

*L** f*(res

_{p}

**(κ(f , g, h))) =**L*p*

^{αα}*(A, ).*(11)

*Deﬁne the diagonal class*

*κ(f, g**α**, h**α**) = ρ**w*_{o}**(κ(f , g**_{α}**, h***α*))

*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:*

*κ(f, g**α**, h**α*)* ∈ Sel(Q, V ).* (12)
Identify

*O*

**f gh**

**with a subring of the power series ring L[[k****− 2, l − 1, m − 1]],**

**where k**− 2 in O

**f***is a uniformiser at the centre 2 of U*

**f***deﬁned similarly. In light of Eq. (12) and Lemma2.1there exist local classesY*

**, and l****− 1 and m − 1 are**

**k***,*Y

**l**and Y**m***in H*^{1}(**Q***p***, V*** _{f}*) satisfying the identity

*p*

*f*(res

*p*

**(κ(f , g, h))) =****u**

Y**u****· (u − u***o**).* (13)

Equation (11) gives

*L**p*^{αα}*(A, ) =*

**u**

*L** f*(Y

*)*

**u**

**· (u − u***o*)

*∈ I*

^{2}

*.*(14) The following key lemma, proved in Part 1 of Proposition 9.3 of [5], gives an explicit description of the linear term of

*L*

*(Y*

**f**

**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*

*corresponds to the arithmetic*

^{−1}*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))*

^{−}*H*^{1}(**Q***p**, V*_{ββ}* ^{−}*) = Hom

_{cont}(

**Q**

*p*

^{∗}*,*

**Q**

*p*)

*⊗*

**Q**

*p*

*V*

_{ββ}

^{−}*,*(15) and the Bloch–Kato dual exponential exp

^{∗}

_{p}*on H*

^{1}(

**Q**

*p*

*, V*

_{ββ}*) satisﬁes*

^{−}exp^{∗}_{p}*(ϕ⊗ v) = ϕ(e(1)) · v*

*in D*_{cris}*(V*_{ββ}^{−}*) = V*_{ββ}^{−}*for each ϕ in Hom*_{cont}(**Q**^{∗}*p**,***Q***p**) and v in V*_{ββ}* ^{−}*, where

*e(1) = (1 + p) ˆ⊗ log*

*p*

*(1 + p)*

^{−1}

**∈ Z**

^{∗}*p*

*ˆ*

**⊗Q***p*

*.*

*For x = ϕ⊗ v in H*^{1}(**Q***p**, V*_{ββ}^{−}*) (with ϕ and v as above) and q in***Q**^{∗}*p** ⊗Q*ˆ

*p*, set

*x(q) = ϕ(q)· v and x(q)*

*f*=

*x(q), η*

*f*

*⊗ ω*

*g*

*α*

*⊗ ω*

*h*

*α*

*fgh*

*.*

**If (ξ, u) denotes one of the pairs (f , k), (g, l) and (h, m), deﬁne***D*˜**u***: H*^{1}(**Q***p***, V***f*)*−→ L*

to be the linear map which on *Y in H*^{1}(**Q***p***, V***f*) takes the value
*D*˜* u*(Y) = (

*−1)*

^{u}

^{o}2 (1*− p** ^{−1}*)

*·*

*y(p** ^{−1}*)

*f*

*− L*

^{an}

**ξ***· y(e(1))*

*f*

*.* (16)

Here *y = ρ**w** _{o}*(

*Y) in H*

^{1}(

**Q**

*p*

*, V*

_{ββ}

^{−}*) is the w*

*o*-specialisation of

*Y, u*

*o*

**= 2 if u = k and***u*

_{o}*L*

**= 1 if u = l, m, and**^{an}

_{ξ}*in L is the analytic*

**L -invariant of ξ, deﬁned by**L^{an}* ξ* =

*−2 · dlog a*

*p*

**(ξ)(u***o*)

*(where dlog a = a*^{}*/a for a in* *O*_{ξ}* ^{∗}*). We can ﬁnally state the aforementioned key
lemma.

**Lemma 2.2. For each local class***Y in H*^{1}(**Q***p***, V***f**) one has*
*L** f*(

*Y) (mod I*

^{2}) =

**u**

*D*˜* u*(

**Y) · (u − u***o*

*).*

**For each pair (u, v) of distinct elements of*** {k, l, m}, deﬁne (cf. Eq. (*13))

*D*˜

**u,u**

**(κ(f , g, h)) = ˜**D*(Y*

**u***) and*

**u***D*˜

**u,v**

**(κ(f , g, h)) = ˜**D*(Y*

**u***) + ˜*

**v***D*

*(Y*

**v**

**u***).*

Equation (14) and Lemma 2.2 give the following lemma (which implies that the
*derivatives ˜D*_{·}* (κ(f , g, h)) are independent of the choice of the classes*Y

*satisfying (13)).*

**u****Lemma 2.3. One has the following equality in***I*^{2}*/I*^{3}*.*
*L**p*^{αα}*(A, ) (mod* *I*^{3}) =

**u,v**

*D*˜**u,v****(κ(f , g, h))****· (u − u***o***)(v**− v*o*)

**2.3. An Exceptional Zero Formula** **`a la Rubin–Perrin–Riou**

**For a positive integer n and each 2n-tuple y = (y**_{1}*, . . . , y** _{2n}*) of elements of ˜

*H*

_{f}^{1}(

*denote by*

**Q, V )***R**p*^{αα}**(y) = Pf**

*⟪y**i**, y**j*⟫_{f gh}

1i,j2n

*∈ 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 –Nekov´aˇr p-adic height pairing*

˜*h*^{αα}* _{p}* : Sel(

**Q, V ) ⊗***L*Sel(

**Q, V ) −→ I**^{2}

*/I*

^{3}

*to be the bilinear form which on y⊗ y** ^{}* in Sel(

**Q, V )***takes the value*

^{⊗2}˜*h*^{αα}_{p}*(y⊗ y** ^{}*) =

*R*

*p*

^{αα}*(q*

*αα*

*, q*

*ββ*

*, y, y*

^{}*).*

The aim of this section is to prove the following proposition.

**Proposition 2.4. Up to sign, one has the equality**

˜*h*^{αα}_{p}*(κ(f, g**α**, h**α*)*⊗ ·) = c**A**· log**αα*(res*p*(*·)) · L**p*^{αα}*(A, ) (modI*^{3})
*of* *I*^{2}*/I*^{3}*-valued L-linear forms on Sel( Q, V ), where c*

*A*=

^{m}

^{p}*(1−p*

^{·}*)*

^{−1}*·ord*

*p*

*(q*

*A*)

*deg(℘** _{∞}*)

*.*