Daniel Delbourgo and Antonio Lei

New York Journal of Mathematics

New York J. Math.26(2020) 496–525.

Heegner cycles and congruences between anticyclotomic p-adic L-functions

over CM-extensions

Daniel Delbourgo and Antonio Lei

Abstract. Let E be a CM-field, and suppose thatf,gare two prim- itive Hilbert cusp forms over E⁺ of weight 2 satisfying a congruence moduloλ^r. Under appropriate hypotheses, we show that the complex L-values of f and g twisted by a ring class character over E, and di- vided by the motivic periods, also satisfy a congruence relation modλ^r (after removing some Euler factors). We treat both the even and odd cases for the sign in the functional equation – this generalizes classi- cal work of Vatsal [23] on congruences between elliptic modular forms twisted by Dirichlet characters. In the odd case, we also show that the p-adic logarithms of Heegner points attached tof and gsatisfy a con- gruence relation moduloλ^r, thus extending recent work of Kriz and Li [17] concerning elliptic modular forms.

Contents

1. Introduction and results for elliptic curves 496

2. The even case: Waldspurger’s formula 502

3. The odd case: p-adic Gross-Zagier formula 509 4. Logarithm maps and Coleman integration 518

References 523

1. Introduction and results for elliptic curves

Fix an odd prime p, and supposeA₁ and A₂ are two elliptic curves defined over Q. Provided that Re(s) > 3/2, their Hasse-Weil L-functions can be expressed in the form of Dirichlet series

L(A₁, s) =

∞

X

m=1

a_m(A₁)·m^−s and L(A₂, s) =

∞

X

m=1

a_m(A₂)·m^−s.

Received August 12, 2019.

2010 Mathematics Subject Classification. Primary 11F33; Secondary 11F41, 11G40, 11R23.

Key words and phrases. Iwasawa theory,p-adicL-functions, Hilbert modular forms.

ISSN 1076-9803/2020

496

Furthermore, both A₁ and A₂ are known to be modular by the deep work in [4] hence these L-functions have an analytic continuation to the whole complex plane.

Definition 1.1. We say the elliptic curves A₁ and A₂ are congruent mod p^r if one has a system ofp-adic congruences

a_m(A₁)≡a_m(A₂) ( mod p^r) for each m∈Nwith gcd(m, N₁N₂) = 1, whereN₁ denotes the conductor ofA₁/Q, andN₂ denotes the conductor of A₂/Q.

In the p-ordinary case, Vatsal proved that the Mazur-Tate-Teitelbaum [19] p-adic L-functions L^MTT_p (A₁) and L^MTT_p (A₂) are congruent modulo p^r·Zp

Γ^cyc

, where Γ^cyc= Gal(Q^cyc/Q) denotes the Galois group of the cy- clotomicZp-extension. Since thesep-adicL-functionsL^MTT_p (A_i) interpolate Dirichlet twists of the Hasse-Weil L-function L(A_i, ψ, s) at s= 1, one can view Vatsal’s result [23] as a statement about congruences between critical L-values divided by the real N´eron periods Ω⁺_A

i. It is therefore natural to ask if this result extends to number fields other thanQ?

To be more specific, letE be a CM-field that is also a solvable extension of Q, and consider the base-change of A₁ and A₂ to E. Throughout this article, we assume that the Leopoldt defect for E is zero. For a character χ : E^×\A^×_E → C^× of finite order, it is reasonable to expect a congruence between the twistedL-values

E_p(A₁/E, χ)· L(A₁/E, χ,1) (Ω⁺_A

1Ω⁻_A

1)^[E:Q]/2 and E_p(A₂/E, χ)· L(A₂/E, χ,1) (Ω⁺_A

2Ω⁻_A

2)^[E:Q]/2 (1.1) modulo p^r, for a suitable choice of factor E_p(A_i/E, χ) and N´eron periods Ω^±_A

i ∈C^×.

For example, if E is an imaginary quadratic field over which the prime p splits then Choi and Kim [6] have established a congruence for the two- variablep-adicL-function overEat cusp forms of different weight. Alterna- tively, ifE =Q(µ_pⁿ) andr = 1, then various types of congruence have been proved in [3, 9,10,22]. With the exception of [6], all these aforementioned congruences above are purely cyclotomic in their nature,so in this paper we shall deal exclusively with the anticyclotomic case.

Throughout we assume that A₁ and A₂ have good ordinary reduction at p, which means p - ap(A₁)·ap(A₂)·N1 ·N2 (although we expect that a version of our results should exist if one allows p to divide N₁·N₂, whilst still ensuring that p - a_p(A₁)·a_p(A₂)). We shall further suppose that the prime p splits inside E. Let ΓE = Gal(E∞/E) be the Galois group of the compositum, E∞ say, of all theZp-extensions of E, which can then be decomposed into ΓE = Γ^cyc_E ×Γ^anti_E where Γ^cyc_E (resp. Γ^anti_E ) is the Galois group of the cyclotomic (resp. full anti-cyclotomic) extension inE∞.

DANIEL DELBOURGO AND ANTONIO LEI

Building on earlier results in [15, 20], for each base Hecke character χ₀ the work of Disegni [11, Thm 4.3.4] allows the construction of a p-adicL- functionL_p A_i, χ₀

∈Zp

Γ_E [1/p

interpolating the special values given in Equation (1.1) at specialisationsχ=χ0·χ^†, as the character χ^†ranges over Hom Γ_E,Q^×p

tors. For a fixed topological generator γ₀ of Γ^cyc_E ∼= 1 +pZp, one can therefore expand each multi-variable p-adicL-function L_p A_i, χ₀ into a Taylor series of the form

L⁽⁰⁾_p A_i, χ0

+L⁽¹⁾_p A_i, χ0

·(γ0−1) +L⁽²⁾_p A_i, χ0

·(γ0−1)² 2 +· · · for either choice of i∈ {1,2}. It is therefore natural to ask whether:

Question. For every non-negative integer j, are the individual coeffi- cients L^(j)_p A₁, χ₀

and L^(j)_p A₂, χ₀

congruent to each other modulo p^r· Zp

Γ^anti_E

?

To make a precise statement, one divides the problem into three disjoint cases. For the rest of the Introduction, we assume that the base Hecke character χ0 is trivial on F^×\A^×_F, where F = E⁺ denotes the maximal totally real subfield ofE. We also assume that the primes ofF above pare unramified in the extension E/F. Let η_E/F be the quadratic character of E/F, and write Si for the set of F-places

S_i = ν : ν

∞ orη_E/F,ν cond(A_i/F)

=−1 .

Definition 1.2. (a) If the global root numbers satisfy 1/2,A_i/E, χ₀

= +1 for eachi∈ {1,2}and if #S1 ≡#S2≡0 (mod 2), then we call this the even case.

(b) If the global root numbers satisfy 1/2,A_i/E, χ₀

= −1 for each i∈ {1,2} and if #S1 ≡#S2≡1 (mod 2), then we naturally refer to this as theodd case.

(c) If 1/2,A₁/E, χ0

=− 1/2,A₂/E, χ0

or if #S1≡#S2+1 (mod 2), then we shall call this the mixed parity case.

In the first two cases (a) and (b), we extend Vatsal’s main result [23] as follows.

Theorem 1.3. In the even case, if the conductor of the Hecke character χ₀ is coprime to the O_E-ideal Q2

i=1cond(A_i/E), then L⁽⁰⁾_p,Σ0 A₁, χ₀

≡ L⁽⁰⁾_p,Σ0 A₂, χ₀

mod p^r+µ0 ·Zp

Γ^anti_E where µ0 ∈ Z is the largest value for which each L⁽⁰⁾p A_i, χ0

∈ p^µ0 · O_Cp

Γ^anti_E .

Note that in the above result, the subscript ‘_Σ⁰’ indicates that these L- functions have been stripped of their Euler factors at the finite primes con- tained in the set

Σ⁰ =

ν∈Spec(O_F) such thatν divides disc(E/F)·

2

Y

i=1

cond(A_i/F)

.

Theorem 1.4. In the odd case, if the conductor of the Hecke characterχ₀ is coprime to Q₂

i=1cond(A_i/E) and all the primes of F above p split in E, then

(i) L⁽⁰⁾_p,Σ0 A₁, χ0

=L⁽⁰⁾_p,Σ0 A₂, χ0

= 0, and (ii) E_0,Σ⁰(A₁)

E_1,Σ⁰(A₁) ·L⁽¹⁾_p,Σ0 A₁, χ0

is congruent to E_0,Σ⁰(A₂)

E_1,Σ⁰(A₂) ·L⁽¹⁾_p,Σ0 A₂, χ0

modulop^r+µ1·Zp

Γ^anti_E ,

where µ1 ∈ Z is the largest value for which each L⁽¹⁾p A_i, χ0

∈ p^µ1 · O_Cp

Γ^anti_E

, and E_k,Σ⁰(A_i) is an Iwasawa function interpolating the prod- uct of Euler factors Q

ν∈Σ⁰Lν(A_i/E, χ, k) at eachk∈Z.

Recall that a quaternion algebra B is called coherent if its ramification set Σ_B has even cardinality, and B is called incoherent if the set Σ_B has odd cardinality. In the case (c) of mixed parity, we can say nothing about mod p^r congruences as the curves A₁,A₂ cannot be parameterised by the same quaternion algebraB/F, otherwiseBwould have to be simultaneously coherent and incoherent!

There is also a third situation in which one can derivep-adic congruences.

Recall that ifE is an imaginary quadratic field, the work of Bertolini, Dar- mon and Prasanna [1] produces ap-adicL-functionL(A_i)∈Zp

Γ^anti_E [1/p]

interpolating critical values ofL A_i/E, χw, s

at character twistsχwof arith- metic weight w∈ N. Liu, Zhang and Zhang have extended this to general CM-fields E, constructing a p-adic L-function on Γ^anti_E interpolating the complex Rankin-Selberg L-function of each A_i, twisted by characters χw

of positive weight (see [18, Theorem 3.2.10]). The corresponding p-adic L-functionsL(A₁) andL(A₂) exist as elements of

LieA⁺_i ⊗_FM LieA⁻_i

⊗_FM D A_i, M F_p^lt in the specific notation of op. cit, where D A_i, M F_p^lt

is a certain (un- bounded) distribution algebra, andF^M = End(A₁)⊗_QF = End(A₂)⊗_QF.

Aside from the case where E is an imaginary quadratic field, it is not known precisely when L(A_i) arise fromp-bounded measures on Γ^anti_E . How- ever, if A_i has good ordinary reduction at p, one might reasonably expect L(A_i) to be an Iwasawa function for each i∈ {1,2}.

In [17], Kriz and Li studied values of the Bertolini-Darmon-Prasanna p-adic L-function via the p-adic logarithms of Heegner points attached to each A_i. In particular, they showed that up to appropriate Euler factors,

DANIEL DELBOURGO AND ANTONIO LEI

these logarithms satisfy a congruence relation via Coleman integration. We generalize their method to show that thep-adic logarithms of Heegner points (over ring class fields for a general CM-fieldE) attached toA₁andA₂satisfy a similar congruence relation. This allows us to compare special values of L(A_i), and deduce the following result.

Theorem 1.5. Suppose we are in the odd case, that the primes ofF above p split inE, and assuming thatboth L(A₁),L(A₂) are Iwasawa functions, then

L_Σ⁰(A₁) ≡ L_Σ⁰(A₂) mod p^r· L^\_A

1,A2

Γ^anti_E where L^\_A

1,A2 is the O_Cp-submodule generated by the values χ L(A₁) and χ L(A₂)

for χ = χ₀·χ^†, as the character χ^† ranges over the elements of Hom Γ^anti_E ,Q^×_p

.

For the remainder of the article, we will work in a more general setting than elliptic curves and solvable CM-fieldsE. We consider modular abelian varieties A? of GL2-type defined over a totally real field F, parameterised by a common definite quaternion algebra B/F.

Written below is a brief but non-exhaustive summary of our terminology.

• F is a totally real field, E will be a CM-extension of F, and D_E/F (resp. D_E) is the relative (resp. absolute) discriminant ofE;

• η_E/F is the quadratic character over F associated to the extension E/F;

• the symbol p will indicate a distinguished prime ideal of O_F lying overp, and we writePfor any primeO_E-ideal above it (pneeds not split inE);

• we fix embeddingsQ,→CandQ,→Qp, and an isomorphismC−→^∼ Cp under which the O_E-ideal P is sent into the maximal ideal of O_Cp;

• χ always denotes a unitary Hecke character over E (usually a fi- nite order character), which we identify with a Galois character Gal E^ab/E χ

→C^×;

• for an integral domain R, we shall write Rχ for the ring extension ofRwhich is obtained by adjoining all the values of the characterχ above;

• ifM is a module equipped with a Gal(E/E)-action, M_(χ) =M⊗χ denotes the same underlying moduleM but with its Galois action twisted byχ;

• E^cyc indicates the cyclotomic Zp-extension ofE, so that the cyclo- tomic character κ_cy maps Γ^cyc_E := Gal E^cyc/E

onto an open sub- group of 1 +pZp;

• fandgdenote primitive Hilbert cusp forms overF of parallel weight two, Nebentypus character ω, and levels Nf CO_F and Ng CO_F respectively;

• associated to both f and g are their modular abelian varieties of GL2-type, Af and Ag, which are defined over the same totally real number fieldF;

• K=Qp C(n,f), C(n,g)

nCO_F

is the finite extension of Qp gen- erated by the Fourier coefficients of f,g, and λ denotes a local pa- rameter inO_K;

• for an abelian group M, its (finite) adelisation is given by Mc = M ⊗_Z Zb where Zb := lim←−mZ/mZ ∼= Q

primeslZl is the profinite completion ofZ.

For example, if E is a solvable extension of Q and A₁,A₂ are two elliptic curves that are congruent modulo λ^r=p^r, one can take f =BC^F_Q(f₁) and g = BC^F_Q(f2) as their base-changes with each fi ∈ S₂(Γ0(Ni)), so that A_f ∼=A_1/F andA_g∼=A_2/F.

We shall now describe a generalisation of Definition1.1to modular abelian varieties over F. Let Ne denote the O_F-ideal lcm N_f, N_g,Q²

where Q = Q

ν|N_fNgν.For a primeq∈Spec(O_F),T(q) denotes theq-th Hecke operator ifqis coprime to the level of the HMF, whilstU(q) is theq-th Hecke operator ifqdivides the level of the HMF (see for example [20, Chapter 4,§1.3]). We will also require the diamond operators

m

, as well as the degeneracy maps V(m) which act on the Fourier expansions by sendingC(n,h)7→C(nm⁻¹,h) for either choice of formh∈ {f,g}.

Definition 1.6. TheNe-depletion off is the Hilbert cusp formef given by f

Y

q|N_g,q-Nf

1− T(q)◦ V(q) +N_{F /Q}(q) q

◦ V(q²) Y

q||N_f

(1− U(q)◦ V(q)). Similarly, theNe-depletiongeof g is defined by the formula

g

Y

q|N_f,q-Ng

1− T(q)◦ V(q) +N_{F /Q}(q) q

◦ V(q²) Y

q||Ng

(1− U(q)◦ V(q)). In particular, ef,eg ∈ S₂ N , ωe

with L(ef, s) = L_NfNg(f, s) and L(eg, s) = LNfNg(g, s).

Hypothesis. (f ≡ g (λ^r)) There is an identity of depleted Hilbert cusp forms

ef = ge + λ^r·X

j

cj·hj

with each scalar termcj ∈ O_K, and where thehj’s denote normalised eigen- forms of parallel weight two, level dividing into Ne, and with Nebentypus characterω.

To reassure the reader, if A₁ and A₂ are two elliptic curves as before that are congruent modulo p^r, then their base-changes f = BC^F_Q(f1) and

DANIEL DELBOURGO AND ANTONIO LEI

g=BC^F_Q(f₂) automatically satisfy Hypothesis (f ≡g(λ^r))upon choosing the uniformizerλ=p. Indeed to verify this claim, we first observe that

f˜1 = X

gcd(m,N1N2)=1

am(A₁)·q^m and ˜f2 = X

gcd(m,N1N2)=1

am(A₂)·q^m satisfy ˜f₁−f˜₂=p^r·f^\for somef^\ ∈ S₂

Γ₀ lcm N₁, N₂,Q

l|N₁N2l²

∩Z[[q]].

However, this latter module has an integral basis consisting of elements of the type h_j

V(d) whereh_j is a newform of level C_j, and d≥1 ranges over integers such that dC_j divides the common level lcm N₁, N₂,Q

l|N1N2l²

; one can therefore express

f˜₁ = ˜f₂ + p^r·X

j,d

c^(d)_j ·h_j

V(d) where the scalarsc^(d)_j ∈Z. After base-changing each of the cusp forms ˜f₁, ˜f₂ and the h_j|V(d)’s from Q to F, we respectively obtain the HMFs ef, eg and the hj’s in Hypothesis (f ≡g(λ^r)).

The proof of our main results (Theorems 1.3, 1.4 and 1.5) makes heavy use of three recent spectacular but rather technical formulae, due to various authors. To treat the even case, we use a version of the Waldspurger for- mula from [5, 26].To treat the odd case, we apply the p-adic Gross-Zagier formula in [11,12]. Lastly, to prove congruences for the Liu-Zhang-Zhang p-adicL-functions, we use the connection between its special values and the logarithms of Heegner cycles [1, 18]. The demonstrations themselves are written up in Sections 2,3, and 4, respectively.

2. The even case: Waldspurger’s formula

Let B be a totally definite quaternion algebra defined over the totally real fieldF. We suppose thatπf andπgare two cuspidal automorphic representa- tions ofB^×_AF, associated to the Hilbert modular formsf and g respectively under the Jacquet-Langlands correspondence on GL2/F, with a common central characterω onA^×_F,fin. Let us also consider a fixed finite order Hecke character χ defined on E^×\A^×_E, corresponding to a weight one theta-series automorphic representation πχ of B^×_AF.

Hypothesis. (Even)The productω·χ A^×F

is trivial, the three finite sets S_Nf =

ν : ν|∞orη_E/F,ν(N_f) =−1 S_Ng =

ν : ν|∞ orη_E/F,ν(Ng) =−1 S_Ne =

ν : ν|∞or η_E/F,ν(N) =e −1 each have evencardinality, and for all places ν of F

1/2, πf,ν, πχ,ν

= 1/2, πg,ν, πχ,ν

= χν(−1)·η_E/F,ν(−1)·ξ Bν

where the sign ξ Bν

= −1 if Bν is a division algebra, and ξ Bν

= +1 otherwise.

Here we have written 1/2, π?,ν, πχ,ν

for the local root number associ- ated to the complex tensor product L-series L s, π? ×πχ

, for each choice of ? ∈ {f,g}. The above hypothesis then implies that both the global root numbers 1/2, π_f, πχ

and 1/2, πg, πχ

in the Rankin L-functions are equal to +1, and there is an F-embedding of E into B that identifiesE^× with a sub-torus inB^×.

Proposition 2.1. If the Hypotheses (f ≡ g (λ^r)) and (Even) both hold, and if f_χ := cond(χ) is coprime to N_fN_g · O_E, there is a congruence of p-integral elements

q

|D_E| · ||c_χ||²·L_Σ(1/2, π_f ×π_χ) Ω^aut,(0)_∞,K (f)

≡

q

|D_E| · ||c_χ||²·L_Σ(1/2, π_g×π_χ) Ω^aut,(0)_∞,K (g)

mod λ^rO_K,χ where c_χis the largestO_F-ideal so that χis trivial onQ

ν-cχO^×_E,ν×Q

ν|c 1 + cO_E,ν

, ||c_χ|| denotes the norm N_F/Q(c_χ), the finite set Σ consists of the places ofF dividingNf·Ng·D_E/F·c_χ· ∞, andΩ^aut,(0)_∞,K (?)is the automorphic period (see Equation (2.1)) associated to each ?∈ {f,g}.

Proof. The key ingredient is the generalised Waldspurger formula in [5, 26]. In particular, we shall take as our common level structure Ne :=

lcm N_f, N_g,Q²

where Q = Q

ν|N_fNgν CO_F. Firstly, one defines a finite subset of Spec(O_F) by

Σ1 :=

n ν

eN where η_E/F(ν) =−1 and ord_ν(cχ)<ordν Neo and next constructs a pair ofO_F-ideals via

c1 := Y

ν|cχ, ν6∈Σ1

ν^ordν(cχ) and N1:= Y

ν|N ,e ν6∈Σ1

ν^ordν(Ne).

Now letRbe an admissibleO_F-order for the pairs (π_f, χ) and (π_g, χ) in the sense of [5, Sect 1], so that in addition Rhas discriminantNe and R ∩E= O_F +c1O_E. We shall also fix a compact open subgroup U =Q

νUν ⊂B^×_AF such thatUν =R^×_ν at all finite placesνofF, and moreover if the placeν|N₁ thenBν must be split. The (zero-dimensional) Shimura varietyX=X_U(B) is then defined by

X_U(B) := B^× Bb^×

Ub

and letg1, . . . , gn∈ Bb^× be a complete set of representatives for X, so that [gi]∈X.

DANIEL DELBOURGO AND ANTONIO LEI

If Z[X] denotes the free Z-module consisting of formal sums P

ia_i[g_i], then there is a height pairing

−,−

X : Z[X]×Z[X] → C[X] from [14], sending each pair P

ia_i[g_i],P

ib_i[g_i]

to the element P

ia_ib_iw_i with w_i =

# B^×∩g_iRb^×g_i⁻¹

O^×_F. There exists a canonical direct sum decomposition Z[X] = M

c∈C₊

Z Xc

where Xc is the preimage of c ∈ C+ := F₊^× Fb^×

Ob^×_F under the natural surjection X_U(B) = B^×

Bb^×

Ub F₊^× Fb^×

Ob_F^×. One may also consider the submodules Z

X_c0

⊂ Z X_c

containing degree zero classes, and set Z[X]⁰ :=L

c∈C+Z X_c0

.

For each choice of ? ∈ {f,g}, let V(π?, χ) indicate the space of ‘test vectors’ in the sense of [5, Defn 3.6]. Because we are working at level Ne rather than level N?, it is no longer true in general that V(π?, χ) is one- dimensional overC; in fact

dim_C V(π?, χ)

= Y

ν|Ne

1 + ordν Ne

−ordν N?

(of course, if Ne =N_f =N_g then bothV(π_f, χ) andV(π_g, χ) correspond to C-lines). There are injections V(π_?, χ) ,→ Z[X]⁰⊗C obtained from Φ 7→

P

iΦ [gi]

w⁻¹_i [gi], which respect the natural action of the Hecke algebra on both C-vector spaces.

Remarks. (a) Considering the Ne-depletions ef,eg ∈ S₂ N , ωe

in Definition 1.6, the images of C·ef and C·eg insideZ[X]⁰⊗Cdefine unique dimension one subspaces.

(b) The action of the Hecke operatorsT(n) onC·ef (resp. C·g) coincidee with their action on C·f (resp. C·g) if n is coprime to Ne, whilst the U(q)-operators annihilate both of the depleted linesC·ef andC·eg whenever gcd q,Ne

6=O_F.

(c) In the notation of [5, Thm 1.9], we can take as test vectors anyf₁⁰, f₂⁰ ∈ C·ef(resp. f₁⁰, f₂⁰ ∈C·eg) viewed insideZ[X]⁰⊗C, and then apply the variation of Waldspurger’s formula tof₁⁰ ∈V(π_?, χ),f₂⁰ ∈V(π_?^∨, χ⁻¹) for ?=f (resp.

?=g).

We now relate the Rankin L-function to twisted CM-cycles living in O_K,χ[X]⁰. Recall the fixed embedding E ,→ B induces a group homo- morphism Pic(O_cχ) → X sending t 7→ x_t where O_cχ denotes the order O_F +c_χO_E, with c_χCO_F indicating the largest O_F-ideal such that χ be- comes trivial on Q

ν-cχO^×_E,ν ×Q

ν|cχ 1 +c_χO_E,ν

. One defines a pair of

(Ne-depleted) CM-cycles by Pe_χ(f) := X

t∈Pic(Ocχ)

χ⁻¹(t)·ef(x_t) and Pe_χ(g) := X

t∈Pic(Ocχ)

χ⁻¹(t)·eg(x_t) which `a priori lie inside O_K,χ[X]. However, if χ is a non-trivial character then P

t∈Pic(O_cχ)χ⁻¹(t) = 0, so clearly Pe_χ(f),Pe_χ(g) ∈ O_K,χ[X]⁰ both have degree zero¹.

We initially focus on the HMF f, and its depleted CM-cycle Peχ(f) ∈ O_K,χ[X]⁰. Viewing f as a holomorphic function φ_f : H^[F:Q] → C, let us denote by hφ_f, φfi_Pet the Petersson self-product of φf, computed using the invariant measure induced on

PGL2(F)

H^[F:Q]×PGL2(AF,fin) U0 Ne

from the standard hyperbolic volume dxdy/y² on the extended upper half- plane. Applying Waldspurger’s formula in the format of [5, Thm 1.9] and [26, Thm 7.1],

L_Σ⁰ 1/2, πf ×πχ

=2^−#ΣD ·(8π²)^[F:Q]·¹₂Vol X_U

0(N)e

· hφ_f, φ_fi_Pet u²p

|D_E| · ||c_χ||²

·

Pe_χ(f),Pe_χ(f)

X,

where Σ⁰ consists of those primes dividing gcd N_f ·N_g,c_χ·D_E/F

· ∞ such that ifν||Ne thenν -D_E/F, whilst ΣD denotes the set of primes ofF dividing gcd N , De _E/F

.

Furthermore, we claim thatu:= #Ker Pic(O_F)→Pic(O_cχ)

×[O^×_c

χ :O_F^×] is always a p-adic unit. To see why this is so, observe that Ker Pic(O_F)→ Pic(O_cχ)

is either 1 or 2 by [24, Theorem 10.3]. Writing W_E for the roots of unity of E, then [W_EO_F^×:O^×_F] is coprime top as the primes of F above p are unramified in E. Moreover [O^×_E : W_EO^×_F] is either 1 or 2 by [24, Theorem 4.12], consequently both [O_E^× : O^×_F] and hence [O^×_c

χ : O^×_F] are coprime to p.

It is also easy to check that there is an inclusion of sets of finite places Σ⁰ ,→Σ. If we now attach a (complex) automorphic period to f overK by setting

Ω^aut,(0)_∞,K (f) := (8π²)^[F:Q]·Vol X_U

0(Ne)

· hφ_f, φfi_Pet (2.1) then rearranging Waldspurger’s formula yields the equality

Pe_χ(f),Pe_χ(f)

X = u²·2^#ΣD+1× q

|D_E| · ||c_χ||²·L_Σ⁰(1/2, π_f ×π_χ) Ω^aut,(0)_∞,K (f)

.

1If χ is trivial then one takes instead

Peχ(f)−deg(Peχ)·ξ ,

Peχ(g)−deg(Peχ)·ξ

∈ Pic(X)⊗C, whereξ denotes the absolute Hodge class [26, Eqn (6.8)] which has degree one on each component.

DANIEL DELBOURGO AND ANTONIO LEI

An entirely similar argument, applied to g and Pe_χ(g) ∈ O_K,χ[X]⁰, estab- lishes that

Peχ(g),Peχ(g)

X = u²·2^#ΣD+1× q

|D_E| · ||c_χ||²·LΣ⁰(1/2, πg×πχ) Ω^aut,(0)_∞,K (g)

.

Crucially, for each eigenform h lying in the (f,g)-isotypic component, the depleted cycles Peχ(h) belongs to the dual lattice O_K,χ[X]⁰∨

under the pairing [−,−]_X. Using the O_K,χ-bilinearity of this pairing, it therefore suf- fices to show that

Pe_χ(f) = Pe_χ(g) +λ^r·Q for someQ∈ O_K,χ[X]⁰ because if this is indeed the case, then as a direct corollary,

Peχ(f),Peχ(f)

X =

Peχ(g),Peχ(g)

X+λ^r× 2·

Peχ(g), Q

X+λ^r· Q, Q

X

,

so that

Pe_χ(f),Pe_χ(f)

X ≡

Pe_χ(g),Pe_χ(g)

X modλ^r.

We now exploit the relation between ef and eg given in Hypothesis (f ≡ g(λ^r)), observing that this relation is preserved when we apply the Jacquet- Langlands correspondence and shift to the quaternion algebra B. One thereby deduces that

Pe_χ(f) = X

t∈Pic(O_cχ)

χ⁻¹(t)·ef(x_t)

= X

t∈Pic(O_cχ)

χ⁻¹(t)·



eg(x_t) +λ^r·X

j

c_j·h_j(x_t)





= Peχ(g) + λ^r·X

j

cj · X

t∈Pic(Ocχ)

χ⁻¹(t)·hj(xt), and setting Q=P

jc_j·P

tχ⁻¹(t)·h_j(x_t)∈ O_K,χ[X]⁰, the result follows at

once.

LetE∞ denote the maximalZp-power extension of E unramified outside p, so Γ_E := Gal(E∞/E)∼=Z^1+[Fp ^:Q]+δwhereδ ≥0 is the defect in Leopoldt’s conjecture. If we choose a base characterχ0 such thatω·χ0

A^×_F is trivial, it follows that the family of characters

χ₀·χ^†

χ^†: Γ^anti_E →µ_p^∞ also satisfies Hypothesis(Even). Henceforth we defineρ₀:= Ind^F_E(χ₀) :G_F →GL₂(O_χ) which is a two-dimensional Artin representation, as are ρ_χ := Ind^F_E(χ) for every character χ = χ0 ·χ^† as above. For the rest of this section, we shall assume that all the primes of F lying above p split in the CM-extension E.

Remarks. (a) Building on earlier work of Hida and Panchishkin [15,20] for the cyclotomic deformation, Disegni [12] has attached p-adicL-functions to GL2×GL2 interpolating the Rankin productL-functionsL(s, π_f ×πχ) and

L(s, π_g×π_χ) at the critical point s= 1/2: since we are taking the unita- rizations, let us identify theseL-values (respectively) withL f⊗Ind^F_E(χ), s and L g⊗Ind^F_E(χ), s

ats= 1.

(b) Let Γ^anti_E denote the Galois group of the anticyclotomic extension of E inside E∞, which by definition is the (−1)-eigenspace of the com- plex conjugation c ∈ Gal(E/F) inside ΓE. For a topological generator γ₀ of Γ^cyc_E and the particular choice ? = f say, one expands the 1 + [F :Q]

-variable Disegni-Hida-Panchishkin p-adic L-function L_p,Σ f, ρ₀

∈ O_K

Γ_E ][1/λ

into a Taylor series of the form Lp,Σ f, ρ0

= L⁽⁰⁾_p,Σ f, ρ0

+L⁽¹⁾_p,Σ f, ρ0

·(γ₀−1)+1

2 L⁽²⁾_p,Σ f, ρ0

·(γ₀−1)² +· · · where L⁽ⁱ⁾_p,Σ f, ρ0

∈ OK Γ^anti_E

[1/λ

under the decomposition ΓE = Γ^cyc_E × Γ^anti_E . Here the subscript ‘_Σ’ above indicates that the p-adic L-function Lp,Σ f, ρ0

has been completely stripped² of its Euler factors at those finite placesν ∈Σ, ν -p.

(c) Note also the condition (Even) implies either L⁽⁰⁾_p,Σ f, ρ0

6= 0, or instead that L⁽⁰⁾_p,Σ f, ρ₀

=L⁽¹⁾_p,Σ f, ρ₀

= 0, because the global root number 1/2, π_f, π_χ

is equal to +1 under our assumptions.

Ifχ=χ₀·χ^† whereχ^† is anticyclotomic, then χ^† L_p,Σ f, ρ₀

=χ^† L⁽⁰⁾_p,Σ(f, ρ₀)

asχ^†(γ₀−1) = 0. The exact interpolation rule from [11, Thm 4.3.4] states that

χ^†

L_p,Σ f, ρ0

=

χ d^(p)_F

·G χ

·q N_F/

Q D_E/F · N_E/F(f_χ)

·χ(D_E/F) Q

p|pαp(f)^{ordp(NE/F(fχ))}

× Y

p|p

Y

P|p

1−χ(P) αp(f)

× LΣ\{p|p} f⊗Ind^F_E(χ),1 Ω^aut,(0)_∞,K (f)

.

(2.2) An analogous formula holds for the value of L_p,Σ g, ρ₀

at each twist χ = χ₀·χ^†.

Theorem 2.2. Assuming Hypothesis (f ≡ g (λ^r)), and that Hypothesis (Even)for the base characterχ₀ holds true with the conductor ofχ₀ coprime

2We have deliberately removed the Euler factors fromLp,Σ f, ρ0

at the finite places in Σ, so that we can obtain a congruence moduloλ^r; it follows that thep-adicL-functions we are considering correspond to Σ-imprimitive versions of the Disegni-Hida-Panchishkin construction.

DANIEL DELBOURGO AND ANTONIO LEI

to N_fN_g· O_E, there is a congruence of p-adic L-functions L⁽⁰⁾_p,Σ f, ρ₀

≡ L⁽⁰⁾_p,Σ g, ρ₀

modλ^r· O_K Γ^anti_E

.

If either λ^r - ^{(ρ0,0)·LΣ(f,ρ0)}

Ω^aut,(0)_∞,K (f) or λ^r - ^{(ρ0,0)·LΣ(g,ρ0)}

Ω^aut,(0)_∞,K (g) with (ρ0, s) the -factor for ρ0, then both sides of this anticyclotomic congruence must be non-trivial moduloλ^r.

Proof. To establish this p-adic congruence, clearly it is sufficient to prove thatχ^†

L⁽⁰⁾_p,Σ f, ρ₀

andχ^†

L⁽⁰⁾_p,Σ g, ρ₀

are congruent moduloλ^r, atχ= χ₀ ·χ^† where χ^† ranges over finite order characters on the anticyclotomic component Γ^anti_E . Because

G χ)

−1 p =

N_E/Q(f_χ)

−1/2

p =

||c_χ||

−1/2

p , the ratio of algebraic numbers

rχ :=

χ d^(p)_F

·G χ

·q

N_F/Q D_E/F · N_E/F(fχ)

·χ(D_E/F) p|D_E| · ||c_χ||²

is a p-adic unit, independent of choosing ? ∈ {f,g} but dependent on χ obviously. From the interpolation in Equation (2.2), and after replacing the Hecke character χ by its dual χ, one can reinterpret the congruence in Proposition2.1as the statement:

Q

p|pα_p(f)^{ordp(NE/F(fχ))} Q

p|p

Q

P|p

1−_α^χ(P)

p(f)

×r⁻¹_χ ·χ^†

L⁽⁰⁾_p,Σ f, ρ₀

≡ Q

p|pαp(g)^{ordp(NE/F(fχ))} Q

p|p

Q

P|p

1−_α^χ(P)

p(g)

×r⁻¹_χ ·χ^†

L⁽⁰⁾_p,Σ g, ρ₀

mod λ^r· O_K,χ. However, for ?∈ {f,g}, we can identify αp(?) with the eigenvalue of Frobp

acting on the maximal unramified quotient of Ta_p(A_?) as a G_Fp-module, in which case α_p(f) ≡ α_p(g) modλ^r· OK,χ since Ta_p(A_f)

λ^r ∼= Tap(A_g) λ^r asG_Fp-modules. Consequently, the reciprocals of these extra terms satisfy



 Q

p|pαp(f)^{ordp(NE/F(fχ))} Q

p|p

Q

P|p

1−_α^χ(P)

p(f)





−1

≡



 Q

p|pα_p(g)^{ordp(NE/F(fχ))} Q

p|p

Q

P|p

1−_α^χ(P)

p(g)





−1

mod λ^r· O_K,χ, which completes the proof of the main congruence.

Finally, identifying G χ₀

·q

N_{F /Q} D_E/F · N_E/F(fχ0)

with the factor (ρ₀,0), the non-triviality of eitherχ^†

L⁽⁰⁾_p,Σ f, ρ₀

orχ^†

L⁽⁰⁾_p,Σ g, ρ₀ mod

λ^r at χ^† = 1, directly implies L⁽⁰⁾_p,Σ f, ρ0

≡ L⁽⁰⁾_p,Σ g, ρ0

6≡ 0 modλ^r · O_K

Γ^anti_E

.

3. The odd case: p-adic Gross-Zagier formula We now treat the opposite situation, where

1/2, πf, πχ

= 1/2, πg, πχ

=−1.

In particularL⁽⁰⁾_p,Σ ?, ρ₀

is identically zero, whence L_p,Σ ?, ρ₀

γ0−1 = L⁽¹⁾_p,Σ ?, ρ₀ + 1

2 L⁽²⁾_p,Σ ?, ρ₀

·(γ₀−1) + O (γ₀−1)² so that χ^†

L⁽¹⁾_p,Σ ?, ρ0

= log_pκcy(γ0)−1

·χ^† _dκs−1

cy Lp,Σ(?,ρ0) ds

s=1

for?∈ {f,g}. Therefore, our goal is to establish a congruence moduloλ^r·log_pκcy(γ0) between χ^†_dκ^s−1

cy Lp,Σ(f,ρ0) ds

s=1

and χ^†_dκ^s−1

cy Lp,Σ(g,ρ0) ds

s=1

under Hypoth- esis (f ≡ g (λ^r)). Again B denotes a totally definite quaternion algebra overF, with the property that the automorphic representations πf and πg

are both parameterised byB^×_AF. Likewiseχ:E^×\A^×_E →C^× will be a fixed Hecke character of finite order, as before.

Hypothesis. (Odd)The productω·χ

A^×F is trivial, the three finite sets S_Nf =

ν : ν|∞orη_E/F,ν(N_f) =−1 SNg =

ν : ν|∞ orη_E/F,ν(Ng) =−1 SNe =

ν : ν|∞or η_E/F,ν(N) =e −1 each have oddcardinality, and for all places ν ofF

1/2, π_f,ν, π_χ,ν

= 1/2, π_g,ν, π_χ,ν

= χ_ν(−1)·η_E/F,ν(−1)·ξ Bν

. The above hypothesis implies that both the global root numbers 1/2, π_f, π_χ

and 1/2, π_g, π_χ

in the RankinL-functions are equal to

−1, and that the quaternion algebra B is incoherent. Henceforth, we shall further assume that all primes of F above p split inside the extension E.

As explained in [12, Sect 1.1], one can interpret the modular parameteri- zations of the abelian varietiesA_f andAg in terms of Shimura curves. For a compact open subgroupU ofB^×_AF, the complex points of the algebraic curve X_U are given by

XU(C) = B\H^±×Bb^×/U.

In fact, there exists an infinite tower of Shimura curves

X_{U U} indexed by the compact open subgroups U ⊂B^×_AF, and we shall set X(B) := lim←−UX_U.

DANIEL DELBOURGO AND ANTONIO LEI

The canonical Hodge class ξ_U ∈ Pic X_U

⊗Q which has degree one on each component induces an embeddingXU

ι_ξU

,→ JacXU. Because the HMFs f and g are parameterised by B^×_AF, the End⁰(A_?)-vector spaces

lim−→

U

Hom⁰_ξU JacXU, Af

and lim−→

U

Hom⁰_ξU JacXU, Ag

are both non-empty; let πA? ∈lim−→UHom⁰ JacXU, A?

be the smooth irre- ducible representation of B^×_AF corresponding to π?, for each choice of cusp form ?∈ {f,g}. Taking Uf =U0(Nf),Ug =U0(Ng) and Ue =U0 Ne

, there exists a factorisation

X(B) →^∼ lim←−

U

X_U −→^ιξ lim←−

U

JacX_U JacX

Ue

JacX_Uf A_f

JacXUg Ag (3.1) and the top sequence of maps yieldsπAf ◦ιξ, whilst the bottom maps yield π_Ag◦ι_ξ.

Before we state our main result below, for each choice of HMF?∈ {f,g}

let us introduce the ratio of Euler factors E

Ne(?, χ) := Y

q|Ne

L_q(?⊗Ind^F_E(χ), s−1) L_q(?⊗Ind^F_E(χ), s)

s=1

.

Whilst the denominator can never vanish, the numerator can sometimes vanish (for example, ifq||N_?andC(q, ?) =χ(Q) for some placeQofE lying above q). Furthermore, these algebraic values can be interpolated by the ratio of two elements of O_K,χ

Γ^anti_E

, denoted by E_0,

Ne(?) and E_1,

Ne(?), so that

χ^† E_0,

Ne(?) χ^† E_1,

Ne(?) = Y

q|Ne

Lq(?⊗Ind^F_E(χ),0) L_q(?⊗Ind^F_E(χ),1) for all characters χ=χ0·χ^† in the standard formulation above.

Theorem 3.1. Assume Hypothesis (f ≡ g (λ^r)), and that Hypothesis (Odd) for the base character χ0 holds true with the conductor of χ0 co- prime to N_fN_g· O_E. Then one has the twin relations

(i) L⁽⁰⁾_p,Σ f, ρ₀

=L⁽⁰⁾_p,Σ g, ρ₀

= 0 , and (ii) E_0,

Ne(f) E_1,

Ne(f)L⁽¹⁾_p,Σ f, ρ₀

≡ E_0,

Ne(g) E_1,

Ne(g)L⁽¹⁾_p,Σ g, ρ₀

modλ^r−r0+δE·log_pκ_cy(γ₀).

Here r₀ := 2·P

P|pord_λ

#Ae_? O_E/P

with ? ∈ {f,g}³, while δ_E ∈ Q^× depends on the CM-extension E/F but does not depend on either f,g, nor on the prime p.

3Note thatAef OE/P

[λ^r]∼=Aeg OE/P

[λ^r] since we are assuming(f≡g(λ^r))holds here.