p -adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou
pp -ADIC GROSS–ZAGIER FORMULA AT CRITICAL SLOPE ANDA CONJECTURE OF PERRIN-RIOU K ˆAZIM B ¨UY ¨UKBODUK, ROBERT POLLACK, AND SHU SASAKI
Abstract.
Let p be an odd prime. Given an imaginary quadratic field K = Q ( √− D K ) where p splits with D K >
3, and a p -ordinary newform f ∈ S k (Γ ( N )) such that N verifies the Heegner hypothesis relative to K , weprove a p -adic Gross–Zagier formula for the critical slope p -stabilization of f (assuming that it is non- θ -critical). In the particular case when f = f A isthe newform of weight 2 associated to an elliptic curve A that has good ordi-nary reduction at p , this allows us to verify a conjecture of Perrin-Riou. The p -adic Gross–Zagier formula we prove has applications also towards the Birchand Swinnerton-Dyer formula for elliptic curves of analytic rank one. Contents
1. Introduction 21.1. Results 3Acknowledgements 102. Notation and Set up 102.1. Modular curves, Hecke correspondences and the weight space 122.2. The Coleman family ( f , β ) 123. Selmer complexes and p -adic heights in families 133.1. Cohomology of ( ϕ, Γ cyc )-modules 133.2. Selmer complexes 143.3. A -adic cyclotomic height pairings 153.4. Specializations and comparison with Nekov´aˇr’s heights 163.5. Universal Heegner points 194. p -adic L -functions over the imaginary quadratic field K p -adic L -functions 214.2. Na¨ıve p -adic L -functions over K Key words and phrases.
Heegner cycles, Families of p -adic modular forms, Birch andSwinnerton-Dyer conjecture. a r X i v : . [ m a t h . N T ] O c t K. B¨UY¨UKBODUK, ROBERT POLLACK, AND SHU SASAKI
5. Proofs of Theorem 1.1.1 and Corollary 1.1.2 255.1. p -adic Gross–Zagier formula for non-ordinary eigenforms at non-criticalslope 265.2. A -adic Gross–Zagier formula 265.3. Proof of Theorem 1.1.1 275.4. Proof of Corollary 1.1.2 276. Applications 286.1. Perrin-Riou’s big logarithm map and p -adic L -functions 286.2. Birch and Swinnerton-Dyer formula for analytic rank one (Proof ofTheorem 1.1.8) 296.3. Proof of Theorem 1.1.5 (Perrin-Riou’s conjecture) 31References 331. Introduction
Fix forever an odd prime p as well as embeddings ι ∞ : Q (cid:44) → C and ι p : Q (cid:44) → Q p .Let N be an integer coprime to p . We let v p denote the valuation on Q p , normalizedso that v p ( p ) = 1.Let f = ∞ (cid:80) n =1 a n q n ∈ S k (Γ ( N )) be a newform of even weight k ≥ N ≥
3. Let K f := ι − ∞ ( Q ( · · · , a n , · · · )) denote the Hecke field of f and P the primeof K f induced by the embedding ι p . Let E denote an extension of Q p that contains ι p ( K f ). We shall assume that v p ( ι p ( a p )) = 0, namely that f is P -ordinary. Let α, β ∈ Q denote the roots of the Hecke polynomial X − ι − ∞ ( a p ) X + p k − of f at p . We assume that E is large enough to contain both ι p ( α ) and ι p ( β ). Since weassume that f is P -ordinary, precisely one of ι p ( α ) and ι p ( β ) (say, without loss ofgenerality, ι p ( α )) is a p -adic unit. Then v p ( ι p ( β )) = k −
1. To ease our notation,we will omit ι p and ι ∞ from our notation unless there is a danger of confusion.The p -stabilization f α ∈ S k (Γ ( N p )) of f is called the ordinary stabilization and f β is called the critical-slope p -stabilization. We shall assume throughout that f β is not θ -critical (in the sense of Definition 2.12 in [Bel12]).Our main goal in the current article is to prove a p -adic Gross–Zagier formula forthe critical-slope p -stabilization f β . This is Theorem 1.1.1. In the particular casewhen f has weight 2 and it is associated to an elliptic curve A / Q , this result allowsus to prove a conjecture of Perrin-Riou. This is recorded below as Theorem 1.1.5;it can be also translated into the statement of Theorem 1.1.4, which is an explicitconstruction of a point of infinite order in A ( Q ) in terms of the two p -adic L -functions associated to f A (under the assumption that A has analytic rank one, ofcourse). As a by product of Theorem 1.1.1, we may also deduce that at least one ofthe two p -adic height pairings associated to A is non-degenerate. This fact yields -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 3 the proof of the p -part of the Birch and Swinnerton-Dyer formula for A ; this isTheorem 1.1.8 below.Before we discuss these results in detail, we will introduce more notation. Let S denote the set consisting of all rational primes dividing N p together with thearchimedean place. We let W f denote Deligne’s (cohomological) p -adic represen-tation associated to f (so that the Hodge–Tate weights of W f are (1 − k, V f = W f ( k/ V f the central critical twist of W f . Both W f and V f areunramified outside S and they are crystalline at p .Let D cris ( V f ) denote the crystalline Dieudonn´e module and D † rig ( V f ) Fontaine’s(´etale) ( ϕ, Γ cyc )-module associated to V f | G Q p . We let D α , D β denote the eigenspacesof D cris ( V f ) for the action of the crystalline Frobenius ϕ ; so that ϕ | D α = p − k/ α and ϕ | D β = p − k/ β .Let K = Q ( √− D K ) be an imaginary quadratic field and let H ( K, V f ) denotethe Bloch-Kato Selmer group associated to V f . For each λ ∈ { α, β } the submodule D λ ⊂ D cris ( V f ) defines a canonical splitting of the Hodge filtration on D cris ( V f ),namely that we have D cris ( V f ) = D λ ⊕ Fil D cris ( V f ) . as E -vector spaces. Note that our assumption that f β is non- θ -critical is necessaryto ensure this splitting when λ = β (see [Bel12, Proposition 2.11(iv)]). We let h Nek λ,K : H ( K, V f ) × H ( K, V f ) −→ E denote the p -adic height pairing that Nekov´aˇr in [Nek93] has associated to thissplitting.Suppose that the prime p splits in K and write ( p ) = pp c . Assume also that K verifies the Heegner hypothesis relative to N . Let (cid:15) K denote the quadratic Dirichletcharacter associated to K/ Q . The Heegner hypothesis ensures that ord s = k L ( f /K , s )is odd and there exists a Heegner cycle z f ∈ H ( K, V f ).1.1. Results.
Let L Kob p,β ( f /K , s ) be the p -adic L -function given as in (4.3.1). It isthe critical slope counterpart of Nekov´aˇr’s p -adic L -function associated to the p -ordinary stabilization f α . It follows from its interpolation property that L Kob p,β ( f /K ,
1) =0. As its predecessors, our p -adic Gross–Zagier formula expresses the first derivativeof L Kob p,β ( f /K , s ) in terms of the p -adic height of the Heegner cycle z f : Theorem 1.1.1.
Let f ∈ S k (Γ ( N )) be a newform with N ≥ . Suppose f is p -ordinary with respect to the embedding ι p and let f β denote its critical-slope p -stabilization ( of slope v p ( β ) = k − . Assume also that f β is not θ -critical. Let K = Q ( √− D K ) be an imaginary quadratic field where the prime p splits and thatsatisfies the Heegner hypothesis relative to N . Then, dds L Kob p,β ( f /K , s ) (cid:12)(cid:12) s = k = (cid:32) − p k − β (cid:33) · h Nek β,K ( z f , z f )(4 | D K | ) k − . Under the additional hypothesis that A be semistable, this has been proved in [BBV16,JSW17, Zha14] using different techniques. We do not need to assume that A is semistable. K. B¨UY¨UKBODUK, ROBERT POLLACK, AND SHU SASAKI
This theorem is proved by appealing to the existence of p -adic families of finiteslope modular forms, which allows us, using the existence of a suitable two-variable p -adic L -function and Theorem 3.5.4 to reduce to the case of non-critical slope.Such a non-critical slope result is precisely Theorem 5.1.1 ( p -adic Gross-Zagierformula for non-ordinary eigenforms of arbitrary weight; which is work in progressby Kobayashi [Kob19]). More precisely, Kobayashi’s method only establishes a p -adic Gross–Zagier formula in the non-ordinary case for one of the two p -stabilizationof a given form (the one of smaller slope). However, this result is sufficient for ourmethod and moreover, our method not only yields the Gross–Zagier formula in thecase of critical slope, but also allows us to handle the case of the other non-ordinary p -stabilization. See Theorem 1.1.11 for an even more general statement.1.1.1. Abelian varieties of GL -type. We assume until the end of this introduc-tion that f has weight 2. Let A f / Q denote the abelian variety of GL -type thatthe Eichler-Shimura congruences associate to f . This means that there existsan order O f ⊂ K f and an embedding O f (cid:44) → End Q ( A f ). We shall assume thatord s =1 L ( f / Q ,
1) = 1 and we choose K (relying on [BFH90]) in a way to ensurethat ord s =1 L ( f /K ,
1) = 1 as well. In this scenario, the element z f ∈ H ( Q , V f )is obtained as the Kummer image of the f -isotypical component P f of a Heegnerpoint P ∈ J ( N )( K ). Here, J ( N ) is the Jacobian variety of the modular curve X ( N ) and we endow it with the canonical principal polarization induced by theintersection form on H ( X ( N ) , Z ). This equips A f with a canonical polarizationas well.We let (cid:104) , (cid:105) J ( N ) ∞ denote the N´eron-Tate height pairing on the abelian variety J ( N ). Nekov´aˇr’s constructions in [Nek93] gives rise to a pair of E -equivariant p -adic height pairings h Nek λ, Q : ( A f ( Q ) ⊗ O f E ) × ( A f ( Q ) ⊗ O f E ) −→ E for each λ = α, β . We set c ( f ) := − L (cid:48) ( f / Q , (cid:104) P f , P f (cid:105) J ( N ) ∞ πi Ω + f ∈ K × f where Ω + f is a choice of Shimura’s period. We note that K f -rationality of c ( f ) isproved in [GZ86]. Corollary 1.1.2.
In addition to the hypotheses of Theorem 1.1.1, suppose that k = 2 and ord s =1 L ( f / Q ,
1) = 1 . Then for A f , P f ∈ A f ( Q ) and c ( f ) ∈ K × f as inthe previous paragraph we have L (cid:48) p,β ( f / Q ,
1) = (1 − /β ) c ( f ) h Nek β, Q ( P f , P f ) . Remark . The version of Theorem 1.1.1 above for the p -ordinary stabilization f α is due to Perrin-Riou (when k = 2) and Nekov´aˇr (when k is general). Theversion of Corollary 1.1.2 concerning the p -adic L -function L p,α ( f / Q , s ) follows fromPerrin-Riou’s p -adic Gross–Zagier theorem. The construction of this p -adic L -function follows from the work of Loeffler [Loe17] andLoeffler-Zerbes [LZ16]. More precisely, P ∈ J ( N )( K ) is given as the trace of a Heegner point y ∈ J ( N )( H K ) whichis defined over the Hilbert class field H K of K . Our restriction on the sign of the functionalequation (for the Hecke L -function of f ) shows that that P f ∈ J ( N )( Q ) ⊗ K f . -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 5 Elliptic curves.
In this subsection, we will specialize to the case when K f = Q , so that A = A f is an elliptic curve defined over Q of conductor N and analyticrank one, with good ordinary reduction at p and without CM. We note that itfollows from [Eme04, Theorem 1.3] that f β is not θ -critical.We still assume that ord s =1 L ( f / Q ,
1) = 1 and we choose K as in Section 1.1.1.We assume that the mod p representation ρ A : G Q −→ Aut F p ( A [ p ]) ∼ −→ GL ( F p )is absolutely irreducible. We fix a Weierstrass minimal model A / Z of A and let ω A denote the N´eron differential normalized as in [PR95, § + A is positive. Set V = T p ( A ) ⊗ Q p and we let ω cris ∈ D cris ( V )denote the element that corresponds to ω A under the comparison isomorphism.Extending scalars (to a sufficiently large extension E of Q p ) if need be, we shalldenote by D α , D β ⊂ D cris ( V ) the corresponding eigenspaces as before. Set ω cris = ω α + ω β with ω α ∈ D α and ω β ∈ D β . We let[ − , − ] : D cris ( V ) × D cris ( V ) −→ E denote the canonical pairing (induced from the Weil pairing) and we set δ A :=[ ω β , ω α ] /c ( f ). We let ω ∗A ∈ D cris ( V ) / Fil D cris ( V ) denote the unique element suchthat [ ω A , ω ∗A ] = 1. We remark that D cris ( V ) / Fil D cris ( V ) may be identified withthe tangent space of A ( Q p ) and the Bloch-Kato exponential mapexp V : D cris ( V ) / Fil D cris ( V ) −→ H ( Q p , V ) = A ( Q p ) ⊗ Q p with the exponential map for the p -adic Lie group A ( Q p ). Theorem 1.1.4.
Suppose A = A f is in the previous paragraph ( so k = 2 , K f = Q and ρ f is absolutely irreducible ) . In addition to all the hypotheses of Theorem 1.1.1,assume that ord s =1 L ( A / Q ,
1) = 1 . Then exp V (cid:32) ω ∗A · (cid:114) δ A (cid:16) (1 − /α ) − · L (cid:48) p,α ( f / Q , − (1 − /β ) − · L (cid:48) p,β ( f / Q , (cid:17)(cid:33) is a Q -rational point on the elliptic curve A of infinite order. The theorem above asserts the validity of a conjecture of Perrin-Riou. We alsonote that this theorem allows for the explicit computation of rational points onelliptic curves. Indeed one can compute the expression appearing in Theorem 1.1.4to very high p -adic accuracy by using the methods of [PS11] where algorithmsare given to compute the derivatives of both ordinary and critical slope p -adic L -functions. Such computations should be compared to the analogous computationsin [KP07] in the non-ordinary case.Theorem 1.1.4 may be deduced from the next result we present (in a manneridentical to the argument in [B¨uy17, § H ( Q , V ) :the Beilinson-Kato element BK and the Heegner point P f given as above, foran appropriate choice of the imaginary quadratic field K . Notice that under ourrunning hypotheses H ( Q , V ) = A ( Q ) ⊗ Q p K. B¨UY¨UKBODUK, ROBERT POLLACK, AND SHU SASAKI and it is a one dimensional Q p -vector space. Note that P f ∈ A ( Q ) is a rationalpoint on A and as such, it is a genuinely algebraic object, whereas BK ∈ A ( Q ) ⊗ Q p is a transcendental object that relates to both p -adic L -functions. The proofof Theorem 1.1.4 boils down to setting up an explicit comparison between BK and P f . This is precisely the content of Theorem 1.1.5. It was conjectured byPerrin-Riou and was proved independently by Bertolini–Darmon–Venerucci in theirpreprint [BDV19] (their approach is different from ours). Theorem 1.1.5.
Suppose A / Q is an elliptic curve as in Theorem 1.1.4 and let P ∈ A ( Q ) be a generator of the free part of its Mordell-Weil group. We have log A (res p (BK )) = − (1 − /α )(1 − /β ) · c ( f ) · log A (res p ( P )) , where log A stands for the coordinate of the Bloch-Kato logarithm associated to A with respect to the basis (of the tangent space) dual to that given by the N´erondifferential ω A . One key result that we rely on establishing Theorem 1.1.5 is the following con-sequence of our p -adic Gross–Zagier formula. We record it in Section 1.1.3 as webelieve that it is of independent interest; while we re-iterate that a proof of The-orem 1.1.4 is not written down explicitly in this article as it follows verbatim asin [B¨uy17].1.1.3. p -adic heights on Abelian varieties of GL -type and the conjecture of Birchand Swinnerton-Dyer. Throughout this section, we still assume that f ∈ S (Γ ( N ))has weight two; but we no longer assume that K f = Q . We retain our hypothesisthat ord s =1 L ( f / Q ,
1) = 1 and we choose K as in Section 1.1.1. In this situation,it follows from the work of Gross–Zagier and Kolyvagin–Logachev that the Tate-Shafarevich group III( A f / Q ) is finite and the Heegner point P := (cid:88) σ : K f (cid:44) → Q P f σ ∈ A f ( Q )generates A f ( Q ) ⊗ Q as a K f -vector space. Theorem 1.1.6.
Suppose f = (cid:80) a n q n ∈ S (Γ ( N )) is a newform with N ≥ andsuch that • v p ( ι p ( a p )) = 0 , • neither of the p -stabilizations of f is θ -critical, • the residual representation ρ f ( associated to the P -adic representation at-tached to f ) is absolutely irreducible, • ord s =1 L ( f / Q ,
1) = 1 .Then either h Nek α, Q or h Nek β, Q is non-degenerate.Remark . When K f = Q and p is a prime of good supersingular reductionfor the elliptic curve A = A f , a stronger form of Theorem 1.1.6 was proved byKobayashi in [Kob13]. Fortunately, this weaker version is good enough for applica-tions towards the Birch and Swinnerton-Dyer conjecture we discuss below.The final result we shall record in this introduction (Theorem 1.1.8 below)is a consequence of Theorem 1.1.1 and Theorem 1.1.6 towards the Birch and -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 7 Swinnerton-Dyer conjecture for the abelian variety A f . Under the additional hy-pothesis that K f = Q and A be semistable, this has been proved in [BBV16, JSW17,Zha14] using different techniques. Our results here allow us to adapt the proof of[Kob13, Cor. 1.3]) to the current setting to obtain a much simpler proof of (the p -part of) the Birch and Swinnerton-Dyer formula and eliminate the semistabilityhypothesis in [JSW17].Before we state our result, we define the O f -equivariant L -function L ( A f / Q , s )(with values in K f ⊗ C ) by setting L ( A f / Q , s ) := ( L ( f σ / Q , s )) σ ∈ Σ , where Σ = { σ : K f (cid:44) → Q } . For any { x i } ⊂ A f ( Q ) (resp. { y j } ⊂ A ∨ f ( Q )) thatinduces a basis of A f ( Q ) ⊗ Z Q (resp. of A ∨ f ( Q ) ⊗ Q ), the N´eron-Tate regulator R ∞ ( A f / Q ) on A f ( Q ) is given as R ∞ ( A f / Q ) := det( (cid:104) x i , y j (cid:105) ∞ )[ A f ( Q ) : (cid:80) Z x i ] [ A ∨ f ( Q ) : (cid:80) Z y j ] . We let Reg ∞ ,σ ( A f / Q ) denote the σ -component of this regulator, given as in (14),so that we have Reg ∞ ( A f / Q ) = (cid:89) σ ∈ Σ Reg ∞ ,σ ( A f / Q )(see Remark 6.2.2 where we discuss this factorization). We may then write L ∗ ( A f / Q ,
1) := (cid:32) − L (cid:48) ( f σ / Q , ∞ ,σ ( A f / Q ) · πi Ω + f σ (cid:33) σ ∈ Σ ∈ K f ⊗ Q to denote the algebraic part of the leading coefficient of the equivariant L -function L ( A f / Q , s ) at s = 1. Theorem 1.1.8.
Suppose f ∈ S (Γ ( N )) is a newform as in Theorem 1.1.6. Ifthe Iwasawa main conjecture holds true for each f σ/ Q , we have L ∗ ( A f / Q , ∈ | III( A f / Q ) | · Tam( A f / Q ) | A f ( Q ) tor | · | A ∨ f ( Q ) tor | ( O f ⊗ Z p ) × . Here: • Tam( A f / Q ) := (cid:81) (cid:96) | N c (cid:96) ( A f / Q ) and c (cid:96) ( A f / Q ) is the Tamagawa factor at (cid:96) . • A f ( Q ) tor ( resp. A ∨ f ( Q ) tor ) is the torsion subgroup of the Mordell-Weil groupof A f ( resp. of the dual abelian variety A ∨ f ) . Corollary 1.1.9.
Suppose A / Q is a non-CM elliptic curve with analytic rank oneand that (MC1) A has good ordinary reduction at p , (MC2) ρ A is absolutely irreducible, (MC3) one of the following two conditions hold: Besides the assumption that A be semistable, [Zha14, Theorem 7.3] has additional assumptionthat p is coprime to Tamagawa factors and [BBV16, Theorem A] requires p be non-anomalousfor A . In Section 5.6 of [BBV16], the authors explain a strategy to weaken the semistabilityhypothesis. K. B¨UY¨UKBODUK, ROBERT POLLACK, AND SHU SASAKI (MC3.1)
There exists a prime q || N such that p (cid:45) ord q (∆ q ) for a minimal dis-criminant ∆ q of A at q . (MC3.2) We have ρ A ( G Q ) ⊃ SL ( Z p ) and there exists a real quadratic field F verifying the conditions of [Wan15, Theorem 4] .Then the p -part of the Birch and Swinnerton-Dyer formula for A holds true.Remark . The Iwasawa main conjecture for f σ/ Q relates the characteristic idealof a Selmer group of the p -adic Galois representation T σ := lim ←− A f ( Q )[ P nσ ], where P σ is the prime of K f that is induced by the embedding ι p ◦ σ : K f (cid:44) → Q p , to oneof the p -adic L -functions L p,λ σ ( f σ/ Q , s ) (where λ σ := ι p ◦ σ ( λ ) for λ = α or β andwhere we have extended σ to an embedding K f ( α ) (cid:44) → Q in an arbitrary manner).Whether or not T σ is an ordinary Galois representation or not depends on whetheror not ι p ◦ σ ◦ ι − ∞ ( a p ) is a p -adic unit and therefore, the proof of the p -part ofBirch and Swinnerton-Dyer formula for a general GL -type abelian variety requiresthe Iwasawa main conjecture both for primes of good ordinary reduction and goodsupersingular reduction. There has been great progress in this direction; c.f. theworks of Skinner-Urban and Wan.When K f = Q and p is a prime of good ordinary reduction for A = A f , oneonly needs the main conjectures for a good ordinary prime. This has been provedin [SU14] and [Ski16, Theorem 2.5.2] (under the hypotheses (MC1), (MC2) and(MC3.1)) and in [Wan15] (under (MC1) and (MC3.2)).We close this introduction with a brief overview of our strategy to prove The-orem 1.1.1. We remark that the original approach of Perrin-Riou and Kobayashi(which is an adaptation of the original argument of Gross and Zagier) cannot beapplied in our case of interest as there is no Rankin-Selberg construction of thecritical-slope p -adic L -functions L p ( f β/ Q , s ) and L p ( f β/ Q ⊗ (cid:15) K , s ). The main idea is toprove a version of the asserted identity in p -adic families. That is to say, we shallchoose a Coleman family f through the p -stabilized eigenform f β (over an affinoiddomain A ) and we shall consider the following objects that come associated to f : • A two-variable p -adic L -function L p ( f /K , s ). The construction is essentiallydue to Loeffler (and it compares to that due to Bella¨ıche); we recall itsdefining properties in Section 4 below. One subtle point is that this p -adic L -function does not interpolate L Kob p,β ( f /K , s ), but rather an explicitmultiple of it. This extra (non-interpolatable) p -adic multiplier is essentiallythe p -adic interpolation factor for the adjoint p -adic L -function attached to f β . Crucially, the same factor also appears in the height side. • An A -adic height pairing h f ,K that interpolates Nekov´aˇr’s p -adic heightpairings for the members of the Coleman family, in the sense that thediagram (4) below (located just before the start of Section 3.5) commutes.It is important to compare the “correction factor” that appears on the right In fact, it could not: See Remark 4.3.2 below where we explain that L Kob p,β ( f /K , s ) does notvary continuously as f β varies in families. This factor appears as the ratio of the two Poincar´e duality pairings on the the f -directsummand summands of two modular curves of respective levels Γ ( N ) and Γ ( N ) ∩ Γ ( p ). See -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 9 most vertical arrow in the lower right square to the non-interpolatable p -adic multiplier mentioned in the previous paragraph. The construction ofthe A -adic height pairing is due to Benois and it is recalled in Section 3.3below. • A “universal” Heegner point Z f that interpolates the Heegner cycles as-sociated to the central critical twists of the members of the family f . Theconstruction of this class is one of the main ingredients here and it is carriedout in [JLZ19, BL19].Relying on the density of non-critical-slope crystalline points in the family f and a p -adic Gross–Zagier formula for these members (recorded in Theorem 5.1.1, whichis Kobayashi’s work in progress), one may easily deduce an A -adic Gross–Zagierformula for L p ( f /K , s ), expressing its derivative with respect to the cyclotomic vari-able as the A -adic height of the universal Heegner cycle (see Theorem 5.2.2 below).The proof of Theorem 1.1.1 follows, on specializing this statement to weight k .Let g = (cid:80) n =1 a n ( g ) q n ∈ S r (Γ ( N )) be a normalized eigenform. We let a, b ∈ Q denote the roots of its Hecke polynomial X − a p ( g ) X + p r − at p . Suppose that v p ( ι p ( a p ( g ))) > < v p ( ι p ( b )) ≤ v p ( ι p ( a )) . Let g b ∈ S r (Γ ( N p )) denote the p -stabilization corresponding to the Hecke root b .Kobayashi’s forthcoming result (Theorem 5.1.1 below) proves a p -adic Gross–Zagierformula for the p -stabilization g b alone. This is sufficient for our purposes; moreover,the method we present here (without any modification whatsoever) allows one todeduce the following p -adic Gross–Zagier formula at every non- θ -critical point x on the eigencurve of tame level N , that admits a neighborhood with a dense setof crystalline classical points (e.g., any crystalline non- θ -critical classical point x verifies this property). Theorem 1.1.11.
Suppose x is any non- θ -critical point of weight w on the eigen-curve of tame level N ≥ , that admits a neighborhood with a dense set of crys-talline classical points. Set L p ( x, s ) := L RS p ( F , κ, s ) | x , where F is any Colemanfamily over a sufficiently small neighborhood of x and finally L RS p ( F , κ, s ) is as inDefinition 4.1.4. Then, dds L p ( x, s ) | s = w = H x,K ( Z x , Z x ) . Here, H x,K is the specialization of the height pairing H F ,K (given as in Defini-tion 5.2.1) to x and likewise, Z x is the specialization of the universal Heegnercycle Z F to x . In particular, if f ∈ S k (Γ ( N )) is a classical newform and λ is a ϕ -eigenvalue on D cris ( V f ) such that f λ is non- θ -critical, then dds L Kob p,λ ( f /K , s ) (cid:12)(cid:12) s = k = (cid:32) − p k − λ (cid:33) · h Nek λ,K ( z f , z f )(4 | D K | ) k − . Proposition 3.4.5 where we make this discussion precise. We are grateful to D. Loeffler for ex-plaining this to us.
Acknowledgements.
The authors thank Daniel Disegni, Shinichi Kobayashi, DavidLoeffler and Barry Mazur for very helpful conversations. K.B. was hosted at Har-vard University, MPIM-Bonn and Paˇsk¯unas group in Universit¨at Duisburg-Essen(under a Humboldt Fellowship for Experienced Researchers) while preparing thisarticle and its companion. He thanks these institutions (and Vytas Paˇsk¯unas) fortheir amazing hospitality. R.P. also thanks MPIM-Bonn for their strong supportand hospitality throughout his extended visit there. S.S. thanks Vytas Paˇsk¯unasfor his unflagging moral support and helpful conversations. K.B. has received fund-ing from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sk(cid:32)lodowska-Curie Grant Agreement No. 745691 (CriticalGZ). R.P.acknowledges support from NSF grant DMS-1702178 as well as a fellowship fromthe Simons Foundation. S.S. acknowledges financial support from DFG/SFB.2.
Notation and Set up
For any field L , we let L denote a fixed separable closure and let G L := Gal( L/L )denote its Galois group.For each prime λ of a number field F , we fix a decomposition group at λ andidentify it with G λ := G F λ . We denote by I λ ⊂ G λ the inertia subgroup. In themain body of our article, we will only work with the case when F = Q or F = K (the imaginary quadratic field we have fixed above). For any finite set of places S of F , we denote by F S the maximal extension of F unramified outside S and set G F,S := Gal( F S /F ).We set C p := (cid:98) Q p , the p -adic completion of Q p . We fix embeddings ι ∞ : Q (cid:44) → C and ι p : Q (cid:44) → C p . When the prime p is assumed to split in the imaginary quadraticfield K , we let p denote the prime of K corresponding to the embedding ι p .We denote by v p : C p → R ∪ { + ∞} the p -adic valuation on C p which is normal-ized by the requirement that v p ( p ) = 1. Set | x | p = p − v p ( x ) . We fix a system ε = ( ζ p n ) n (cid:62) of primitive p n th roots of the unity in Q such that ζ pp n +1 = ζ p n for all n . We set Γ cyc = Gal( Q ( ζ p ∞ ) / Q ) and denote by χ cyc : Γ cyc ∼ −→ Z × p the cyclotomic character. The group Γ cyc factors canonically as Γ cyc = ∆ × Γ where∆ = Gal( Q ( ζ p ) / Q ) and Γ = Gal( Q ( ζ p ∞ ) / Q ( ζ p )). We let ω denote the Teichm¨ullercharacter (that factors through ∆) and set (cid:104) χ cyc (cid:105) := ω − χ cyc . We let Λ := Z p [[Γ]].We write Λ ι to denote the free Λ-module of rank one, on which G Q acts via G Q (cid:16) Γ ι −→ Γ (cid:44) → Λ × ι : γ (cid:55)−→ γ − By slight abuse, we denote all the objects (Γ cyc , χ cyc , ∆ , Γ , ω, Λ and ι ) introducedin the previous paragraph but defined over the base field Q p (in place of Q ) withthe same set of symbols. -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 11 For any a topological group G and a module M that is equipped with a contin-uous G -action, we shall write C • ( G, M ) for the complex of continuous cochains of G with coefficients in M .Let S be a finite set of places of Q that contains p and prime at infinity. If V isa p -adic representation of G Q ,S with coefficients in an affinoid algebra A , we shalldenote by D † rig ,A ( V ) the ( ϕ, Γ cyc )-module associated to the restriction of V to thedecomposition group at p .Let Σ denote the set of rational primes that divides N p , together with thearchimedean place. We denote the set of places of K above those in Σ also by Σ.For our fixed imaginary quadratic field K , we let O denote the maximal order of K . For any positive integer c , let O c := Z + c O denote the order of conductor c in K and let H c denote the ring class field of K of O c . Write H ( Np ) c for the maximalextension of H c outside N p and G c := Gal( H ( Np ) c /H c ). Fix a positive integer c coprime to N . We also set L cp s := H cp s ( µ p s ).For any eigenform g , we shall write g K in place of g ⊗ (cid:15) K for its twist by thequadratic character associated to K/ Q .For each non-negative real number h , we let D h denote the Q p -vector space of h -tempered distributions on Z p and set D ∞ := ∪ h D h . We also let D denote the Λ-algebra of Q p -valued locally analytic distributions on Z p . The natural map D h → D is an injection (for every h ) since locally analytic functions are dense in the spaceof continuous functions.We let R + denote the Q p -algebra of analytic functions on the open unit ball. Inexplicit terms, R + := (cid:40) ∞ (cid:88) n =0 c n X n : lim n →∞ | c n | p s n = 0 for every s ∈ [0 , (cid:41) . According to [PR94, Proposition 1.2.7], the algebra D is naturally isomorphic viathe Amice transform to R + .On fixing a topological generator γ of Γ (which in turn fixes isomorphisms Γ ∼ = Z p and Λ ∼ = Z p [[ X ]]), we may define the Q p -algebras D h (Γ) ⊂ D ∞ (Γ) ⊂ D (Γ)of distributions on Γ. We also set H := { f ( γ −
1) : f ∈ R + } ⊂ Q p [[Γ]](so that H ∼ = R + via γ (cid:55)→ X ). For H = (cid:80) ∞ n =0 c n ( H )( γ − n ∈ H ,we shall set H (cid:48) := ∞ (cid:88) n =0 ( n + 1) c n +1 ( H )log p χ cyc ( γ ) ( γ − n ∈ H . Notice then that ( H (cid:48) ) = c ( H ) (cid:14) log p χ cyc ( γ ) does not depend on the choice of γ .We shall equip D h (Γ) (0 ≤ h ≤ ∞ ) and H with a Λ-module and Galois modulestructure via the compositum of the mapsΛ (cid:44) → Λ[1 /p ] = D (Γ) (cid:44) → D (Γ) Amice (cid:44) → H . We set H A := H (cid:98) ⊗ A given a Q p -affinoid A . We let H ι := H ⊗ Λ Λ ι andsimilarly define H ι A .2.1. Modular curves, Hecke correspondences and the weight space.
Foreach non-negative integer s ∈ Z ≥ , we let Y s denote the affine modular curve oflevel Γ ( N ) ∩ Γ ( p s ). It parametrizes triples ( E, C, (cid:36) ) where E is an elliptic curve, C is a cyclic group of E of order N and (cid:36) is a point of order p s . We let X s denoteits compactification and J s := Jac( X s ).For each s , we let H s ⊂ End( J s ) denote Z p -the algebra generated by all Heckeoperators { T (cid:96) } (cid:96) (cid:45) Np together with { U (cid:96) } (cid:96) | Np and the diamond operators {(cid:104) m (cid:105) : m ∈ ( Z /p s Z ) × } .We set Λ wt := Z p [[ Z × p ]]. For z ∈ Z × p , we let [ z ] ∈ Λ wt denote the group-likeelement. The Hecke algebra H s comes equipped with a Λ wt -module structure via[ z ] (cid:55)→ (cid:104) z (cid:105) . We let m s denote the maximal ideal of H s that is determined by theresidual representation ρ f associated to our fixed eigenform f . When there is norisk of confusion, we shall abbreviate m := m s .Following [How07], we define the critical weight character Θ : Γ cyc → Λ wt (cen-tered at weight k ) by settingΘ( σ ) := ω k − ( σ )[ (cid:104) χ cyc (cid:105) / ( σ )]for σ ∈ Γ cyc , where ω : Γ cyc → Z × p is the Teichm¨uller character. We let Λ † wt denoteΛ wt as a module over itself, but allowing G Q act via the character Θ − . Let ξ ∈ Λ † wt denote the element that corresponds to 1 ∈ Λ wt .For any H s -module M on which G Q acts, we shall write M † := M ⊗ Λ wt Λ † wt which we equip with the diagonal G Q -action. Here the tensor product is over Λ wt and its action on M is given via the morphism Λ wt → H s (the diamond action).2.2. The Coleman family (f , β ). We fix an isomorphism e k − Λ wt ∼ = Z p [[ w ]] andlet W := Sp Z p [[ w ]] denote the weight space and let U = B ( k, p − r ) ⊂ W denote theclosed disk about k of radius p − r for some positive integer r . We let O ( U ) denotethe ring of analytic functions on the affinoid U ; the ring O ( U ) is isomorphic to theTate algebra A = E (cid:104)(cid:104) w/p r (cid:105)(cid:105) . For each κ ∈ k + p r − Z p , we shall denote by ψ κ the morphism ψ κ : A −→ Ew (cid:55)−→ (1 + p ) κ − k − . Consider the sequence I = { κ ∈ Z ≥ | κ ≡ k (mod ( p − p r − ) } of integers andlet f = ∞ (cid:88) n =1 a n q n ∈ A [[ q ]] -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 13 denote a p -adic family of cuspidal eigenforms passing through f β , in the sense ofColeman [Col97]. This means that for every point κ ∈ I , the formal expression f ( κ ) := ∞ (cid:88) n =1 ψ κ ( a n ) q n is the q -expansion of a cuspidal eigenform of level Γ ( N p ) and weight k , with theadditional property that f ( k ) = f. Let us denote by β = a p for the U p -eigenvaluefor its action on f , so that we have β ( k ) = β . By shrinking the closed ball U ifnecessary and using [Col97, Corollary B5.7.1], we may (and we will henceforth)assume that f is a family of constant slope k − f specialises to aclassical form of weight w and slope k − w > k lying in U ).Let W f denote the big Galois representation associated to the family f withcoefficients in A = O ( U ). We define its twist V f := W f ⊗ Λ wt Λ † wt . We recall that W f comes equipped with a Λ wt -module structure via the diamond action. Notethen that V f is self-dual in the sense that we have a G Q , Σ -equivariant symplecticpairing (that we denote by (cid:104) , (cid:105) Np ∞ )(1) (cid:104) , (cid:105) Np ∞ : V f × V f −→ A (1) . Selmer complexes and p -adic heights in families Cohomology of ( ϕ, Γ cyc ) -modules. In this subsection, we shall review thecohomology of ( ϕ, Γ cyc )-modules. Fix a topological generator γ of Γ. Recall that A stands for the affinoid algebra over E and R A for the relative Robba ring over A . For any ( ϕ, Γ cyc )-module D over R A consider the Fontaine–Herr complex C • ϕ,γ ( D ) : D ∆ d −→ D ∆ ⊕ D ∆ d −→ D ∆ , where d ( x ) = (( ϕ − x ) , ( γ − x ) and d ( y, z ) = ( γ − y ) − ( ϕ − z ) (forfurther details and properties, see [Her98, Liu08, KPX14]). We define H i ( D ) := H i ( C • ϕ,γ ( D )) . It follows from [Liu08, Theorem 0.2] and [KPX14, Theorem 4.4.2] that H i ( D ) is afinitely generated A -module for i = 0 , , D = D † rig , A ( V f ), it follows by [Liu08, Theorem 0.1]and [Pot13, Theorem 2.8]) that there exist canonical (up to the choice of γ ) andfunctorial isomorphisms(2) H i ( D † rig , A ( V f )) (cid:39) H i ( Q p , V f ) . for each i . The following proposition is due to Benois [Ben14, Proposition 2.4.2]and refines the isomorphism (2). Set K • p ( V f ) := Tot (cid:16) C • (cid:16) G p , V f ⊗ A (cid:101) B † rig , A (cid:17) ϕ − −−−→ C • (cid:16) G p , V f ⊗ A (cid:101) B † rig , A (cid:17)(cid:17) , where (cid:101) B † rig , A is the ring of p -adic periods introduced by Berger in [Ber02]. Proposition 3.1.1 (Benois) . We have a diagram C • ( G p , V f ) ξ (cid:39) (cid:47) (cid:47) K • p ( V f ) ,C • ϕ,γ ( D † rig ( V f )) η (cid:39) (cid:79) (cid:79) where the maps η and ξ are both quasi-isomorphisms. Selmer complexes.
Local conditions at primes above p . A result of Liu [Liu08, Theorem 0.3.4]shows that the ( ϕ, Γ cyc )-module D † rig , A ( V f ) admits a triangulation over A . In moreprecise terms, the module D † rig , A ( V f ) sits in an exact sequence(3) 0 → D β → D † rig , A ( V f ) → (cid:101) D β → , where both D β and (cid:101) D β are ( ϕ, Γ cyc )-modules of rank 1.Recall that we have assumed p = pp c splits, so that K q = Q p for each q ∈ { p , p c } .We define U + q ( V f , D β ) := C • ϕ,γ ( D β ). On composing the quasi-isomorphism η ofProposition 3.1.1 with the canonical morphism U + q ( V f , D β ) → C • ϕ,γ ( D † rig , A ( V )), weobtain a map i + q : U + q ( V f , D β ) −→ K • q ( V f )where K • q ( V f ) := Tot (cid:16) C • (cid:16) G q , V f ⊗ A (cid:101) B † rig , A (cid:17) ϕ − −−−→ C • (cid:16) G q , V f ⊗ A (cid:101) B † rig , A (cid:17)(cid:17) as above.3.2.2. Local conditions away from p . For each non-archimedean prime λ ∈ Σ \{ p , p c } of K , we define the complex U + λ ( V ) = (cid:104) V I λ f Fr λ − −−−−→ V I λ f (cid:105) , which is concentrated in degrees 0 and 1 and where Fr λ denotes the geometricFrobenius. We define i + λ : U + λ ( V f ) −→ C • ( G λ , V f )by setting i + λ ( x ) = x in degree 0, i + λ ( x )(Fr λ ) = x in degree 1.In order to have a uniform notation for all primes in Σ we set K • λ ( V f ) := C • ( G λ , V f )and U + λ ( V f , D β ) := U + λ ( V f ) for a non-archimedean prime λ ∈ Σ \ { p , p c } . Since weassume p >
2, we may safely ignore the archimedean places. -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 15
The Selmer complex.
We define the complexes K • Σ ( V f ) := (cid:76) λ ∈ Σ K • λ ( V f ) and U +Σ ( V f , D β ) := (cid:76) λ ∈ Σ U + λ ( V f , D β ) . Observe that we have a diagram C • ( G K, Σ , V f ) res Σ (cid:47) (cid:47) K • Σ ( V f ) U +Σ ( V f , D β ) , i +Σ (cid:79) (cid:79) where i +Σ = ( i + λ ) λ ∈ Σ and res Σ denotes the localization map. Definition 3.2.1.
The Selmer complex associated to these data is defined as S • ( V f /K , D β ) = cone (cid:18) C • ( G K, Σ , V f ) ⊕ U +Σ ( V f , D β ) res Σ − i +Σ −−−−−→ K • Σ ( V f ) (cid:19) [ − . Definition 3.2.2.
We denote by R Γ( V f /K , D β ) the class of S • ( V f /K , D β ) in thederived category of A -modules and denote by H i ( V f /K , D β ) := R i Γ( V f /K , D β ) . its cohomology. A -adic cyclotomic height pairings. We provide in this section an overviewof the construction of p -adic heights for p -adic representations over the affinoidalgebra A , following [Ben14]. We retain our previous notation and conventions.Let J A denote the kernel of the augmentation map H (cid:98) ⊗ A =: R + , A → A which is induced by γ (cid:55)→
1. Note that J A = ( γ − R + , A and J A /J A (cid:39) A as A -modules. The exact sequence0 → V f ⊗ J A /J A −→ V f ⊗ R + , A /J A −→ V f −→ β cyc V f , D β : R Γ( V f /K , D β ) −→ R Γ( V f /K , D β )[1] ⊗ A J A /J A . Definition 3.3.1.
The p -adic height pairing associated to the Coleman family ( f , β ) is defined as the morphism h f , β : R Γ( V f /K , D β ) ⊗ L A R Γ( V f /K , D β ) β cyc V f , Dβ ⊗ id −−−−−−−→ (cid:16) R Γ( V f /K , D β )[1] ⊗ J A /J A (cid:17) ⊗ L A R Γ( V f /K , D β ) ∪ −→ J A /J A [ − where ∪ is the cup-product pairing R Γ( V f /K , D β ) ⊗ L A R Γ( V f /K , D β ) ∪ −→ A [ − which is induced from the G K, Σ -equivariant symplectic pairing (cid:104) , (cid:105) Np ∞ of (1) . In the level of cohomology, h f , β induces a pairing h , f , β : H ( V f /K , D β ) ⊗ A H ( V f /K , D β ) −→ J A /J A . Proposition 3.3.2.
The A -adic height pairing h , f , β is symmetric.Proof. This is a direct consequence of [Ben14, Theorem I] and the fact that thepairing (cid:104) , (cid:105) Np ∞ is symplectic. (cid:3) The map γ − J A ) (cid:55)→ log χ cyc ( γ ) induces an isomorphism ∂ cyc : J A /J A ∼ −→ A . We define the A -valued height pairing h f ,K by setting h f ,K := ∂ cyc ◦ h , f , β . Specializations and comparison with Nekov´aˇr’s heights.
Shrinking U if necessary, we shall assume that 2 k / ∈ I . Throughout this subsection, we fix aninteger κ ∈ I with κ ≥ k and set g := f ( κ ) ∈ S κ (Γ ( N p )) and b = β ( κ ) . The Galois representation V f ⊗ A ,ψ κ E is the central-critical twist V g of Deligne’srepresentation W g associated to the cuspidal eigenform g . Lemma 3.4.1.
The eigenform g is non- θ -critical and old at p .Proof. If κ > k , the eigenform for g is not critical since in this case we have v p ( b ) = k − < κ −
1. If κ = k , then g = f β is non- θ -critical by assumption.If g were new at p , we would have k − v p ( b ) = κ/ − κ = 2 k ,contradicting the choice of I . (cid:3) Corollary 3.4.2.
The Galois representation V g is crystalline at p . Definition 3.4.3.
We let g ◦ ∈ S κ (Γ ( N )) denote the newform such that g = g b ◦ isthe p -stabilization of g ◦ . Consider the Bloch-Kato Selmer group H ( K, V g ). It comes equipped withNekov´aˇr’s p -adic height pairing h Nek b,K : H ( K, V g ) ⊗ H ( K, V g ) −→ E. The height pairing h Nek b,K is associated to the Hodge-splittingD cris ( V g ) = D b ⊕ Fil D cris ( V g )together with the symplectic pairing (cid:104) , (cid:105) N : V g ⊗ V g −→ E (1)that is induced from the Poincar´e duality for the ´etale cohomology of the modularcurve X ( N ), where V g = V g ◦ appears as a direct summand.Our goal in this subsection is to compare these objects to those obtained byspecializing the A -adic objects we have defined in the previous section. Definition 3.4.4. i) We let W Np denote the Atkin-Leher operator of level N p and let (cid:104) , (cid:105) Np denote thePoincar´e duality pairing on the cohomology of the modular curve of level Γ ( N ) ∩ Γ ( p ) . -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 17 ii) Realizing V g as the g -isotypical ( with respect to the Hecke operators T (cid:96) for (cid:96) (cid:45) N p and operators U (cid:96) for (cid:96) | N p ) direct summand in the cohomology of the modularcurve of level Γ ( N ) ∩ Γ ( p ) , we define (cid:104) , (cid:105) (cid:48) Np : V g ⊗ V g −→ E (1) by setting (cid:104) x, y (cid:105) (cid:48) Np := (cid:104) x, W Np y (cid:105) Np and refer to it as the p -stabilized Poincar´e duality pairing on V g . iii) We let Pr ∗ b : V g ∼ −→ V g denote the natural isomorphism appearing in [KLZ17, Proposition 10.1.1/1] . Here V g on the left is the g -isotypical direct summand in the cohomology of X ( N ) ( withrespect to the Hecke operators T (cid:96) for (cid:96) (cid:45) N and operators U (cid:96) for (cid:96) | N ) , whereas V g on the right is the g -isotypical direct summand in the cohomology of the modularcurve of level Γ ( N ) ∩ Γ ( p ) ( with respect to the Hecke operators T (cid:96) for (cid:96) (cid:45) N p andoperators U (cid:96) for (cid:96) | N p ) . We are grateful to D. Loeffler for bringing the following observation to our at-tention.
Proposition 3.4.5. i) (cid:104) ψ κ x, ψ κ y (cid:105) (cid:48) Np = ψ κ ◦ (cid:104) x, y (cid:105) Np ∞ . ii) We have (cid:104) Pr ∗ b x , Pr ∗ b y (cid:105) (cid:48) Np = b λ N ( g ◦ ) E ( g ) E ∗ ( g ) (cid:104) x, y (cid:105) N , where λ N ( g ◦ ) is the Atkin-Lehner pseudo-eigenvalue of g ◦ , E ( g ) = (cid:16) − p κ − b (cid:17) and E ∗ ( g ) = (cid:16) − p κ − b (cid:17) .Proof. The first assertion is well-known; c.f. Proposition 4.4.8 and Theorem 4.6.6of [LZ16]. For the second, we note that (cid:104) Pr ∗ b x , Pr ∗ b y (cid:105) (cid:48) Np = (cid:104) Pr ∗ b x , W Np Pr ∗ b y (cid:105) Np = (cid:104) x , (Pr b ) ∗ W Np Pr ∗ b y (cid:105) N = b λ N ( g ◦ ) E ( g ) E ∗ ( g ) (cid:104) x, y (cid:105) N where the first and second equalities follows from definitions, whereas the third isa consequence of the discussion in the final paragraph of the proof of Proposition10.1.1 of [KLZ17]. (cid:3) Since the g is non- θ -critical (Lemma 3.4.1), the triangulation (3) gives rise to asaturated triangulation 0 −→ D b −→ D † rig ( V g ) −→ (cid:101) D b −→ ϕ, Γ cyc )-module D † rig ( V g ) by base change, where D b := D β ⊗ A ,ψ κ E and (cid:101) D b := (cid:101) D β ⊗ A ,ψ κ E . With this data at hand, one may proceed precisely as inSection 3.2.3 to define a Selmer complex S • ( V g /K , D b ) in the category of E -vector This map would have been denoted by (Pr b ) ∗ in op. cit. spaces. We let R Γ( V g /K , D b ) denote the corresponding object in the derived cate-gory and H i ( V g /K , D b ) denote its cohomology.The general formalism to construct p -adic heights we outlined in Section 3.3(where we utilize the symplectic pairing (cid:104) , (cid:105) (cid:48) Np : V g ⊗ V g −→ E (1)given in Definition 3.4.4 to determine an isomorphism V ∗ g (1) ∼ → V g ) also equips uswith an E -valued height pairing h g,b,K : H ( V g /K , D b ) ⊗ H ( V g /K , D b ) −→ E .
Lemma 3.4.6. i) We have a natural morphism ( which we shall denote by ψ κ , by slight abuse ) ψ κ : H ( V f /K , D β ) ⊗ A ,ψ κ E −→ H ( V g /K , D b ) of E -vector spaces, which is an isomorphism for all but finitely many choices of g . ii) The following diagram commutes: H ( V f /K , D β ) ψ κ (cid:15) (cid:15) ⊗ A H ( V f /K , D β ) ψ κ (cid:15) (cid:15) h f ,K (cid:47) (cid:47) A ψ κ (cid:15) (cid:15) H ( V g /K , D b ) ⊗ E H ( V g /K , D b ) h g,b,K (cid:47) (cid:47) E Proof.
Let ℘ κ := ker( ψ κ ) be the prime of A corresponding to g . Notice then that H ( V f /K , D β ) ⊗ A ,ψ κ E = H ( V f /K , D β ) /℘ κ H ( V f /K , D β )and the general base change principles for Selmer complexes (c.f. [Pot13, Section1]) shows that the sequence0 −→ H ( V f /K , D β ) /℘ κ H ( V f /K , D β ) −→ H ( V g /K , D b ) −→ H ( V f /K , D β )[ ℘ κ ]of E -vector spaces is exact. The first assertion now follows. The second followseasily from definitions. (cid:3) Proposition 3.4.7.
There is a natural isomorphism H ( V g /K , D b ) ∼ −→ H ( K, V g ) . Moreover, the height pairing h g,b,K coincides with h Nek b,K (cid:14) b λ N ( g ◦ ) E ( g ) E ∗ ( g ) .Proof. The proof of the first assertion reduces to [Ben14, Theorem III] once weverify(i) D cris ( V g ) ϕ =1 = 0,(ii) H ( (cid:101) D b ) = 0.Assume first κ (cid:54) = k (so that g (cid:54) = f β ). Let g ◦ be as in Definition 3.4.3. The roots ofthe Hecke polynomial for g ◦ at p could not be the pair { , p κ − } , as otherwise wewould have κ − v p ( b ) = k −
1. This verifies both conditions in this case. -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 19
When κ = k and g = f β , both conditions follow as a consequence of theRamanujan-Petersson conjecture for f (as proved by Deligne), according to whichthe roots of the Hecke polynomial of f at p could not be the pair { , p k − } .The assertion concerning the comparison of two p -adic heights follows from[Ben14, Theorem 11] together with Proposition 3.4.5. (We find it instructive tocompare Benois’ result to [Nek06, Theorem 11.4.6] in the ordinary case.) (cid:3) The following commutative diagram summarizes the discussion in this subsec-tion:(4) H ( V f /K , D β ) ψ κ (cid:15) (cid:15) ⊗ A H ( V f /K , D β ) ψ κ (cid:15) (cid:15) h f ,K (cid:47) (cid:47) A ψ κ (cid:15) (cid:15) H ( V g /K , D b ) ⊗ E H ( V g /K , D b ) h g,b,K (cid:47) (cid:47) E b λ N ( g ◦ ) E ( g ) E ∗ ( g ) (cid:15) (cid:15) H ( K, V g ) ∼ = Pr ∗ b (cid:79) (cid:79) ⊗ E H ( K, V g ) ∼ = Pr ∗ b (cid:79) (cid:79) h Nek b,K (cid:47) (cid:47) E Universal Heegner points.
In this subsection, we shall introduce elementsin the Selmer groups on which we shall calculate the A -adic height h f ,K .3.5.1. Heegner cycles.
We recall the definition of Heegner cycles on Kuga-Sato va-rieties, following the discussion in [Nek95]. Recall that we have fixed an imaginaryquadratic field K such that all primes dividing the tame level N p splits completelyin K/ Q . Let g ∈ S κ (Γ ( N )) be a cuspidal eigenform of weight κ > Y ( N ) denote the modular curve over Q which is the moduli of elliptic curveswith full level N structure and we let j : Y ( N ) → X ( N ) denote its non-singularcompactification. Since we assume N ≥
3, there is a universal generalized ellipticcurve E → X ( N ) that restricts to the universal elliptic curve f : E → Y ( N ). The( κ − E with itself over Y ( N ) has a canonical non-singularcompactification W described in detail in [Del71, Sch90]. We have natural maps(5) H κ − ( W × Q Q , Q p )( κ/ → H ( X ( N ) × Q Q , j ∗ Sym κ − ( R f ∗ Q p ))( κ/ → V g . Scholl defines a projector ε (where his w corresponds to our κ −
2) and proves thatthere is a canonical isomorphism H ( X ( N ) × Q Q , j ∗ Sym κ − ( R f ∗ Q p )) ∼ −→ εH κ − ( W × Q Q , Q p ) . We finally define B := (cid:26)(cid:18) ∗ ∗ ∗ (cid:19)(cid:27) (cid:46) {± } ⊂ GL ( Z /N Z ) (cid:14) {± } and the idempotent ε B := | B | (cid:80) g ∈ B g (which acts on the modular curves Y ( N )and X ( N )). Definition 3.5.1.
We let N be an ideal of O such that O / N ∼ = Z /N Z . For anarbitrarily chosen ideal A ⊂ O , consider the isogeny C / A → C / A N − . It represents the Heegner point y = y A on Y ( N )( C ) . It is defined over the Hilbert class field H of K . Choose any point (cid:101) y ∈ Y ( N ) × Q H over the Heegner point y (viewed as a closedpoint of Y ( N ) × Q H ). The fiber E (cid:101) y is a CM elliptic curve defined over H whoseendomorphism ring is isomorphic to O . We letΓ √ D K ⊂ E (cid:101) y × E (cid:101) y denote the graph of √ D K ∈ O (fix any one of the two square-roots) Definition 3.5.2.
We let Y := Γ √ D K × · · · × Γ √ D K (cid:124) (cid:123)(cid:122) (cid:125) κ/ − times ⊂ E (cid:101) y × · · · × E (cid:101) y (cid:124) (cid:123)(cid:122) (cid:125) κ − times = ( W × Q H ) (cid:101) y and call the cycle (with rational coefficients) that is represented by ε B εY inside of ε B ε CH κ/ ( W × Q H ) ⊗ Q (which we also denote by the same symbol ε B εY ) theHeegner cycle. The cohomology class of εY in H κ ´et ( W × Q Q , Q p )( κ/
2) vanishes, so that one mayapply the Abel-Jacobi mapAJ : CH κ/ ( W × Q H ) ⊗ Q −→ H ( H, H κ − ( W × Q Q , Q p )( κ/ ε B εY . Definition 3.5.3.
We let AJ g : CH κ/ ( W × Q H ) ⊗ Q → H ( H, V g ) denote thecompositum of the map (5) with the Abel-Jacobi map and define the Heegner cycle z g := cor H/K (AJ g ( ε B εY )) ∈ H ( K, V g ) . Since p (cid:45) N , all X ( N ), X ( N ) and W have good reduction at p and it followsfrom [Nek00, Theorem 3.1(i)] that z g ∈ H ( K, V g ) . Heegner cycles in Coleman families.
For a classical weight κ ∈ I and ψ κ asin Section 2.2, we let f ( κ ) ◦ ∈ S κ (Γ ( N )) denote the newform whose p -stabilization(with respect to β ( κ )) is the eigenform f ( κ ).The following result (construction of a big Heegner point along the Colemanfamily f ) is [BL19, Proposition 4.15(iii)] and [JLZ19, Theorem 5.4.1]. Theorem 3.5.4 (B¨uy¨ukboduk–Lei, Jetchev–Loeffler–Zerbes) . There exists a uniqueclass Z f ∈ H ( V f /K , D β ) that is characterized by the requirement that for any κ ∈ I we have ψ κ (cid:0) Z f (cid:1) = u − K (2 (cid:112) − D K ) − κ (cid:18) − p κ − β ( κ ) (cid:19) z f ( κ ) ◦ ∈ H ( K, V f ( κ ) ) , where u K = |O × K | / and − D K is the discriminant of K .Remark . Jetchev–Loeffler–Zerbes in [JLZ19] rely on the overconvergent ´etalecohomology of Andreatta–Iovita–Stevens. The construction of “universal” Heeg-ner cycles in [BL19] exploits the p -adic construction of rational points, a theme -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 21 first observed by Rubin [Rub92], and dwells on the formula of Bertolini–Darmon–Prasanna which relates the Bloch–Kato logarithms of these cycles to appropriateRankin–Selberg p -adic L -values. In [BPS19], we will give another construction of“universal” Heegner cycles in the context of Emerton’s completed cohomology (onrealizing the family f on Emerton’s eigensurface).4. p -adic L -functions over the imaginary quadratic field K We introduce the needed p -adic L -functions for the arguments in this paper.We first discuss a Rankin-Selberg p -adic L -function defined over our imaginaryquadratic field K . We then compare this p -adic L -function to a na¨ıve product of p -adic L -functions defined over Q .4.1. Ranking-Selberg p -adic L -functions. Loeffler and Zerbes in [LZ16] haveconstructed p -adic L -functions (in 3-variables) associated to families of semi-ordinaryRankin-Selberg products f ⊗ f of eigenforms, where f runs through a Colemanfamily and f through a p -ordinary family. (See also [Loe17] where the correctinterpolation property is extended from all crystalline points to all critical points.)We shall let f vary in a (suitable branch of the) universal CM family associated to K (which we shall recall below), and thus we may reinterpret this p -adic L -functionas a p -adic L -function associated to the base change of f to K .4.1.1. CM Hida families.
For a general modulus n of K , let K ( n ) denote themaximal p -extension contained in the ray class field modulo n . We set H ( p ) n :=Gal( K ( n ) /K ). In particular, K ( p ∞ ) := ∪ K ( p n ) is the unique Z p -extension of K which is unramified outside p . We let Γ p := lim ←− n H ( p ) p n denote its Galois groupover K . We fix an arbitrary Hecke character ψ of ∞ -type ( − , p and whose associated p -adic Galois character factors through Γ p . Notice thenthat ψ ≡ p mod m E , where we have let p : ( O / p ) × → O × E denote the trivialcharacter modulo p . Remark . If the class number of K is prime to p , then the Hecke character ψ is unique, as the ratio of two such characters would have finite p -power order andconductor dividing p .The theta-seriesΘ( ψ ) := (cid:88) ( a , p )=1 ψ ( a ) q N a ∈ S (Γ ( | D K | p ) , (cid:15) K ω − )is a newform and it is the weight two specialization (with trivial wild character)of the CM Hida family g with tame level | D K | and character (cid:15) K ω . The weightone specialization of this CM Hida family with trivial wild character equals the p -ordinary theta-series Θ ord ( p ) := (cid:80) ( a , p )=1 q N a ∈ S (Γ ( | D K | p ) , (cid:15) K ) of p . Remark . One may construct the Hida family g as follows. We let T | D K | p denote the Hecke algebra given as in [LLZ15, § I p ⊂ T | D K | p as in [LLZ15, Definition 5.1.1]. Note that in order to determine themap φ p that appears in this definition, we use the algebraic Hecke character ψ wehave chosen above. It follows by [LLZ15, Prop. 5.1.2] that I p is non-Eisenstein, p -ordinary and p -distinguished. By [LLZ15, Theorem 4.3.4], the ideal I p corresponds uniquely to a p -distinguished maximal ideal I of the universal ordinary Hecke algebra T | D K | p ∞ acting on H ( Y ( | D K | p ∞ )) (definitions of these objects may be found in [LLZ15,Definition 4.3.1]). The said correspondence is induced from Ohta’s control theo-rem [Oht99, Theorem 1.5.7(iii)], which also attaches to I p a unique non-Eisenstein, p -ordinary and p -distinguished maximal ideal I p r of T | D K | p r for each r ≥ T | D K | p r φ p r −→ O L [ H p r ] −→ O E −→ O E /(cid:36) E , and therefore with its original form given in [LLZ15, Definition 5.1.1]). The ideal I determines the CM Hida family g alluded to above.We shall henceforth identify the rigid analytic ball Sp Z p [[Γ p ]] with the weightspace for the Hida family g . We let (cid:101) κ ∈ Sp Z p [[Γ p ]] denote the point correspondingto the weight one specialization Θ ord ( ℘ ).4.1.2. The p -adic L -function and interpolation property. We fix an affinoid neigh-borhood Y ⊂ Sp Z p [[Γ p ]] and let L RS p ( f , g | Y ) ∈ O ( Y ) (cid:98) ⊗ H A denote the 3-variable Rankin-Selberg p -adic L -function of Loeffler and Zerbes [LZ16,Loe17]. Since g is a p -ordinary family, we may choose Y as large as we like andobtain a p -adic L -function L RS p ( f , g ) ∈ H (Γ p ) (cid:98) ⊗ H A . As explained in detail in [BL17a], the p -adic L -function may be thought as a relative p -adic L -function for f over K , interpolating the algebraic parts of the L -values L ( f ( κ ) /K , Ψ ,
1) where Ψ runs through the algebraic Hecke characters of K withinfinity type ( a, b ) with 0 ≤ a ≤ b ≤ κ − Definition 4.1.3.
We let D RS f /K ∈ H A denote the p -adic distribution obtainedby specializing L RS p ( f , g ) to the point (cid:101) κ ∈ Sp Z p [[Γ p ]] in the weight space for g ,corresponding to the weight one specialization Θ ord ( ℘ ) . The following interpolation property characterizes the distribution D RS f /K . Theorem 4.1.4 (Loeffler) . For every κ ∈ I , any j ∈ Z ∩ [1 , κ − and all Dirichletcharacters η of conductor p r ( we allow r = 0) we have ( ψ κ ⊗ ηχ j − )( D RS f /K ) = ( − j − × V ( f ( κ ) ◦ , η, j ) p r ( j − ) W ( η ◦ N K/ Q ) β ( κ ) r E ( f ( κ )) E ∗ ( f ( κ )) × i κ − N j − κ +1 Γ( j ) j + κ − π j × L ( f ( κ ) ◦ /K , η − ◦ N K/ Q , j ) (cid:104) f ( κ ) ◦ , f ( κ ) ◦ (cid:105) N where V ( f ( κ ) ◦ , η, j ) is as in Theorem 4.2.1, W ( η ⊗ N K/ Q ) is the root number forthe complete Hecke L -series Λ( η ◦ N K/ Q , s ) ( c.f. [Nek95, Page 626]) and finally, E ( f ( κ )) := (cid:18) − p κ − β ( κ ) (cid:19) , E ∗ ( f ( κ )) := (cid:18) − p κ − β ( κ ) (cid:19) . -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 23 Remark . The p -adic L -function (and its interpolation property) recorded inTheorem 4.1.4 is a slight alteration of Loeffler’s original formulation in [Loe17]. Itcan be obtained following the calculations carried out in [BL18] based on Loeffler’swork in op. cit.4.2. Na¨ıve p -adic L -functions over K . We now consider a na¨ıve version of a p -adic L -function over K by taking the product of two p -adic L -functions over Q .We begin by recalling two-variable p -adic L -functions over the eigencurve. Thisconstruction is due to Glenn Stevens, but first appeared in the literature in [Bel12].Suppose that h is a Coleman family over a sufficiently small affinoid disc Sp( A )about a non- θ -critical point g of weight k on the eigencurve (in the sense of Defini-tion 2.12 in [Bel12]) with U p -eigenvalue α . Let I denote the set of classical weightsof forms occurring in A . Theorem 4.2.1.
There exists a unique p -adic distribution D h ∈ H A which ischaracterized by the following interpolation property: For every κ ∈ I , any j ∈ Z ∩ [1 , κ − and all Dirichlet characters η of conductor p r ≥ , ( ψ κ ⊗ ηχ j − )( D h ) = ( − j Γ( j ) V ( h ( κ ) ◦ , η, j ) τ ( η ) p ( j − r α ( κ ) r L ( h ( κ ) ◦ , η − , j )(2 πi ) j Ω ± h ( κ ) C ± h ( κ ) where, • τ ( η ) is the Gauss sum ( normalized to have norm p r/ ) , • V ( h ( κ ) ◦ , η, j ) = (cid:0) − p j − η ( p ) (cid:14) α ( κ ) (cid:1) (cid:0) − p κ − − j η − ( p ) (cid:14) α ( κ ) (cid:1) , • Ω + h ( κ ) and Ω − h ( κ ) are canonical periods in the sense of [Vat99, § , • C + h ( κ ) and C − h ( κ ) are non-zero constants that only depend on κ and C + h ( k ) = C − h ( k ) = 1 , • the sign ± is determined so as to ensure that ( − ( j − η ( −
1) = ± .Proof. See [Bel12, Theorem 3 and (4)]. (cid:3)
Remark . If the slope of h is smaller than h , then D h /K ∈ H h (cid:98) ⊗ A . Definition 4.2.3.
For the Coleman family f we have fixed above, mimicking Kobayashi [Kob13, Kob12] , we set D naive f /K := D f / Q · D f K/ Q ∈ H A . Here, f K is the family obtained by twisting the Coleman family f by the quadraticcharacter (cid:15) K . We call D f /K the na¨ıve base change p -adic L -function .Remark 4.2.2 tells us that we in fact have D f /K ∈ H k − (cid:98) ⊗ A . The na¨ıve base change p -adic L -function is then characterized by the followinginterpolation property: For every κ ∈ I , any j ∈ Z ∩ [1 , κ −
1] with j ≡ k/ even Dirichletcharacters η of conductor p r (we allow r = 0),( ψ κ ⊗ ηχ j − )( D naive f /K ) = Γ( j ) V ( f ( κ ) ◦ , η, j ) τ ( η ) p r ( j − β ( κ ) r × L ( f ( κ ) ◦ , η − , j )(2 πi ) j (6) × C ε f ( κ ) C ε f ( κ ) K Ω ε f ( κ ) Ω ε f ( κ ) K where ε ∈ {±} is the sign of ( − k/ − .4.3. A factorization formula.
We will be working with the following locallyanalytic functions on U × (1 + p Z p ): Definition 4.3.1.
Given a locally analytic distribution D on Γ cyc , we set L p ( D , s ) := (cid:104) χ cyc (cid:105) s − ω k/ − ( D ) where ω is the Teichm¨uller character. We define L RS p ( f , κ, s ) and L naive p ( f /K , κ, s ) on U × (1 + p Z p ) by setting L RS p ( f /K , κ, s ) := β − λ N ( f ) − i κ − N | D K | − / ( − k/ − L p ( D RS f /K , s ) (cid:12)(cid:12)(cid:12) w =(1+ p ) κ − k − ,L naive p ( f /K , κ, s ) := L p ( D naive f /K , s ) (cid:12)(cid:12)(cid:12) w =(1+ p ) κ − k − . We also set L Kob p, β ( κ ) ( f ( κ ) ◦ /K , s ) := β ( κ ) λ N ( f ( κ ) ◦ ) E ( f ( κ )) E ∗ ( f ( κ )) L RS p ( f /K , κ, s ) for each choice of κ ∈ I . In the particular case when κ = k , we shall write L Kob p,β ( f /K , s ) in place of L Kob p, β ( k ) ( f ( k ) ◦ /K , s ) . See Remark 4.3.7 below for a comparison of L Kob p, β ( κ ) ( f ( κ ) ◦ /K , s ) to Nekov´aˇr’s p -adic L -function in the p -ordinary set up. Remark . Observe that the p -adic multipliers E ( f ( κ )) E ∗ ( f ( κ )) do not varycontinuously, the p -adic L -functions L Kob p, β ( κ ) ( f ( κ ) ◦ /K , s ) do not interpolate as κ ∈ I varies.Let us consider the meromorphic function R f /K := L RS p ( f /K , κ, s ) (cid:14) L naive p ( f /K , κ, s ) . Notice that for any κ ∈ I , the specialization L p ( f /K , κ , s ) is non-zero, so R f /K ( κ , s )is a meromorphic function in s . Lemma 4.3.3.
The meromorphic function r ( κ ) := R f /K ( κ, k/
2) ( in the variable κ ) specializes to (cid:104) N (cid:105) k − κ (cid:104) (cid:105) − κ + k − k ( − k/ | D K | / β ( κ ) λ N ( f ( κ )) E ( f ( κ )) E ∗ ( f ( κ )) (cid:104) f ( κ ) ◦ , f ( κ ) ◦ (cid:105) N × Ω ε f ( κ ) Ω ε f ( κ ) K C ε f ( κ ) C ε f ( κ ) = (cid:104) N (cid:105) k − κ ( − k/ | D K | / (cid:104) (cid:105) κ − k k − (cid:104) f ( κ ) , f ( κ ) (cid:105) N,p × Ω ε f ( κ ) Ω ε f ( κ ) K C ε f ( κ ) C ε f ( κ ) K (7) whenever κ ∈ I . Here, (cid:104) , (cid:105) N,p is the Petersson inner product at level Γ ( N p ) . -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 25 Proof.
This is immediate on comparing the interpolation formulae for L RS p ( f /K , κ, s )and L naive p ( f /K , κ, s ) at s = k/ κ ∈ I . The equality in (7) follows from thefollowing well-known comparison of Petersson inner products: (cid:104) f ( κ ) , f ( κ ) (cid:105) N,p = β ( κ ) λ N ( f ( κ )) E ( f ( κ )) E ∗ ( f ( κ )) (cid:104) f ( κ ) ◦ , f ( κ ) ◦ (cid:105) N . (cid:3) Lemma 4.3.4. R f /K = (cid:104) N (cid:105) s − k r ( κ ) .Proof. The interpolation formulae for L RS p ( f /K , κ, s ) and L naive p ( f /K , κ, s ) (given bytaking η = and j ≡ k/ p −
1) in Theorems 4.2.1 and 4.1.4, so that thecharacter η (cid:104) χ cyc (cid:105) j − ω k/ − is crystalline at p ) together with Lemma 4.3.3 showthat L RS p ( f /K , κ, j ) = (cid:104) N (cid:105) j − κ r ( κ ) L naive p ( f /K , κ, j )for every κ ∈ I and j ∈ Z ∩ [1 , κ −
1] with j ≡ k/ p − (cid:3) Corollary 4.3.5. R f /K ( k, s ) = (cid:104) N (cid:105) s − k ( − k/ | D K | / (cid:104) f β , f β (cid:105) N,p Ω εf Ω εf K . In particular, R f /K ( k, k/
2) = ( − k/ Ω εf Ω εf K | D K | / (cid:104) f β , f β (cid:105) N,p . Corollary 4.3.6. L Kob p,β ( f /K , s ) = (cid:104) N (cid:105) s − k ( − k/ | D K | / (cid:104) f, f (cid:105) N Ω εf Ω εf K L p ( f β/ Q , s ) L p ( f β,K/ Q , s ) . Proof.
This is an immediate consequence of Lemma 4.3.3 and 4.3.4, on recallingthat our choices enforce the requirement that C ± f β = C ± f β,K = 1. (cid:3) Remark . Only in this remark, h denotes a primitive Hida family of tame level N and U p -eigenvalue α . We let h denote its specialization to weight 2 r ; suppose h is old at p and let us write α for the U p -eigenvalue on h . In this situation, Nekov´aˇrin [Nek95, I.5.10] constructed a two-variable p -adic L -function associated to h . Welet L Nek p ( h /K , s ) denote its restriction to cyclotomic characters.In this particular case, the distribution D RS h was constructed by Hida and itenjoys an interpolation property that is identical to one recorded in Theorem 4.1.4.One may specialize L RS p ( h /K , κ, s ) to the p -stabilized form h and obtain a p -adic L -function L Kob p,α ( h ◦ /K , s ) as above. One may compare the interpolation formulaefor the respective distributions giving rise to L Nek p ( h /K , s ) and L Kob p,α ( h ◦ /K , s ) todeduce that L Kob p,α ( h ◦ /K , s + r −
1) = L Nek p ( h /K , s ) . Proofs of Theorem 1.1.1 and Corollary 1.1.2
We shall assume until the end of this article that K (cid:54) = Q ( i ) , Q ( √− u K = 1. p -adic Gross–Zagier formula for non-ordinary eigenforms at non-critical slope. Suppose g = (cid:80) n =1 a n ( g ) q n ∈ S r (Γ ( N )) is a normalized eigen-form. We let a, b ∈ Q denote the roots of its Hecke polynomial X − a p ( g ) X + p r − at p . Suppose that v p ( ι p ( a p ( g ))) > < h := v p ( ι p ( b )) < v p ( ι p ( a ))so that we have 2 h < r −
1. Let g b ∈ S r (Γ ( N p )) denote the p -stabilizationcorresponding to the Hecke root b and let g be a Coleman family which admits g b as its specialization in weight 2 r . Theorem 4.1.4 applies and equips us with atwo-variable p -adic L -function L RS p ( g /K , κ, s ). Let us set L Kob p,b ( g /K , s ) := b λ N ( g ) E ( g ) E ∗ ( g ) L RS p ( g /K , r, s )as in Definition 4.3.1. The following p -adic Gross–Zagier formula is Kobayashi’swork [Kob19] in progress. Theorem 5.1.1 (Kobayashi) . dds L Kob p,b ( g /K , s ) (cid:12)(cid:12) s = r = (cid:18) − p w − b (cid:19) h Nek b,K ( z g , z g )(4 | D K | ) w − . The corollary below is a restatement of Theorem 5.1.1, taking the diagram (4)and Theorem 3.5.4 into account. Recall that we have to assume K (cid:54) = Q ( i ) , Q ( √− u K = 1. Corollary 5.1.2.
For each κ ∈ I as in Section 3.4 with κ ≥ k , dds L RS p ( f /K , κ, s ) (cid:12)(cid:12) s = κ = (cid:16) − p κ − b (cid:17) (4 | D K | ) κ − · h f ( κ ) ◦ , β ( κ ) ,K ( z f ( κ ) ◦ , z f ( κ ) ◦ )= ψ κ ◦ h f ,K (cid:0) Z f , Z f (cid:1) . Remark . Note that the assumption that κ ≥ k guarantees that we have2 v p ( β ( κ )) < κ −
1, as required to apply Kobayashi’s Theorem 5.1.1.The reason why we record this trivial alteration of Theorem 5.1.1 here is becauseboth sides of the asserted equality interpolate well as κ varies (unlike its predecessorTheorem 5.1.1). See Remark 4.3.2.5.2. A -adic Gross–Zagier formula. Recall the Coleman family f over the affi-noid algebra A = A ( U ) from Section 2.2. Recall also the A -valued cyclotomicheight pairing h f ,K we have introduced in Section 3.3 and the universal Heegnerpoint Z f ∈ ( V f /K , D β ) given as in Theorem 3.5.4. Recall finally also the two-variable p -adic L -function L RS p ( f /K , κ, s ) from Section 4. Definition 5.2.1.
Let us write H f ,K for the Amice transform of the height pairing h f ,K . In explicit terms, H f ,K ( x, y ) := h f ,K ( x, y ) (cid:12)(cid:12) w =(1+ p ) κ − k − . -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 27 Theorem 5.2.2 ( A -adic Gross–Zagier formula) . With the notation as above, thefollowing identity is valid in A : ∂∂s L RS p (cid:18) f /K , κ, s + κ − k (cid:19) (cid:12)(cid:12)(cid:12) s = k = H f ,K ( Z f , Z f ) . Proof.
Consider the difference D ( κ ) := ∂∂s L RS p (cid:18) f /K , κ, s + κ − k (cid:19) (cid:12)(cid:12)(cid:12) s = k − H f ,K ( Z f , Z f ) . It follows from the interpolative properties of L RS p ( f /K , κ, s ), the A -adic heightpairing h f ,K outlined in (4) and that of the universal Heegner cycle (Theorem 3.5.4)together with Corollary 5.1.2 show that D ( κ ) = 0 , ∀ κ ∈ I ∩ Z ≥ k . By the density of I ∩ Z ≥ k in the affinoid U , we conclude that D is identicallyzero, as required. (cid:3) Proof of Theorem 1.1.1.
On specializing the statement of Theorem 5.2.2( A -adic Gross–Zagier formula) to κ = k and relying once again on the interpolativeproperties of the A -adic height pairing h f ,K and that of the universal Heegner cycle Z f , the proof of Theorem 1.1.1 follows at once. (cid:3) Proof of Corollary 1.1.2.
Recall that we are assuming that the weight k = 2.Recall also that A f / Q stands for the abelian variety of GL -type that the Eichler-Shimura congruences associate to f and that we assume that L ( f / Q , s ) has a simplezero at s = 1.It follows from the classical (complex) Gross–Zagier formula and Theorem 1.1.1that(8) dds L Kob p,β ( f /K , s ) (cid:12)(cid:12) s =1 = (cid:18) − β (cid:19) L (cid:48) ( A f /K, | D K | / (cid:104) P f , P f (cid:105) ∞ ,K π (cid:104) f, f (cid:105) N h Nek β,K ( P f , P f )where we recall that P f ∈ A f ( K ) is the Heegner point and (cid:104) , (cid:105) ∞ ,K is the N´eron-Tateheight pairing over K . Since we know in our set up that Tr K/ Q P f is non-torsion, itis a non-zero multiple of P within the one-dimensional Q -vector space A f ( Q ) ⊗ Q .We may therefore replace in (8) the height pairings of P f over K with those of P over Q to deduce that(9) dds L Kob p,β ( f /K , s ) (cid:12)(cid:12) s =1 = (cid:18) − β (cid:19) L (cid:48) ( A f /K, | D K | / (cid:104) P, P (cid:105) ∞ π (cid:104) f, f (cid:105) N h Nek β, Q ( P, P )On the other hand we have(10) L (cid:48) ( A f /K,
1) = L (cid:48) ( A f / Q , L ( A Kf / Q , dds L Kob p,β ( f /K , s ) (cid:12)(cid:12) s =1 = − π Ω + f Ω + f K | D K | / π (cid:104) f, f (cid:105) N × dds L p,β ( f / Q , s ) (cid:12)(cid:12) s =1 (11) × (cid:18) − β (cid:19) L ( A Kf / Q , − πi Ω + f K (12) by the definition of L Kob p,β ( f /K , s ) and the interpolation property of L p,β ( f K/ Q , s ).Plugging the identities (10) and (11) in (9), the desired equality follows. (cid:3) Applications
We shall illustrate various applications of the p -adic Gross–Zagier formula atcritical slope (Theorem 1.1.1 and Corollary 1.1.2). These were already recorded inSection 1 as Theorems 1.1.5, 1.1.6 and 1.1.8. Before we give proofs of these results,we set some notation and record a number of preliminary results.6.1. Perrin-Riou’s big logarithm map and p -adic L -functions. Until the endof this article, we assume that f ∈ S (Γ ( N )) is a newform that does not admita θ -critical p -stabilization. Recall that A f / Q denotes the abelian variety of GL -type that the Eichler-Shimura congruences associate to f . Our assumption that L ( f / Q , s ) has a simple zero at s = 1 is still in effect. We assume also that theresidual representation ρ f (associated to the P -adic representation attached to f )is absolutely irreducible.Let P denote the prime of K f above that is induced by the embedding ι p andset T := (lim ←− A f ( Q )[ P n ]) ⊗ O f, P O and V = T ⊗ O E . Here, we recall that E isa finite extension of K f, P that contains both α and β (where α is the root of X − a p ( f ) X + p which is a p -adic unit and β = p/α is the other root) and O isits ring of integers. Since we assumed p >
2, the Fontaine-Laffaille condition holdstrue for V . In particular, there is an integral Dieudonn´e module D cris ( T ) ⊂ D cris ( V )constructed as in [Ber04, § IV]. We fix a ϕ -eigenbasis { ω α , ω β } of D cris ( T ). Let Log V : H ( Q p , V ⊗ Λ ι ) ⊗ H −→ D cris ( V ) ⊗ H denote Perrin-Riou’s big dual exponential map. We let L PR := Log V ◦ res p ( BK ) ∈ D cris ( V ) ⊗ H denote Perrin-Riou’s vector valued p -adic L -function, where BK ∈ H ( Q , V ⊗ Λ ι )is the Beilinson-Kato element. Set H E := H ⊗ Q p E and define L ( α )PR , L ( β )PR ∈ H E asthe coordinates of L PR with respect to the basis { ω α , ω β } , so that we have L PR = L ( α )PR ω α + L ( β )PR ω β . Note that L ( α )PR and L ( β )PR are well-defined only up to multiplication by an elementof O × . Definition 6.1.1.
Set D f β/ Q := ψ ( D f / Q ) ∈ D k − (Γ) ⊗ E . Associated to the other( p -ordinary) stabilization f α of f , we also have the Mazur–Swinnerton-Dyer mea-sure D f α/ Q ∈ D (Γ) ⊗ E . We remark that the measure D f α/ Q is characterized by aninterpolation formula identical to that for D f β/ Q (which does not characterize D f β/ Q itself ), exchanging every α in the formula with β and vice versa.For λ = α, β , we set L p,λ ( f, s ) := L p ( D f λ/ Q , s ) . The following result is due to Kato (when λ = α ) and has been announced byHansen (when λ = β ). See also related work by Ochiai [Och18]. -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 29 Theorem 6.1.2.
Suppose that f β is non-critical. Then for each λ ∈ { α, β } , thereexists c λ ∈ O × with L ( λ )PR = c λ · D f λ/ Q . Remark . When K f = Q and A f is an elliptic curve, there is a choice for the ϕ -eigenbasis of D cris ( V ) with which we can take c λ = 1 for λ = α, β . In this remark,we explain how to make this choice.Let A / Z be a minimal Weierstrass model of the elliptic curve A f . Let ω A denote a N´eron differential that is normalized as in [PR95, § + A f := (cid:82) A f ( R ) ω A >
0. Let ω cris ∈ D cris ( V ) denote the element that corresponds to ω A under the comparison isomorphism. The eigenbasis { ω α , ω β } is then given bythe requirement that ω α + ω β = ω cris . Proof of Theorem 1.1.6 (non-triviality of p -adic heights). Suppose on thecontrary that both h Nek α, Q and h Nek β, Q were trivial. It follows from Corollary 1.1.2and Perrin-Riou’s p -adic Gross–Zagier formula for the slope-zero p -adic L -function L p ( D f α / Q , s ) that ( L (cid:48) PR ) = 0 . Using [PR93, Proposition 2.2.2], we conclude that log V (BK ) = 0, or equiva-lently, that res p (BK ) = 0. Since ord s =1 L ( f / Q , s ) = 1, the theorem of Kolyvagin-Logachev shows that the compositum of the arrows A f ( Q ) ⊗ O f ,ι p E ∼ −→ H ( Q , V ) res p −→ H f ( Q p , V ) = A f ( Q p ) ⊗ O f, P E is injective. It follows that BK ∈ H ( Q , V ) is a torsion class, contradicting [B¨uy17,Theorem 1.2] . (cid:3) Birch and Swinnerton-Dyer formula for analytic rank one (Proof ofTheorem 1.1.8).
Recall the set Σ := { σ : K f (cid:44) → Q } of embeddings of K f into Q . Each embedding σ extends to σ : K f ( α ) (cid:44) → Q ; fix one such extension. Recallthat P is the prime induced by the embedding ι p : Q (cid:44) → Q p , which we extend to anisomorphism ι p : C ∼ −→ C p . To save ink, let us set λ σ in place of ι p ◦ σ ( λ ), where λ ∈ { α, β } .For each σ ∈ Σ, the field σ ( K f ) is the Hecke field K f σ of f σ and let P σ ⊂ σ ( K f )denote its prime induced by ι p . Let E = K f, P ( ι p ( α )) denote the extension of K f, P generated by ι p ( α ) and let O denote its ring of integers, m its maximal ideal. Weshall set E σ := K f σ , P σ ( α σ ) to ease notation and write O σ for its ring of integers, m σ for its maximal ideal. Let us write T σ := lim ←− A f ( Q )[ P nσ ], where the action of P σ act on A f is induced from σ ( K f ) σ − −→ K f , and we set V σ = T σ ⊗ E σ .We retain the set up in the previous section, except that we write for each σ ∈ Σ L σ PR ∈ D cris ( V σ ) ⊗ H E σ for Perrin-Riou’s vector valued p -adic L -function associated to f σ and the prime P σ of K f . The proof of this result is provided in op. cit. only when K f = Q , but the argument carriesover to treat the general case. Proposition 6.2.1.
Suppose the Iwasawa main conjecture holds true for each f σ/ Q .Then for each σ ∈ Σ , there exists λ σ ∈ { α σ , β σ } such that p -adic height pairing h Nek λ σ , Q : H ( Q , V σ ) ⊗ H ( Q , V σ ) −→ E σ is non-trivial.Proof. If ι p ◦ σ ◦ ι − ∞ ( a p ) is a p -adic unit, then this assertion is already provedin Theorem 1.1.6. Otherwise, the assertion follows (still as in the proof of The-orem 1.1.6) from Kobayashi’s p -adic Gross-Zagier formulae [Kob12, Theorem 3],validity of main conjectures up to µ -invariants in that set up (which we assume)and [B¨uy17, Theorem 1.2]. (cid:3) Proof of Theorem 1.1.8.
Since we assume the validity of main conjectures for f σ ,Perrin-Riou’s leading term formulae for her module of p -adic L -functions in[PR93, §
3] together with Theorem 6.1.2 and Proposition 6.2.1 show that the m σ -adic Birch and Swinnerton-Dyer conjecture (which corresponds to the statementBSD D λ ( V ) in [PR93, Proposition 3.4.6]) for A f is true up to m σ -adic units:ord m σ (cid:18) (1 − /λ σ ) − L (cid:48) p,λ σ ( f σ / Q , P σ ( A f / Q ) (cid:19) = length O σ (III( A f / Q )[ P ∞ σ ])(13) + ord m σ Tam( A f / Q )for every σ ∈ Σ. Here,(14) Reg P σ ( A f / Q ) = h Nek λ σ , Q ( P f σ , P f σ )[ A f ( Q ) ⊗ O f O σ : O σ · P f σ ] . and notice that the terms concerning the torsion groups A f ( Q )[ P ∞ σ ] and A ∨ f ( Q )[ P ∞ σ ]are omitted from this formula, as they are both trivial since we assume that ρ f = A f ( Q )[ P σ ] is absolutely irreducible.Applying either Corollary 1.1.2 (if the p -adic valuation of λ σ is 1), or Perrin-Riou’s p -adic Gross–Zagier formula at slope-zero (if λ σ is a p -adic unit) or Kobayashi’s p -adic Gross-Zagier formula at supersingular primes (if the p -adic valuation of λ σ is positive but less than 1), we see that(15) − L (cid:48) ( f σ / Q , ∞ ,σ ( A f / Q ) 2 πi Ω + f σ = (1 − /λ σ ) − L (cid:48) p,λ σ ( f σ , P σ ( A f / Q )where Reg ∞ ,σ ( A f / Q ) := (cid:104) P f σ , P f σ (cid:105) ∞ / [ A f ( Q ) ⊗ O f O σ : O σ · P f σ ]. We remark thatthis equality takes place in the field E σ = σ ( K f ) P σ ( α σ ). Combining (13) and (15),we infer thatord m σ (cid:32) − L (cid:48) ( f σ / Q , ∞ ,σ ( A f / Q )2 πi Ω + f σ (cid:33) = length O σ (III( A f / Q )[ P ∞ σ ])+ ord m σ Tam( A f / Q ) . Perrin-Riou’s Proposition 3.4.6 in [PR93] is written for the p -adic Tate module of an ellipticcurve, but it works verbatim for the Galois representation T σ (the P σ -adic Tate-module of A f ). Note that the element L σ PR we have introduced above is a generator of this module since weassume the truth of main conjectures. -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 31 The proof of the first assertion in Theorem 1.1.8 follows.We now explain the proof of its second portion; that the Birch and Swinnerton-Dyer formula for an elliptic curve A/ Q (satisfying the conditions of the secondportion of our theorem) is valid up to p -adic units. Let f denote the newformassociated to A . Based on our results above in the general case, we only needto prove that the rational number 2 πi Ω + f (cid:14) Ω + A is a p -adic unit. This amounts toshowing that the Manin constant c A is a p -adic unit. In our setting, this followsfrom [Maz78, Corollary 4.1], which states that if p | c A , then p | N . (cid:3) Remark . Recall the σ -part Reg ∞ ,σ ( A f / Q ) of the regulator Reg ∞ ( A f / Q )which we defined asReg ∞ ,σ ( A f / Q ) := (cid:104) P f σ , P f σ (cid:105) ∞ [ A f ( Q ) ⊗ O f O σ : O σ · P f σ ] , where the ring O σ is given as above. Note that the set { P f σ } σ ∈ Σ ⊂ A f ( Q ) givesrise to an orthogonal basis of A f ( Q ) ⊗ Q (with respect to the archimedean heightpairing), so that we have the factorizationReg ∞ ( A f / Q ) = (cid:89) σ ∈ Σ Reg ∞ ,σ ( A f / Q ) . Proof of Theorem 1.1.5 (Perrin-Riou’s conjecture).
We assume through-out Section 1.1.5 that f = f A is an eigenform of weight 2, which is associated tothe elliptic curve A / Q that has good ordinary reduction at p and that has analyticrank one. We also assume throughout that ρ A is absolutely irreducible.We shall follow the argument in the proof of Theorem 2.4(iv) of [B¨uy17] veryclosely, where the analogous assertion has been verified in the case when the prime p is a prime of good supersingular reduction. Essentially, the argument in op. cit.works verbatim, on replacing all references to Kobayashi’s work with referencesto Corollary 1.1.2, Perrin-Riou’s p -adic Gross–Zagier formula at slope zero andTheorem 1.1.6. We summarize it here for the convenience of the readers.Let us write Log V = Log
V,α · ω α + Log
V,β · ω β . Recall that ω ∗ A ∈ D cris ( V ) / Fil D cris ( V ) stands for the unique element such that[ ω A , ω ∗ A ] = 1. We define log A (res p (BK )) according to the identitylog V (res p (BK )) = log A (res p (BK )) · ω ∗ A . The dual basis of { ω α , ω β } with respect to the pairing [ , ] is { ω ∗ β , ω ∗ α } , where ω ∗ β (respectively, ω ∗ α ) is the image of ω ∗ A under the inverse of the isomorphism s D β : D β ∼ → D cris ( V ) / Fil D cris ( V ) (respectively, under the inverse of s D α ). Let λ ∈ { α, β } be such that the heightpairing h Nek λ, Q is non-trivial and let λ ∗ be given so that { λ, λ ∗ } = { α, β } . Then,(1 − /λ ) · c ( f ) · h Nek λ, Q ( P, P ) = L (cid:48) p,λ ( f, (cid:0) Log
V,λ ( ∂ λ BK ) (cid:1) = (cid:2) exp ∗ ( ∂ λ BK ) , (1 − p − ϕ − )(1 − ϕ ) − · ω ∗ λ ∗ (cid:3) = (1 − p − λ ∗ )(1 − /λ ∗ ) − [exp ∗ ( ∂ λ BK ) , ω ∗ A ]= (1 − /λ )(1 − /λ ∗ ) − [exp ∗ ( ∂ λ BK ) , log V (res p (BK ))]log A (res p (BK ))= − (1 − /λ )(1 − /λ ∗ ) − h Nek λ, Q (BK , BK )log A (res p (BK )) . Here: • The first equality follows from Perrin-Riou’s p -adic Gross–Zagier formula if λ = α , or else it is Corollary 1.1.2. • The second equality follows from the definition of
Log
V,λ and the fact thatit maps to Beilinson-Kato class to D f λ/ Q (Theorem 6.1.2); as well as thedefinition of the derived Beilinson-Kato class ∂ λ BK ∈ H ( (cid:101) D λ )which is given within the proof of Theorem 2.4 in [B¨uy17]. • The element ∂ λ BK ∈ H / f ( Q p , V ) := H ( Q p , V ) /H ( Q p , V )is the projection of the derived Beilinson-Kato class ∂ λ BK under the nat-ural map pr : H ( (cid:101) D λ ) −→ H / f ( Q p , V )and the third equality follows from the explicit reciprocity laws of Perrin-Riou (as proved by Colmez) (c.f. the discussion in [BL17b, Section 2.1]). • Fourth and fifth equalities follow from definitions (and using the fact that λλ ∗ = p ). • The final equality follows from the Rubin-style formula proved in [BB17,Theorem 4.13] and the comparison of various p -adic heights summarized inthe diagram (4).We therefore infer that(16) h Nek λ, Q (BK , BK )log A (res p (BK )) = − (1 − /α )(1 − /β ) · c ( f ) · h Nek λ, Q ( P, P ) . The fact that h Nek λ, Q ( ∗ , ∗ ) and (log A ◦ res p ( ∗ )) are both non-trivial quadratic formson the one dimensional Q p -vector space A ( Q ) ⊗ Q p and combining with (16), weconclude that h Nek λ, Q ( P, P )log A (res p ( P )) = h Nek λ, Q (BK , BK )log A (res p (BK )) = − (1 − /α )(1 − /β ) · c ( f ) · h Nek λ, Q ( P, P )log A (res p (BK )) . (cid:3) -ADIC GROSS–ZAGIER AT CRITICAL SLOPE 33 References [BB17] Denis Benois and Kazım B¨uy¨ukboduk,
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