Indivisibility of Heegner points and arithmetic applications
aa r X i v : . [ m a t h . N T ] A ug INDIVISIBILITY OF HEEGNER POINTS AND ARITHMETICAPPLICATIONS
ASHAY A. BURUNGALE, FRANCESC CASTELLA, AND CHAN-HO KIM
Abstract.
We upgrade Howard’s divisibility towards Perrin-Riou’s Heegner point main con-jecture to the predicted equality. Contrary to previous works in this direction, our main resultallows for the classical Heegner hypothesis and non-squarefree conductors. The main ingredi-ents we exploit are W. Zhang’s proof of Kolyvagin’s conjecture, Kolyvagin’s structure theoremfor Shafarevich–Tate groups, and the explicit reciprocity law for Heegner points.
Contents
1. Introduction 12. Selmer structures 43. Heegner point Kolyvagin systems 54. Equivalent main conjectures 75. Equivalent special value formulas 76. Skinner–Urban lifting lemma 97. Proof of the main results 9References 101.
Introduction
Let E/ Q be an elliptic curve of conductor N and let K be an imaginary quadratic field ofdiscriminant D K with ( D K , N ) = 1. Then K determines a factorization N = N + N − with N + (resp. N − ) divisible only by primes which are split (resp. inert) in K . Throughoutthis paper, the following hypothesis will be in force: Assumption 1.1 (Generalized Heegner hypothesis) . N − is the square-free product of an even number of primes. Let p > E with ( p, D K ) = 1, and let K ∞ be the anti-cyclotomic Z p -extension of K . Under Assumption 1.1, the theory of complex multiplicationprovides a collection of CM points on a Shimura curve with “Γ ( N + )-level structure” attachedto the quaternion algebra B/ Q of discriminant N − defined over ring class extensions of K .By modularity, these points give rise to Heegner points on E defined over the ring class exten-sions, and exploiting the p -ordinarity assumption these can be turn into a norm-compatiblesystem of Heegner points on E over the Z p -extension K ∞ /K .Let T be the p -adic Tate module of E , and set V := T ⊗ Q p and A := V /T ≃ E [ p ∞ ]. LetΛ = Z p J Gal( K ∞ /K ) K be the anticyclotomic Iwasawa algebra, and let T and A be the Λ-adic Date : August 23, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Iwasawa theory, Heegner points, Kolyvagin systems. ersions of T and A , respectively, recalled in Section 2. LetSel Gr ( K, T ) ⊂ lim ←− H ( K n , T ) , Sel Gr ( K, A ) ⊂ lim −→ H ( K n , A )be Λ-adic Greenberg ordinary Selmer groups defined in [How04a, How04b] (and recalled inSection 2 below), where K n is the unique subextension of K ∞ with [ K n : K ] = p n . LettingSel p m ( E/K n ) ⊂ H ( K n , E [ p m ]) be the p m -th descent Selmer groups fitting into the fundamen-tal exact sequence0 −→ E ( K n ) ⊗ Z /p m Z −→ Sel p m ( E/K n ) −→ X ( E/K n )[ p m ] −→ , there are Λ-module pseudo-isomorphismsSel Gr ( K, T ) ∼ lim ←− n lim ←− m Sel p m ( E/K n ) , Sel Gr ( K, A ) ∼ lim −→ n lim −→ m Sel p m ( E/K n ) . Via the Kummer map, the norm-compatible sequence of Heegner points on E along K ∞ /K gives rise to a Λ-adic Heegner cohomology class κ ∞ ∈ Sel Gr ( K, T ) which was first shown to benon-torsion by Cornut–Vatsal [CV07]. After Kolyvagin [Kol88], the non-triviality of a Heegnerpoint over a ring class field H/K implies that the Mordell–Weil rank of the underlying abelianvariety over H being one and also the finiteness of the corresponding Tate–Shafarevich group,with the index of the Heegner point in the Mordell–Weil group being closely to the size ofthe Tate–Shafarevich group (essentially, the latter is the square of the former). After Perrin-Riou [PR87, Conj. B] and Howard [How04b], a Λ-adic analogue of this result takes the formof the following “Heegner point main conjecture”, where we let ι : Λ → Λ be the involutioninduced by the inversion in Gal( K ∞ /K ). Conjecture 1.2 (Perrin-Riou, Howard) . The Λ -modules Sel Gr ( K, T ) and Sel Gr ( K, A ) haverank and corank 1, respectively. Letting X = Hom Z p (Sel Gr ( K, A ) , Q p / Z p ) be the Pontrjagin dual of Sel Gr ( K, A ) , there is a torsion Λ -module M ∞ with: (1) char( M ∞ ) = char( M ∞ ) ι , (2) X ∼ Λ ⊕ M ∞ ⊕ M ∞ , (3) char( M ∞ ) = char (cid:18) Sel Gr ( K, T )Λ κ ∞ (cid:19) . The third statement is a form of Iwasawa main conjecture involving zeta elements whichsimilarly appears in other settings, notably [Rub90, Thm. 5.1] and [Kat04, Conj. 12.10]. Here,we assume that the Manin constant is 1 and that O × K = {± } for notational simplicity.1.1. Main result.
Similarly as in [Zha14, Notations], we consider the following condition onthe triple (
E, p, K ): Assumption 1.3 (Condition ♥ ) . Denote by
Ram( ρ ) the set of primes ℓ dividing exactly N such that the G Q -module E [ p ] is ramified at ℓ . Then: (1) Ram( ρ ) contains all primes ℓ k N + . (2) Ram( ρ ) contains all primes ℓ | N − . (3) If N is not square-free, then ρ ) > , and either Ram( ρ ) contains a prime ℓ | N − or there are at least two primes ℓ k N + . (4) If ℓ | N + , then H ( Q ℓ , ρ ) = 0 . Remark 1.4.
This is a slight strengthening of Condition ♥ in [Zha14], where in part (2) E [ p ]is only required to be ramified at the primes ℓ | N − with ℓ ≡ ± p ).Under Condition ♥ (and other hypotheses recalled in Theorem 3.2 below), W. Zhang [Zha14]has recently obtained a proof of Kolyvagin’s conjecture [Kol91a]. Concerning the nature ofthis conjecture, let us just mention here that it concerns the p -indivisibility of so-called derivedHeegner classes on E , and as such it does not seem to have an Iwasawa-theoretic flavour. n this note, we shall build on W. Zhang’s result to prove the following theorem towardsConjecture 1.2, where in addition to Assumptions 1.1 and 1.3, the following is in force: Assumption 1.5. (1) p = pp splits in K . (2) ρ : G K → Aut F p ( E [ p ]) is surjective. Theorem 1.6 (Main result) . Suppose that the triple ( E, K, p ) satisfies Assumptions 1.1, 1.3,and 1.5, and assume in addition that ord s =1 L ( E/K, s ) = 1 . Then Conjecture 1.2 holds.
Outline of the proof.
After Perrin-Riou’s work, the first results towards Conjecture 1.2were due to Bertolini [Ber95] and Howard [How04a, How04b], which under mild hypothesesestablished one of the divisibilities predicted by the third statement in the conjecture. Moreprecisely, adapting to the anticyclotomic setting the Kolyvagin system machinery of Mazur–Rubin [MR04], Howard constructed a Λ -adic Kolyvagin system κ ∞ whose base class κ ∞ couldbe shown to be non-trivial by Cornut–Vatsal [CV07], yielding a proof of all the statements inConjecture 1.2 except for the divisibility “ ⊆ ” in the third part.Later, the first cases of the full Conjecture 1.2 were obtained in [Wana, Thm. 1.2] and [Cas17,Thm. 3.4]. These were obtained by building on X. Wan’s work [Wanb], which when combinedwith the reciprocity law for Heegner points [CH18] yields a proof of the missing divisibility “ ⊆ ”.However, these results excluded the case N − = 1 (i.e. the “classical” Heegner hypothesis) and N is assumed to be square-free. In contrast, our proof of Theorem 1.6 is based on a differentidea, dispensing with the use of [Wanb] and allowing for those excluded cases. Moreover, weexpect the analytic rank 1 hypothesis made in Theorem 1.6 to not be essential to our method(see Remark 1.8).As alluded to above, Howard’s results in [How04a, How04b] are based on the Mazur–Rubinmachinery of Kolyvagin systems, suitably adapted to the anticyclotomic setting. As essentiallyknown to Kolyvagin [Kol91a], the upper bound provided by this machinery can be shown to be sharp under a certain nonvanishing hypothesis; in the framework of [MR04], this correspondsto the Kolyvagin system being primitive [MR04, Def. 4.5.5 and 5.3.9].Even though primitivity was not incorporated into Howard’s treatment [How04a,How04b] ,we shall upgrade his divisibility to an equality by building on W. Zhang’s proof of Kolyvagin’sconjecture [Zha14]. In order to carry out this strategy, we consider a different anticyclotomicmain conjecture. Under Assumption 1.5, Bertolini–Darmon–Prasanna [BDP13] (as extendedby Brooks [HB15] for N − = 1) have constructed a p -adic L -function L BDP p ∈ Λ ur := Z ur p ˆ ⊗ Λ,where Z ur p is the completion of the maximal unramified extension of Z p , p -adically interpolatinga square-root of certain Rankin–Selberg L -values. A variant of Greenberg’s main conjectures[Gre94] relates L BDP p to the characteristic ideal of an “ N − -minimal” anticyclotomic Selmergroup Sel N − ∅ , ( K, A ) ⊂ lim −→ H ( K n , A )defined in [JSW17, § Gr ( K, A ) bythe defining local conditions at the primes dividing p (and possibly N − ): Conjecture 1.7.
The Pontrjagin dual X N − ∅ , of Sel N − ∅ , ( K, A ) is Λ -torsion, and we have char( X N − ∅ , )Λ ur = ( L BDP p ) . As a key step in our proof, in Section 4 we establish the equivalence between Conjecture 1.2and Conjecture 1.7. In particular, we show that Howard’s divisibility implies the divisibility“ ⊇ ” in Conjecture 1.7. In Section 5, assuming(1) rank Z E ( K ) = 1 , See [How06] however, esp. Theorem 3.2.3, although we will have no use for any of the results in that paper. y a useful commutative algebra lemma from [SU14] and the “anticyclotomic control theorem”of [JSW17], we reduce the proof of the opposite divisibility to the proof of the equality(2) [ E ( K ) : Z .P ] = X ( E/K )[ p ∞ ] Y ℓ | N + c ℓ up to a p -adic unit, where P ∈ E ( K ) is a p -primitive generator of E ( K ) up to torsion, and c ℓ is the Tamagawa number of E/ Q ℓ . Under the hypotheses of Theorem 1.6, equalities (1) and(2) follow from the Gross–Zagier formula [GZ86, YZZ13], and the work of Kolyvagin [Kol91a],and W. Zhang [Zha14], with P ∈ E ( K ) given by the trace of a Heegner point defined over theHilbert class field of K , yielding our main result. Remark 1.8.
By the work of Cornut–Vatsal [CV07], the Heegner points y n ∈ E ( K n ) definedover the n -th layer of the anticyclotomic Z p -extension are non-torsion for n sufficiently large.Taking one such n , and letting y n,χ ∈ E ( K n ) χ ⊂ E ( K n ) ⊗ Z [Gal( K n /K )] Z [ χ ]be the image of y n in the χ -isotypical component of for a primitive character χ : Gal( K n /K ) → Z [ χ ] × , one can use Kolyvagin’s methods (as extended in [BD90]) to establish the rank oneproperty of E ( K n ) χ , and the Gross–Zagier formula [YZZ13] combined with a generalization ofKolyvagin’s structure theorem for Shafarevich–Tate groups [Kol91b] should yield an analogueof (2) in terms of the index of y n,χ .2. Selmer structures
We keep the notations from the Introduction. In particular, E/ Q is an elliptic curve ofconductor N with good ordinary reduction at a prime p >
3, and K is an imaginary quadraticfield of discriminant prime to N p in which p = pp splits. Throughout the rest of this paper,we also fix once and for all a choice of complex and p -adic embeddings C ι ∞ ← ֓ Q ι p ֒ → Q p .Let Σ be a finite set of places of K including the places lying above p , ∞ and the primesdividing N . For a finite extension F of K , let F Σ denote the maximal extension of F unramifiedoutside the places lying above Σ. Following [MR04], given a Selmer structure F = {F w } w | v,v ∈ Σ on a G K -module M , we define the associated Selmer group Sel F ( F, M ) bySel F ( F, M ) := ker (cid:18) H ( F Σ /F, M ) −→ Y w H ( F w , M )H F ( F w , M ) (cid:19) . If M is a G K -module and L/K a finite Galois extension, we have the induced representationInd
L/K M := { f : G K → M : f ( σx ) = f ( x ) σ for all x ∈ G K , σ ∈ G L } , which is equipped with commuting actions of G K and Gal( L/K ). Consider the modules T := lim ←− (cid:0) Ind K n /K T (cid:1) , A := lim −→ (cid:0) Ind K n /K A (cid:1) ≃ Hom( T , µ p ∞ ) , where the limits are with respect to the corestriction and restriction maps, respectively. Theseare finitely and cofinitely generated over Λ, respectively.We recall the ordinary filtrations at p . Let G Q p := Gal( Q p / Q p ), viewed as a decompositiongroup at p inside G Q via ι p . By p -ordinarity, there is a one-dimensional G Q p -stable subspaceFil + V ⊂ V such that the G Q p -action on the quotient Fil − V := V /
Fil + V is unramified. SetFil + T := T ∩ Fil + V, Fil − T := T /
Fil + T, Fil + A := Fil + V /
Fil + T, Fil − A := A/ Fil + A, and define the submodules Fil + T ⊂ T and Fil + A ⊂ A byFil + T := lim ←− Ind K n /K Fil + T, Fil + A := lim −→ Ind K n /K Fil + A, and set Fil − T := T / Fil + T and Fil − A := A / Fil + A . ollowing the terminology introduced in [Cas17, § M denotes any of the G K -modulesabove, we consider the following three local conditions at a place v lying above p :H ∅ ( F v , M ) := H ( F v , M ) , H ( F v , M ) := ker (cid:0) H ( F v , M ) −→ H ( F v , Fil − M ) (cid:1) , H ( F v , M ) := 0 . We also recall two local conditions at a place v not lying above p :H ( F v , M ) := 0 , H ( F v , M ) := H ( F v /I v , M I v ) . If F = K and M = A , then H ( F v , M ) = H ( F v , M ) unless v divides N − and ρ is unramifiedat v as in [PW11, Page 1362].Using these local conditions, for a, b ∈ {∅ , Gr , } we defineSel a,b ( K, T ) := ker (cid:18) H ( K Σ /K, T ) → H ( K p , T )H a ( K p , T ) × H ( K p , T )H b ( K p , T ) × Y v ∈ Σ ,v ∤ p H ( K v , T )H ( K v , T ) (cid:19) . In particular, Sel Gr ( K, T ) := Sel Gr , Gr ( K, T ) coincides with the Λ-adic Selmer group of [How04a,Def. 2.2.6], which we shall denote by H F Λ ( K, T ) following the notation in [How04a]. The samedefinitions and notational convention applies to A . Putting the trivial local condition at primesdividing N − , we can also define the N − -minimal variant of discrete Selmer group bySel N − a,b ( K, A ) := ker (cid:18) Sel a,b ( K, A ) → Y v | N − H ( K v , A ) (cid:19) . Remark 2.1. If v divides N − and ρ is unramified at v , then H ( K v , A ) = H ( K v , A ) since v splits completely in K ∞ /K . See [PW11, Lem. 3.4] for the exact difference. Indeed, the N − -minimal Selmer groups are practically preferred in the anticyclotomic Iwasawa theory formodular forms since the mod p n Selmer groups with the N − -ordinary local condition [PW11, § n under the tame ramificationcondition described in Remark 1.4. See [PW11, Prop. 3.6] for detail.3. Heegner point Kolyvagin systems
In this section we briefly recall the statement of Howard’s theorems towards Conjecture 1.2,as well as the results from Wei Zhang’s proof of Kolvyagin’s conjecture that we shall use toupgrade Howard’s divisibility to an equality.Let X := Hom Z p (Sel Gr ( K, A ) , Q p / Z p )be the Pontrjagin dual of the Λ-adic Greenberg Selmer group. Since we shall not directlyneed it here, we refer the reader to [How04a, § κ ∞ = { κ ∞ n } n ∈N (attached to a G K -module M together with a Selmer structure F ), where n runs over the set of square-free products of certain primes inert in K , with the conventionthat 1 ∈ N . Theorem 3.1.
Assume that p > is a good ordinary prime for E , D K is coprime to pN ,and ρ is surjective. Let F Λ be the Selmer structure for the Greenberg Selmer group. Then: (1) There exists a Λ -adic Kolyvagin system κ ∞ for ( T , F Λ ) with κ ∞ = 0 , (2) Sel Gr ( K, T ) is a torsion-free, rank one Λ -module. (3) There is a torsion Λ -module M ∞ such that char( M ∞ ) = char( M ∞ ) ι and a pseudo-isomorphism X ∼ Λ ⊕ M ∞ ⊕ M ∞ . (4) char( M ∞ ) divides char (cid:0) Sel Gr ( K, T ) / Λ κ ∞ ) . roof. This is [How04a, Thm. 2.2.10], as extended in [How04b, Thm. 3.4.2] to the case N − = 1.The non-triviality of κ ∞ follows from the work of Cornut–Vatsal [CV07]. (cid:3) Following [Zha14], we say that a prime ℓ is called a Kolyvagin prime if ℓ is prime to pN D K ,inert in K , and the index M ( ℓ ) := min p { v p ( ℓ + 1) , v p ( a ℓ ) } is strictly positive, where a ℓ = ℓ + 1 − E ( F ℓ ). Let δ w : E ( F w ) ⊗ Z Z p −→ H ( F w , T )be the local Kummer map, and let F be the Selmer structure on T given by H F ( F w , T ) :=im( δ w ). As explained in [How04a, § § N − = 1),Heegner points give rise to a (mod p M ) Kolyvagin system κ = (cid:8) κ n = c M ( n ) ∈ H ( K, E [ p M ]) : 0 < M M ( n ) , n ∈ N (cid:9) for ( T /p M T, F ), where N denotes the set of square-free products of Kolyvagin primes, andfor n ∈ N we set M ( n ) := min { M ( ℓ ) : ℓ | n } , with M (1) = ∞ by convention. Theorem 3.2.
Assume that: • p > is a good ordinary prime for E , • D K is coprime to pN , • Condition ♥ holds for ( E, p, K ) , • G K → Aut F p ( E [ p ]) is surjective.Then the collection of mod p cohomology classes (3) κ = { κ n = c ( n ) ∈ H ( K, E [ p ]) : n ∈ N } is nonzero. In particular, κ n = 0 for some n .Proof. This is [Zha14, Thm. 9.3]. (cid:3)
Remark 3.3.
In the terminology of [MR04], Wei Zhang’s Theorem 3.2 may be interpretedas establishing the primitivity of the system κ . Mazur–Rubin also introduced the (weaker)notion of Λ- primitivity for the cyclotomic analogue of κ ∞ (see [MR04, Def. 5.3.9]), and insome sense our main result in this paper may be seen as a realization of the implications κ is primitive = ⇒ κ ∞ is Λ-primitive = ⇒ Conjecture 1.2 holds , where κ ∞ is Howard’s Heegner point Λ-adic Kolyvagin system from Theorem 3.1.Combined with Kolyvagin’s work, Theorem 3.2 yields the following exact formula the orderof X ( E/K )[ p ∞ ] that we shall need. Corollary 3.4.
Let the hypotheses be as in Theorem 3.2. If ord s =1 L ( E/K, s ) = 1 , then ord p ( X ( E/K )[ p ∞ ]) = 2 · ord p [ E ( K ) : Z .y K ] where y K ∈ E ( K ) is a Heegner point.Proof. After Theorem 3.2 (more precisely, the non-vanishing of (3)), this follows from Kolyva-gin’s structure theorem from X ( E/K ) [Kol91b] (see also [McC91]), using that y K has infiniteorder by the Gross–Zagier formula [GZ86, YZZ13] ( cf. [Zha14, Thm. 10.2]). (cid:3) . Equivalent main conjectures
In this section we establish the equivalence between Conjecture 1.2 (the Heegner point mainconjecture) and Conjecture 1.7 (the Iwawawa–Greenberg main conjecture for L BDP p ) in theIntroduction.To ease the notation, for a, b ∈ {∅ , Gr , } we let X a,b denote the Pontrjagin dual of thegeneralized Selmer group Sel a,b ( K, A ), keeping the earlier convention that X := X Gr , Gr . Theorem 4.1.
Suppose E [ p ] is ramified at all primes ℓ | N − . Then Conjectures 1.2 and 1.7are equivalent. More precisely, X has Λ -rank if and only X ∅ , is Λ -torsion, and one-sideddivisibility holds in Conjecture 1.2(3) if and only if the same divisibility holds in Conjecture 1.7.Proof. This is essentially shown in the Appendix of [Cas17] ( cf. [Wana, § a,b ( K, A ) =Sel N − a,b ( K, A ) by assumption. If X has Λ-rank 1, then Sel Gr ( K, T ) has Λ-rank 1 by Lemma 2.3(1),and hence X ∅ , is Λ-torsion by Lemma A.4. Conversely, assume that X ∅ , is Λ-torsion. Then X Gr , is also Λ-torsion (see eq. (A.7)), and so X Gr , ∅ has Λ-rank 1 by Lemma 2.3(2). Now,global duality yields the exact sequence(4) 0 −→ coker(loc p ) −→ X ∅ , Gr −→ X −→ , where loc p : Sel Gr ( K, T ) → H ( K p , T ) is the restriction map. Since H ( K p , T ) has Λ-rank1, the leftmost term in (4) is Λ-torsion by Theorem A.1 and the nonvanishing of L BDP p (seeTheorem 1.5); since X Gr , ∅ ≃ X ∅ , Gr by the action of complex conjugation, we conclude from(4) that X has Λ-rank 1.Next, assume that X has Λ-rank 1. By Lemma 2.3(1), this amounts to the assumption thatSel Gr ( K, T ) has Λ-rank 1, and so by Lemmas A.3 and A.4 for every height one prime P of Λwe have(5) length P ( X ∅ , ) = length P ( X tors ) + 2 length P (coker(loc p )) , where X tors denotes the Λ-torsion submodule of X , and for every height one prime P ′ of Λ ur (6) ord P ′ ( L BDP p ) = length P ′ (coker(loc p )Λ ur ) + length P ′ (cid:18) Sel Gr ( K, T )Λ ur Λ ur κ ∞ (cid:19) . Thus for any height one prime P of Λ, letting P ′ denote its extension to Λ ur , we see from (5)and (6) thatlength P ( X tors ) P (cid:18) Sel Gr ( K, T )Λ κ ∞ (cid:19) ⇐⇒ length P ( X ∅ , ) P ′ ( L BDP p ) , and similarly for the opposite inequalities. The result follows. (cid:3) Remark 4.2.
Accounting for the difference between the unramified (as implicitly used here)and the strict local conditions in H ( F w , A ) for w | ℓ | N − in terms of p -parts of the correspondingTamagawa numbers (see e.g. [PW11, § E [ p ]. Indeed, the difference will only affect the µ -invariants of both sides due to [PW11, Lem. 3.4].5. Equivalent special value formulas
The goal of this section is to establish Corollary 5.4 below, which is a manifestation of theequivalence of Theorem 4.1 after specialization at the trivial character.
Theorem 5.1.
Assume that rank Z E ( K ) = 1 , X ( E/K ) < ∞ , and E [ p ] is irreducibleas G Q -module. Then X N − ∅ , is Λ -torsion, and letting f ∅ , ( T ) ∈ Z p J T K be a generator of its haracteristic ideal, the following equivalence holds: f ∅ , (0) ∼ p (cid:18) − a p + pp (cid:19) · log ω E ( P ) ⇐⇒ [ E ( K ) : Z .P ] ∼ p X ( E/K )[ p ∞ ] Y ℓ | N + c ℓ , where P ∈ E ( K ) is any generator of E ( K ) ⊗ Z Q , c ℓ is the Tamagawa number of E/ Q ℓ , and ∼ p denotes equality up to a p -adic unit. Remark 5.2.
Note that no Tamagawa defect at the primes dividing N − is assigned in theRHS due to the N − -minimal local condition of the Selmer group. Indeed, c ℓ for ℓ dividing N − becomes trivial in our setting due to Condition ♥ (Condition 1.3.(2)). See [PW11, Prop.3.7] for the definite case. Proof.
As shown in [JSW17, p. 395-6], our assumptions imply hypotheses (corank 1), (sur),and (irred K ) of [JSW17, § loc. cit. , Thm. 3.3.1] (with S = S p the set of primesdividing N and Σ = ∅ ) the module X N − ∅ , is Λ-torsion, and(7) Z p /f ∅ , (0) = N − ∅ , ( K, E [ p ∞ ]) · C ( E [ p ∞ ]) , where C ( E [ p ∞ ]) := ( K p , E [ p ∞ ]) · ( K p , E [ p ∞ ]) · Y w | N + ( K w , E [ p ∞ ]) . On the other hand, from [JSW17, (3.5.d)] we have(8) N − ∅ , ( K, E [ p ∞ ]) = X ( E/K )[ p ∞ ] · (cid:18) Z p / ( − a p + pp ) · log ω E P )[ E ( K ) : Z .P ] p · ( K p , E [ p ∞ ]) (cid:19) , where P ∈ E ( K ) is any generator of E ( K ) ⊗ Z Q , and [ E ( K ) : Z .P ] p denotes the p -part of theindex [ E ( K ) : Z .P ]. Combining (7) and (8) we thus arrive at Z p /f ∅ , (0) = X ( E/K )[ p ∞ ] · (cid:18) Z p / ( − a p + pp ) · log ω E P )[ E ( K ) : Z .P ] p (cid:19) · Y w | N + ( K w , E [ p ∞ ]) . Since the order of H ( K w , E [ p ∞ ]) is the p -part of the Tamagawa number of E/K w , the resultfollows. (cid:3) The fundamental p -adic Waldspurger formula due to Bertolini–Darmon–Prasanna [BDP13]will allow us to relate the left-hand side of Theorem 5.1 to the anticyclotomic main conjecture. Theorem 5.3 (Bertolini–Darmon–Prasanna) . The following special value formula holds: L BDP p (0) = (cid:18) − a p + pp (cid:19) · (cid:0) log ω E y K (cid:1) , where y K ∈ E ( K ) is a Heegner point.Proof. This is a special case of [BDP13, Theorem 5.13] ( cf. [BDP12, Theorem 3.12]) and [HB15,Theorem 1.1]. (cid:3)
Corollary 5.4.
With notations and hypotheses as in Theorem 5.1, assume in addition that ( E, p, K ) satisfies Condition ♥ . Then the following equivalence holds: f ∅ , (0) ∼ p L BDP p (0) ⇐⇒ [ E ( K ) : Z .P ] ∼ p X ( E/K )[ p ∞ ] for P a p -unit multiple of the Heegner point y K ∈ E ( K ) .Proof. Since Condition ♥ forces all the Tamagawa numbers c ℓ for the primes ℓ | N + to be p -adicunits, the result follows from Theorem 5.1 and Theorem 5.3. (cid:3) Remark 5.5.
Here we require P to be a p -unit multiple of the Heegner point y K , as otherwisethe logarithm log ω E P and the index [ E ( K ) : Z .P ] can be divisible by an extra power of p . . Skinner–Urban lifting lemma
We recall the following “easy lemma” in [SU14].
Lemma 6.1.
Let A be a ring and a be a proper ideal contained in the Jacobson radical of A .Assume that A/ a is a domain. Let L ∈ A be such that its reduction modulo a is non-zero. Let I ⊆ ( L ) be an ideal of A and I be the image of I in A/ a . If L (mod a ) ∈ I , then I = ( L ) .Proof. This is [SU14, Lem. 3.2]. (cid:3)
For our application, we shall set A := Λ, a := ( γ −
1) the augmentation ideal of Λ, L := f ∅ , ( T ) a generator of the characteristic ideal of X ∅ , , and I the ideal generated by L BDP p .The divisibility I ⊆ ( L ) will be a consequence of Theorem 3.1 and Theorem 4.1, and (assuminganalytic rank 1) the relations 0 = f ∅ , (0) ∼ p L BDP p (0) will be deduced from Corollary 3.4 andCorollary 5.4; the equality I = ( L ) will then follow. Remark 6.2.
Note that the roles of algebraic and analytic p -adic L -functions are switchedin our setting comparing with those of [SU14]. This is possible since Λ is a UFD, and so thecharacteristic ideal of a finitely generated Λ-module is principal.7. Proof of the main results
We are now ready to prove our main results.
Theorem 7.1.
Let E/ Q be an elliptic curve of conductor N , let p > be a good ordinaryprime for E , and let K be an imaginary quadratic field of discriminant D K with ( D K , N p ) = 1 .Assume that: • N − is the square-free product of an even number of primes. • ( E, p, K ) satisfies Condition ♥ . • G K → Aut F p ( E [ p ]) is surjective. • p = pp splits in K .In addition, assume that ord s =1 L ( E/K, s ) = 1 . Then Conjecture 1.2 holds.Proof.
By Theorem 3.1, the Pontryagin dual X of Sel Gr ( K, A ) has Λ-rank 1, and its Λ-torsionsubmodule X tors is such thatchar( X tors ) ⊇ char (cid:18) Sel Gr ( K, T )Λ κ ∞ (cid:19) . By Theorem 4.1, it follows that the Pontrjagin dual X N − ∅ , of Sel N − ∅ , ( K, A ) is Λ-torsion, andwe have(9) ( f ∅ , ) ⊇ ( L BDP p ) , where f ∅ , ∈ Λ is a generator of the characteristic ideal char( X N − ∅ , ). On the other hand, by thework of Gross–Zagier and Kolyvagin, the assumption that ord s =1 L ( E/K, s ) = 1 implies theHeegner point y K ∈ E ( K ) is non-torsion, and we have rank Z E ( K ) = 1 and X ( E/K ) < ∞ ;while by Corollary 3.4 we have[ E ( K ) : Z .y K ] ∼ p X ( E/K )[ p ∞ ] . In light of Corollary 5.4, the last equality (up to a p -adic units) amounts to the equality(10) f ∅ , (0) ∼ p L BDP p (0) , and given (9) and (10), the result follows from Lemma 6.1. (cid:3) Corollary 7.2.
Let the hypotheses be as in Theorem 7.1. Then Conjecture 1.7 holds.Proof.
Since E [ p ] is ramified at all primes ℓ | N − by hypothesis, the result follows from Theo-rem 4.1 and Theorem 7.1. (cid:3) eferences [BD90] Massimo Bertolini and Henri Darmon, Kolyvagin’s descent and Mordell-Weil groups over ring classfields , J. Reine Angew. Math. (1990), 63–74.[BD05] ,
Iwasawa’s main conjectures for elliptic curves over anticyclotomic Z p -extensions , Ann. ofMath. (2) (2005), no. 1, 1–64.[BDP12] Massimo Bertolini, Henri Darmon, and Kartik Prasanna, p -adic Rankin L -series and rational pointson CM elliptic curves , Pacific J. Math. (2012), no. 2, 261–303, Jonathan Rogawski MemorialIssue.[BDP13] , Generalized Heegner cycles and p -adic Rankin L -series , Duke Math. J. (2013), no. 6,1033–1148, Appendix by Brian Conrad.[Ber95] Massimo Bertolini, Selmer groups and Heegner points in anticyclotomic Z p -extensions , Compos.Math. (1995), no. 2, 153–182.[Cas17] Francesc Castella, p -adic heights of Heegner points and Beilinson-Flach classes , J. Lond. Math. Soc.(2) (2017), no. 1, 156–180.[CH18] Francesc Castella and Ming-Lun Hsieh, Heegner cycles and p -adic L -functions , Math. Ann. (2018), no. 1-2, 567–628.[CV07] Christophe Cornut and Vinayak Vatsal, Nontriviality of Rankin-Selberg L -functions and CM points , L -functions and Galois representations (Cambridge) (David Burns, Kevin Buzzard, and Jan Nekov´aˇr,eds.), London Math. Soc. Lecture Note Ser., vol. 320, Cambridge University Press, 2007, pp. 121–186.[Gre94] Ralph Greenberg, Iwasawa theory and p -adic deformations of motives , Motives (Seattle, WA, 1991),Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 193–223.[GZ86] Benedict Gross and Don Zagier, Heegner points and derivatives of L -series , Invent. Math. (1986),no. 2, 225–320.[HB15] Ernest Hunter Brooks, Shimura curves and special values of p -adic L -functions , Int. Math. Res. Not.IMRN (2015), no. 12, 4177–4241.[How04a] Benjamin Howard, The Heegner point Kolyvagin system , Compos. Math. (2004), no. 6, 1439–1472.[How04b] ,
Iwasawa theory of Heegner points on abelian varieties of GL type , Duke Math. J. (2004), no. 1, 1–45.[How06] , Bipartite Euler systems , J. Reine Angew. Math. (2006), 1–25.[JSW17] Dimitar Jetchev, Christopher Skinner, and Xin Wan,
The Birch and Swinnerton-Dyer formula forelliptic curves of analytic rank one , Camb. J. Math. (2017), no. 3, 369–434.[Kat04] Kazuya Kato, p -adic Hodge theory and values of zeta functions of modular forms , Ast´erisque (2004), 117–290.[Kol88] Victor Kolyvagin, On the Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves (Rus-sian) , Izv. Akad. Nauk SSSR Ser. Mat. (1988), no. 6, 1154–1180, 1327, Math. USSR-Izv. (1989), no. 3, 473–499.[Kol91a] , On the structure of Selmer groups , Math. Ann. (1991), no. 2, 253–259.[Kol91b] ,
On the structure of Shafarevich-Tate groups , Algebraic Geometry (Spencer Bloch, IgorDolgachev, and William Fulton, eds.), Lecture Notes in Math., vol. 1479, Springer, 1991, Proceedingsof the US-USSR Symposium held in Chicago, June 20–July 14, 1989, pp. 94–121.[Lan90] Serge Lang,
Cyclotomic Fields I and II , combined 2nd ed., Grad. Texts in Math., vol. 121, Springer-Verlag, 1990.[McC91] William McCallum,
Kolyvagin’s work on Shafarevich-Tate groups , L -functions and Arithmetic (JohnCoates and M.J. Taylor, eds.), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge UniversityPress, 1991, Proceedings of the Durham Symposium, July, 1989, pp. 295–316.[MR04] Barry Mazur and Karl Rubin, Kolyvagin Systems , Mem. Amer. Math. Soc., vol. 168, AmericanMathematical Society, March 2004.[PR87] Bernadette Perrin-Riou,
Fonctions
L p -adiques, th´eorie d’Iwasawa et points de Heegner , Bull. Soc.Math. France (1987), no. 4, 399–456.[PW11] Robert Pollack and Tom Weston,
On anticyclotomic µ -invariants of modular forms , Compos. Math. (2011), 1353–1381.[Rub90] Karl Rubin, The Main Conjecture , combined 2nd ed., Grad. Texts in Math., vol. 121, ch. Appendix,Springer-Verlag, 1990, Appendix to [Lan90].[SU14] Christopher Skinner and Eric Urban,
The Iwasawa main conjectures for GL , Invent. Math. (2014), no. 1, 1–277.[Wana] Xin Wan, Heegner point Kolyvagin system and Iwasawa main conjecture , preprint.[Wanb] ,
Iwasawa main conjecture for Rankin-Selberg p -adic L -functions , preprint.[YZZ13] Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang, The Gross-Zagier formula on Shimura curves , Ann.of Math. Stud., vol. 184, Princeton University Press, 2013. Zha14] Wei Zhang,
Selmer groups and the indivisibility of Heegner points , Camb. J. Math. (2014), no. 2,191–253.(Burungale) School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton,NJ 08540, USA
E-mail address : [email protected] (Castella) Department of Mathematics, Princeton University, Washington Road, Princeton,NJ 08544-1000, USA
E-mail address : [email protected] (Kim) School of Mathematics, Korea Institute for Advanced Study, 85 Hoegi-ro, Dongdaemun-gu, Seoul 02455, Republic of Korea
E-mail address : [email protected]@gmail.com