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Perrin-Riou's Main Conjecture for Elliptic Curves at Supersingular

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Perrin-Riou's Main Conjecture for Elliptic Curves at Supersingular PERRIN-RIOU'S MAIN CONJECTURE FOR ELLIPTIC CURVES AT SUPERSINGULAR PRIMES FRANCESC CASTELLA AND XIN WAN Abstract. Let E=Q be a semistable elliptic curve with good supersingular reduction at a prime p > 3. In this paper, we prove the implication 1 corankZp Selp (E=Q) = 1 =) ords=1L(E; s) = 1; which is a p-converse to the theorem of Gross{Zagier and Kolyvagin. The implication for good ordinary primes p is due to Chris Skinner and Wei Zhang by different methods. Our proof rests on a new version of Perrin-Riou's main conjecture for Heegner points over the anticyclotomic Zp-extension of an imaginary quadratic field in which p splits that we formulate in this paper, and prove under mild hypotheses. Contents 1. Introduction 1 2. p-adic L-functions 4 3. Local results 6 3.1. Local points 6 3.2. The plus/minus Coleman maps 9 3.3. The plus/minus Logarithm maps 10 4. Heegner point main conjecture 11 4.1. The plus/minus Heegner classes 12 4.2. The main conjecture 14 5. Main results 15 5.1. First divisibility 15 5.2. Explicit reciprocity law 17 5.3. Second divisibility 19 5.4. Converse to GZK for supersingular primes 23 References 25 1. Introduction 1.1. Perrin-Riou's Heegner point main conjecture. Since one of the goals of this paper is to formu- late an analogue of Perrin-Riou's Heegner point main conjecture [PR87, Conj. B] for supersingular primes, let us begin by briefly recalling the formulation of the original conjecture in the ordinary case. P1 n Let E=Q be an elliptic curve of conductor of N, and let f = n=1 anq 2 S2(Γ0(N)) be the associated newform. Let K be a quadratic imaginary field satisfying the classical Heegner hypothesis: (Heeg) every prime `jN splits in K. ac Fix a prime p - 6N, and let K1 be the anticyclotomic Zp-extension of K, i.e., the unique Zp-extension of ac ac K that is Galois over Q with complex conjugation τ 2 Gal(K=Q) acting as inversion on Γ = Gal(K1=K). 2010 Mathematics Subject Classification. 11R23 (primary); 11G05, 11G40 (secondary). 1 2 F. CASTELLA AND X. WAN For an algebraic extension L=Q let Selp1 (E=L) and Sˇp(E=L) be the Selmer groups fitting into the descent exact sequences 1 0 ! E(L) ⊗ Qp=Zp ! Selp1 (E=L) ! Ø(E=L)[p ] ! 0; (1.1) 0 ! E(L) ⊗ Zp ! Sˇp(E=L) ! TpØ(E=L) ! 0; ac ac where TpØ(E=L) is the p-adic Tate module of the Tate{Shafarevich group of E=L. Let Λ = Zp Γ ac ac ac J nK be the anticyclotomic Iwasawa algebra and, denoting by Kn the subfield of K1 with [Kn : K] = p , consider the Λac-modules ac 1 ac ac ac (1.2) Sel 1 (E=K ) = lim Sel(p )(E=K ); Sˇ (E=K ) = lim Sˇ (E=K ): p 1 −! n p 1 − p n n n Upon the choice of a modular parametrization π : X0(N) ! E, the construction of Heegner points on ac X0(N) gives rise to a system of points yn 2 E(Kn ) satisfying (1.3) Tr ac ac (y ) = a y − y Kn+1=Kn n+1 p n n−1 2 −1 for all n > 0. Letting α be one of the roots of X −apX +p, the \p-stabilized" element yn,α := yn −α yn−1 therefore satisfies Tr ac ac (y ) = αy : Kn+1=Kn n+1,α n,α When p - ap (i.e., in the ordinary case), α can be chosen to be a p-adic unit, and from the Kummer −n ˇ ac images of the points α ·yn,α one obtains a compatible system z1 2 Sp(E=K1). (The ordinary hypothesis −n is essential here, as otherwise the denominators of α yn,α grow without bound.) Perrin-Riou's Heegner ac ˇ ac point main conjecture [PR87, Conj. B] then predicts that z1 is not Λ -torsion, that both Sp(E=K1) and ac _ ac the Pontryagin dual X = Selp1 (E=K1) have Λ -rank one, and that ˇ ac 2 ? Sp(E=K1) ac ac (1.4) charΛ (Xtors) = charΛ ac Λ · z1 as ideals in Λac, where the subscript tors denotes the Λac-torsion submodule. (In general, the left-hand side 2 2 × of (1.4) should include the factor cEuK , with cE the Manin constant associated to π and uK = jOK j=2, but for p - 6N this is a p-adic unit; see [Maz78, Cor. 3.1].) We note that the non-triviality of z1 (also predicted by Mazur's conjecture on higher Heegner points [Maz84]) follows from the work of Cornut{Vatsal [Cor02, Vat02]. Moreover, after the work of several authors [Ber95, How04a, How04a, Wan14, Cas17, BCK20, Zan19, BT20, CGLS20], conjecture (1.4) in the ordinary case is known under relatively mild hypotheses, with K just assumed to satisfy the generalized Heegner hypothesis: writing N = N +N − with N + (resp. N −) divisible only by primes which are split or ramified (resp. inert) in K, (gen-H) N − is the squarefree product of an even number of primes. Our formulation of Perrin-Riou's Heegner point main conjecture for supersingular primes will be in this ac more general context, where analogues of the Heegner points yn 2 E(Kn ) as above are obtained from the parametrization of E by Shimura curves attached to indefinite quaternion algebras. 1.2. The conjecture for supersingular primes. Assume that K satisfies hypothesis (gen-H), and that p 6N is a prime of good supersingular reduction for E. By the Hasse bound, this forces a = 0, and so - p p α = ± −p. As already noted, in this case the p-stabilized points yn,α do not give rise to an element in ˇ ac −n Sp(E=K1), since the denominators of α yn,α grow without bound. This phenomenon is reminiscent of the unboundedness of the p-adic L-function of [MTT86] attached to the pair (f; α). In that context, this ± was beautifully understood by Pollack [Pol03] as giving rise to a pair of plus/minus p-adic L-functions Lp for which a well-behaved Iwasawa main conjecture could be formulated in terms of Kobayashi's plus/minus Selmer groups [Kob03]. Inspired by these works, in this paper we deduce from the norm-relation Tr ac ac (y ) = −y Kn+1=Kn n+1 n−1 PERRIN-RIOU'S MAIN CONJECTURE FOR ELLIPTIC CURVES AT SUPERSINGULAR PRIMES 3 a construction of \plus/minus Heegner classes" z± 2 H1 (Kac;T E) = lim H1(Kac;T E), where T E is 1 Iw 1 p −n n p p the p-adic Tate module of E. Moreover, assuming that (spl) p = pp¯ splits in K, ± and building on the local results established in this paper x3, we show that the classes z1 land in a certain ˇ± ac 1 ˇ ac Selmer group Sp (E=K1) ⊂ HIw(K1;TpE) differing from Sp(E=K1) in its defining local conditions at ± the places above p. We can show that the classes z1 are non-trivial (independently of Cornut{Vatsal), ˇ± ac ± and we conjecture that Sp (E=K1) and the Pontryagin dual X of a corresponding discrete Selmer group ± ac 1 ac 1 ac Selp1 (E=K1) ⊂ H (K1;E[p ]) have both Λ -rank one, with ˇ± ac 2 ± ? Sp (E=K1) (1.5) char ac (X ) = char ac Λ tors Λ ac ± Λ · z1 as ideals in Λac (see Conjecture 4.8). This conjecture is our natural extension for supersingular primes of Perrin-Riou's Heegner point main conjecture [PR87, Conj. B]. 1.3. Main results and outline of the proofs. We can prove the following result toward conjecture (1.5) (see Theorem 5.9). Theorem A. Let E=Q be an elliptic curve of conductor N, p > 3 a prime of good supersingular reduction for E, and K an imaginary quadratic field satisfying (gen-H) and (spl). Assume in addition that: • ρ : GK ! AutZp (TpE) is surjective; • N is squarefree; • E[p] is ramified at every prime `jN −; • N − 6= 1. ˇ± ac ± ac Then Sp (E=K1) and X both have Λ -rank one, and conjecture (1.5) holds. It should be possible to relax some of the hypotheses in Theorem A, which we leave for future work. As a consequence of Theorem A, we deduce the following p-converse to the theorem of Gross{Zagier{Kolyvagin. Theorem B. Let E=Q be a semistable elliptic curve and p > 3 a prime of good supersingular reduction. Then 1 corankZp Selp (E=Q) = 1 =) ords=1L(E; s) = 1: 1 1 Ø In particular, if corankZp Selp (E=Q) = 1 then # (E=Q)[p ] < 1. In particular, Theorem B leads to new mod p criteria for elliptic curves E=Q to satisfy the Birch and Swinnerton-Dyer conjecture (see Corollary 5.21). Remark 1.1. For good ordinary primes, Theorem B is due to Skinner [Ski20] and Zhang [Zha14] by different methods. Our method of proof of closer in spirit to Skinner's, although the main result in [Ski20] assumes the finiteness of Ø(E=Q)[p1], whereas our result deduces it as a consequence. Remark 1.2. Our idea of deducing a p-converse to Gross{Zagier{Kolyvagin from a divisibility in a Perrin- Riou Heegner point main conjecture1 has inspired subsequent work, notably the proof of the p-converse to GZK in the ordinary CM case [BT20] and in the residually reducible case [CGLS20].
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