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Let E /Q be a CM elliptic curve and p a prime of good ordinary reduction for E.Weshowthat if Sel (E /Q) has Z -corank one, then E (Q) has a point of inﬁnite order. The non-torsion p p point arises from a Heegner point, and thus ord L(E , s) = 1, yielding a p-converse to s=1 a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For p > 3, this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real ﬁelds [4]. Résumé Soit E /Q une courbe elliptique à multiplication complexe et p un nombre premier de bonne réduction ordinaire pour E. Nous montrons que si corank Sel (E /Q) = 1, alors E Z p a un point d’ordre inﬁni. Le point de non-torsion provient d’un point de Heegner, et donc ord L(E , s) = 1, ce qui donne une réciproque à un théorème de Gross–Zagier, Kolyvagin, s=1 et Rubin dans l’esprit de [49, 54]. Pour p > 3, cela donne une nouvelle preuve du résultat principal de [12], que notre approche étend à tous les nombres premiers. L’approche se généralise aux courbes elliptiques à multiplication complexe sur les corps totalement réels [4]. To Bernadette Perrin-Riou, with admiration. Francesc Castella castella@ucsb.edu Ashay Burungale ashayburungale@gmail.com Christopher Skinner cmcls@princeton.edu Ye Tian ytian@math.ac.cn California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA The University of Texas at Austin, 2515 Speedway, Austin, TX 78712, USA University of California Santa Barbara, South Hall, Santa Barbara, CA 93106, USA Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA Academy of Mathematics and Systems Science, MCM, HLM, Chinese Academy of Sciences, Beijing 100190, China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 123 326 A. Burungale et al. Keywords P-adic · L-functions · Selmen groups · Elliptic waves · Heegen points Mathematics Subject Classiﬁcation Primary 11G05; Secondary 11G40 Contents 1 Introduction .............................................. 326 2 Preliminaries .............................................. 328 2.1 CM abelian varieties ....................................... 329 2.2 Heegner points .......................................... 329 2.3 Heegner point main conjecture .................................. 330 3 Selmer groups ............................................. 331 3.1 Selmer groups of certain Rankin–Selberg convolutions ..................... 332 3.2 Selmer groups of characters .................................... 333 3.3 Decomposition .......................................... 334 4 p-Adic L-functions .......................................... 334 4.1 The BDP p-adic L-function .................................... 334 4.2 Katz p-adic L-functions ...................................... 336 4.3 Factorization ............................................ 337 5 Explicit reciprocity law ........................................ 338 6 Twisted anticyclotomic main conjectures for K ............................ 340 7 The main results ............................................ 341 8The p-converse ............................................. 342 References ................................................. 345 1 Introduction Starting with [49, 54], several works have been devoted to p-converses to a celebrated theorem of Gross–Zagier, Kolyvagin, and Rubin: If the p -Selmer group Sel (E /Q) has Z -corank p p 1 for an elliptic curve E /Q,thenord L(E , s) = 1. Besides being an evidence for the Birch s=1 and Swinnerton-Dyer conjecture, an important impetus for the p-converse theorems has come from recent developments in arithmetic statistics. For instance, such p-converse theorems have led to the proof [10] that a large proportion of elliptic curves over Q—and conditionally, 100% of them—satisfy the Birch and Swinnerton-Dyer conjecture. The p-converse theorems of [49, 54] are obtained by exhibiting a certain Heegner point on E with inﬁnite order, and hold for primes p > 3 of good ordinary reduction of E,and under certain hypotheses that excluded the CM elliptic curves. Our main result is the following CM p-converse theorem. For primes p > 3, the result wasﬁrstprovedin[12]. Theorem A Let E /Q be an elliptic curve with complex multiplication by an order of an imaginary quadratic ﬁeld K of discriminant −D < 0. Assume that the Hecke character associated to E has conductor exactly divisible by d := ( −D ). Let p be a prime of good K K ordinary reduction for E. Then corank Sel ∞ (E /Q) = 1 ⇒ ord L(E , s) = 1. Z p s=1 In particular, if corank Sel (E /Q) = 1 then rank E (Q) = 1 and #W(E /Q)< ∞. Z p Z Note that the ‘in particular’ clause in Theorem A follows from combining its conclusion with the fundamental work of Gross–Zagier, Kolyvagin, and Rubin. In turn, this consequence yields the following mod p criterion for analytic rank one. 123 ∞ p -Selmer groups and rational points on CM elliptic curves 327 Corollary B Let (E , K) be as in Theorem A, and let p be a prime of good ordinary reduction for E such that: (i) E (Q)[p]= 0; (ii) Sel (E /Q) Z/pZ, where Sel (E /Q) ⊂ H (Q, E [p]) is the p-Selmer group of E. p p Then ord L(E , s) = 1 and W(E /Q)[p ]= 0. s=1 More generally, we prove a p-converse for CM abelian varieties B /K associated with Hecke characters λ over K of inﬁnity type (−1, 0) (see Theorem 8.2). Our approach to the CM p-converse differs from [12]. A salient feature is that the approach generalizes to CM elliptic curves over totally real ﬁelds: It sidesteps the inherence of elliptic units in [12], and leads to the ﬁrst p-converse theorems over general totally real ﬁelds. This note might thus be viewed as a prelude to [4]. The conductor hypothesis in Theorem A arises from an appeal to [19] which supposes a classical Heegner hypothesis. This hypothesis may be removed (cf. [3, 4]) via the p-adic Waldspurger formula of Liu–Zhang–Zhang [34], whose habitat is the general Yuan–Zhang– Zhang framework [53]. (The hypothesis is not present in [12], thanks to Disegni’s p-adic Gross–Zagier formula [22], also in the framework of [53].) Remark C The p-converse as in Theorem A is independently due to Ressler and Yu [47, 52]. Complementing the present note, their approach builds on [12] and does not require the conductor hypothesis. A key new element in their work is a counterpart of the main results of Agboola–Howard [1] for small primes. Remark D A spectacular result of Smith [50] reduces Goldfeld’s conjecture [24]for CM elliptic curves E /Q with E (Q)[2] (Z/2Z) and admitting no rational cyclic 4-isogeny to the CM p-converse and its rank zero analogue for the prime p = 2. The unconditional rank zero CM p-converse is in fact proved in [11, 13]. Unfortunately, Theorem A falls short of providing the desired rank one CM p-converse: First, E should be allowed to have just potentially good ordinary reduction at p (this might be approachable by our strategy); the second—and the main—hindrance is that Q( −7) is the only imaginary quadratic ﬁeld of class number 1 with 2 split, and incidentally E (Q)[2] Z/2Z for all elliptic curves E /Q with CM by Q( −7) (see e.g. the table in [38, p. 2]). The approach Assuming #W(E /Q)[p ] < ∞ and p #O ,the p-converse as in The- orem A goes back to Rubin [45, Thm. 4]. Around the same time, Rubin proved a striking formula [44] which expresses the p-adic formal group logarithm of a point P ∈ E (Q) ⊗ Z Z p in terms of the value of a Katz p-adic L-function outside its range of interpolation. Our approach to Theorem A is inspired by the p-adic Waldspurger formula of Bertolini–Darmon– Prasanna [7], which is a remarkable generalization of Rubin’s formula, and sheds new light on [44](cf.[6]). Let E /Q be an elliptic curve with CM by an imaginary quadratic ﬁeld K,and p a prime of good ordinary reduction. Let λ be the Hecke character over K associated with E so that L(E , s) = L(λ, s). In view of a non-vanishing result of Rohrlich [42], we pick a pair (ψ, χ ) of Hecke characters over K with χ of ﬁnite order such that ψχ = λ, L(ψ χ, 1) = 0, (1.1) where ψ := ψ ◦ c is the composition of ψ with the non-trivial automorphism of K/Q.For f = θ the theta series associated to ψ, the main result of [7] relates the p-adic formal group 123 328 A. Burungale et al. logarithm of a Heegner point P ∈ B (K) to a value (outside the range of interpolation) ψ,χ ψ,χ of a p-adic Rankin L-series L ( f ,χ).Here B is a CM abelian variety over K endowed v ψ,χ with a K-rational map i : B → E. By the Gross–Zagier formula [53], P is non-torsion λ ψ,χ ψ,χ if and only if L ( f ,χ, 1) = 0. Setting P := i (P ) ∈ E (K), K λ ψ,χ one thus obtains a point on E which, in light of (1.1) and the factorization L( f ,χ, s) = L(λ, s) · L(ψ χ, s), (1.2) is non-torsion if and only if ord L(E , s) = 1. s=1 Now, Theorem A is equivalent to: If Sel (E /Q) has Z -corank 1, then P is non-torsion. p p K In [12] the non-triviality is shown via the following. (1) The anticyclotomic Iwasawa main conjecture (IMC) for (Hecke characters over) K in the root number +1 case [43]; (2) The anticyclotomic IMC for K in the root number −1 case [1, 2]; (3) The non-vanishing of the -adic regulator appearing in (2) [14]; (4) The -adic Gross–Zagier formula [22]. Here denotes the anticyclotomic Iwasawa algebra over K (with certain coefﬁcients). Bypassing (2), (3), and (4), our approach builds on the explicit reciprocity law [19], which realizes L ( f ,χ) as the image of a -adic Heegner class z under a Perrin-Riou big v f ,χ logarithm map. Similarly as in [6], we establish a factorization 2 ∗ ∗ ∗ L (θ ,χ) = L (ψ χ ) · L (ψ χ ) v ψ v v in Sect. 4 relating L (θ ,χ) to the product of two Katz p-adic L-functions, mirroring (1.2). v ψ Along with an analogous decomposition for Selmer groups shown in Sect. 3, the Iwasawa– Greenberg main conjecture for L (θ ,χ) is readily seen to be a consequence of the main v ψ results of [31, 43]. Building on the -adic explicit reciprocity law, in Sect. 5 we prove the equivalence between the main conjecture for the p-adic L-function L (θ ,χ) and a v ψ different main conjecture formulated in terms of the zeta element z . Finally, the latter f ,χ yields the implication corank Sel ∞ (E /Q) = 1 ⇒ P = 0 ∈ E (K) ⊗ Q Z p K Z via a variant of Mazur’s control theorem. Dedication The p-converse for CM elliptic curves E /Q is due to Rubin if #W(E /Q)[p ] < ∞. The essence of our removal of this hypothesis is Iwasawa theory of Heegner points, as pioneered by Perrin-Riou [40]. The theory of big logarithm maps [41], another major contribution of Perrin-Riou, is also elemental to our approach. It is a great pleasure to dedicate this note to Bernadette Perrin-Riou as a humble gift on the occasion of her 65th birthday. 2 Preliminaries Fix throughout a prime p, an algebraic closure Q of Q, and embeddings C ← Q → C . Fix also an imaginary quadratic ﬁeld K of discriminant −D < 0 and ring of integers O . K K 123 ∞ p -Selmer groups and rational points on CM elliptic curves 329 2.1 CM abelian varieties × × 2 We say that a Hecke character ψ : K \A → C has inﬁnity type (a, b) ∈ Z if, writing a b ψ = (ψ ) with v running over the places of K, the component ψ satisﬁes ψ (z) = z z ¯ v v ∞ ∞ × × for all z ∈ (K ⊗ R) C , where the identiﬁcation is made via ı . Hence in particular Q ∞ the norm character N ,given by q → #(O /q) on ideals of O , has inﬁnity type (−1, −1). K K K The central character of such ψ is the character ω on A deﬁned by −(a+b) ψ | × = ω · N , where N is the norm on A . Our ﬁxed embedding ı deﬁnes a natural map σ : K ⊗ Q → C , and we let σ : p Q p p K ⊗ Q → C be the composition of σ with the non-trivial automorphism of K.The p- Q p p × × adic avatar of a Hecke character ψ of inﬁnity type (a, b) is the character ψ : K \A → C K,f given by −1 a b ψ(x ) = ı ◦ ı (ψ (x ))σ (x ) σ(x ) p p p for all x ∈ A ,where x ∈ (K ⊗ Q ) is the p-component of x. p p K,f Throughout the following, we shall often omit the notational distinction between an alge- braic Hecke character and its p-adic avatar, as it will be clear from the context which one is meant. Let ψ be an algebraic Hecke character of K inﬁnity type (−1, 0) with values in a number ﬁeld F ⊂ Q with ring of integer O .Let P be the prime of F above p induced by ı , ψ ψ ψ p and denote by the completion of F at P and by O the ring of integers of .By ψ ψ ψ ψ a well-known theorem of Casselman’s (see [6, Thm. 2.5] and the reference [48,Thm.6] therein), attached to ψ there is a CM abelian variety B , unique up to isogeny over K, ψ/K with the property that −1 V B ψ P ψ as one-dimensional -representations of G ,where V B = lim B [P ] ⊗ is ψ K P ψ ψ O ψ ← − the rational P-adic Tate module of B . 2.2 Heegner points Let f ∈ S ( (N )) be a normalized eigenform of weight 2, level N prime to p, and neben- 2 1 typus ε . We assume that K satisﬁes the Heegner hypothesis relative to N : there is an ideal N ⊂ O withO /N Z/N Z, (Heeg) K K and ﬁx once and for all an ideal N as above. We assume also that pO = vv splits in K, (spl) with v the prime of K above p induced by our ﬁxed embedding ı .Let F ⊂ Q be the number ﬁeld generated by the Fourier coefﬁcients of f .Denoteby P the prime of F above p induced by ı , and assume that f is P-ordinary, i.e. v (a ( f )) = 0, where v is the P-adic p P p P valuation on F. Let A /Q be the abelian variety of GL -type associated to f , determined up to isogeny f 2 over Q by the equality of L-functions L(A , s) = L( f , s), τ :F →C 123 330 A. Burungale et al. where f runs over all the conjugates of f .Denoteby the completion of F at P,and let O be the ring of integers of .Let T A := lim A [P ] be the P-adic Tate module of A , P f f f ← − which is free of rank two over O. For every positive integer c,let K be the ring class ﬁeld of K of conductor c,so Gal(K /K) Pic(O ) by class ﬁeld theory, where O = Z + cO is the order of K of c c c K conductor c.For every c > 0 prime to N and every ideal a of O , we consider the CM point ˜ ˜ x ∈ X (N )(K ) constructed in [19, §2.3], where K is the compositum of K and the ray a 1 c c c class ﬁeld of K of conductor N.Let be the class of the degree 0 divisor (x ) − (∞) in a a J (N ) = Jac(X (N )), and denote by z = δ( ) its image under the Kummer map 1 1 a a ˜ ˜ δ : J (N )(K ) → H (K , T J (N )). 1 c c p 1 Fix a parametrization π : J (N ) → A ,and let y ∈ H (K , T A ) be the image of 1 f f ,a c P f y under the natural projection 1 1 1 ˜ ˜ ˜ H (K , T J (N )) − → H (K , T A ) → H (K , T A ). c p 1 c p f c P f For the ease of notation, we set y = y for a = O . A standard calculation shows f ,c f ,a c that if p c, then for every n > 0we have a ( f ) · y − ε (p) · y if n > 1, n−1 n−2 p f f ,cp f ,cp Cor (y ) = (2.1) f ,cp ˜ ˜ K n /K n−1 −1 cp cp u a ( f ) − σ − σ · y if n = 1, p v v f ,c × × where u := [O : O ] and σ ,σ ∈ Gal(K /K) are Frobenius elements at the primes of c v v c c cp K above p (see [19, Prop. 4.4]). Let α be the P-adic unit root of x − a ( f )x + ε (p)p, and for any positive integer c p f prime to N deﬁne the α-stabilized Heegner class y by f ,c,α −1 y − ε (p)α · y if p | c, f ,c f f ,c/ p y := f ,c,α −1 −1 −1 u 1 − σ α − σ α · y if p c. v v f ,c This deﬁnition is motivated by the following result. Lemma 2.1 For all positive integers c prime to N , we have Cor (y ) = α · y . ˜ ˜ f ,cp,α f ,c,α K /K cp c Proof This follows immediately from (2.1). 2.3 Heegner point main conjecture ˜ ˜ Fix a positive integer c prime to Np, and put K ∞ = K m . The Galois group cp cp m≥0 G = Gal(K /K) decomposes as c cp G × , c c where is the maximal torsion-free quotient of G , giving the Galois group of the anticyclo- tomic Z -extension K /K,and is a ﬁnite abelian group. p ∞ c −1 Let χ be a ﬁnite order Hecke character of K with χ | × = ε and of conductor dividing cN. Upon enlarging F is necessary, assume that contains the values of χ. For each n,take m 0so that K ⊃ K ,and set cp n 123 ∞ p -Selmer groups and rational points on CM elliptic curves 331 −m σ z := α χ(σ ) · y m . (2.2) f ,χ ,n f ,cp ,α σ ∈Gal(K m /K ) cp In view of Lemma 2.1, the deﬁnition of z does not depend on the choice of m. Moreover, f ,χ ,n letting A be the Serre tensor A ⊗ χ,wesee that z deﬁnes a class f ,χ f f ,χ ,n z ∈ H (K , T A ). f ,χ ,n n P f ,χ Let = O , = ⊗ (2.3) 0 0 O be the anticyclotomic Iwasawa algebras. From their construction, the classes z are f ,χ ,n contained in the pro-P Selmer group S (A /K ) ⊂ H (K , T A ), and by Lemma 2.1 P f ,χ n n P f ,χ they are norm-compatible, hence deﬁning a class z ={z } in the compact -adic f ,χ f ,χ ,n n 0 Selmer group S (A /K ) := lim S (A /K ). f ,χ ∞ P f ,χ n ← − On the other hand set, X (A /K ) := Hom lim Sel (A /K ), /O , f ,χ ∞ Z f ,χ n p P − → 1 ∞ ∞ where Sel (A /K ) ⊂ H (K , A [P ]) is the P -Selmer groups of A . Set also f ,χ n n f ,χ f ,χ S(A /K ) = S (A /K ) ⊗ , X(A /K ) = X (A /K ) ⊗ , f ,χ ∞ f ,χ ∞ O f ,χ ∞ f ,χ ∞ O which are ﬁnitely generated -modules. The following conjecture in a natural extension of Perrin-Riou’s Heegner point main conjecture [40,Conj.B]. Conjecture 2.2 The modules S(A /K ) and X(A /K ) have both -rank one, and f ,χ ∞ f ,χ ∞ char (X(A /K ) ) = char S(A /K )/ · z , f ,χ ∞ -tors f ,χ ∞ f ,χ where the subscript -tors denotes the maximal -torsion submodule. In [12] a conjecture similar to Conjecture 2.2 is formulated in terms of a -adic Heegner class deduced from work of Disegni [22](see[12, Conj. 2.2]). Similarly as in [12], our proof of Theorem A is based on a study of Conjecture 2.2. The novelty in our approach is in the proof of cases of this conjecture. 3 Selmer groups In this section we introduce the different Selmer groups entering in our arguments. In partic- ular, the decomposition in Proposition 3.4 will play a key role. As well as in other results on the p-converse theorem in rank 1 without a ﬁniteness condition on the Tate–Shafarevich group that appeared after [49]: [18, 21, 51], etc. 123 332 A. Burungale et al. 3.1 Selmer groups of certain Rankin–Selberg convolutions As in Sect. 2.2,let f ∈ S ( (N )) be a P-ordinary newform with nebentypus ε ,and let K 2 1 f be an imaginary quadratic ﬁeld satisfying (Heeg)and (spl). Let c > 0 be a positive integer prime to N . Similarly as in [6, Def. 3.10], we say that a Hecke character ξ of inﬁnity type (2 + j , − j ), with j ∈ Z,has ﬁnite type (c, N,ε ) if it satisﬁes: (a) ω · ε = 1,where ω is the central character of ξ ; ξ f ξ (b) f = c · N ,where N is the unique divisor of N with norm equal to the conductor of ε ; ξ f (c) the local sign ( f ,ξ) is +1 for all ﬁnite primes q. Condition (a) implies that the Rankin–Selberg L-function L( f ,ξ, s) is self-dual, with s = 0 as the central critical point, and by (c) the sign in the functional equation is +1 (resp. −1) when j ≥ 0 (resp. j < 0).Denoteby (c, N,ε ) the set of such characters ξ,and cc f put (1) (c, N,ε ) = ξ ∈ (c, N,ε ) | j < 0 , f cc f cc (2) (c, N,ε ) = ξ ∈ (c, N,ε ) | j ≥ 0 . f cc f cc Denote by ρ : G → Aut (V ) the P-adic Galois representation associated to f ,so f Q f that V (1) ⊗ T A . f O P f (1) −1 Let χ be a ﬁnite order character of K such that χ N ∈ (c, N,ε ), and consider the cc f conjugate self-dual G -representation V := V (1)| ⊗ χ. (3.1) f ,χ f G For any -module M,let M = Hom (M , Q /Z ) be the Pontryagin dual. Fix a 0 cts p p G -stable lattice T ⊂ V , and deﬁne the G -module K f ,χ f ,χ K W := T ⊗ , (3.2) f ,χ f ,χ O where the tensor product is endowed with the diagonal Galois action, with G acting on via the inverse of the tautological character : G Gal(K /K)→ . K ∞ Deﬁnition 3.1 Fix a ﬁnite set of places of K containing ∞ and the primes dividing Np, and denote by K the maximal extension of K unramiﬁed outside .The Selmer group S (W ) is deﬁned by v f ,χ 1 1 1 S (W ) := ker H (K /K, W ) → H (K , W ) × H (K , W ) . v f ,χ f ,χ v f ,χ w f ,χ w∈,w p We also set X ( f ,χ) := Hom (S (W ), Q /Z ) ⊗ , v cts v f ,χ p p O which is independent of the lattice T . f ,χ Note that X ( f ,χ) and the Selmer group X(A /K ) deﬁned in Sect. 2.3 differ only v f ,χ ∞ in their deﬁning local conditions at the primes above p.Moreprecisely,by P-ordinarity, for every prime w of K above p there is a G -module exact sequence + − 0 → F T → T → F T → 0 (3.3) f ,χ f ,χ f ,χ w w 123 ∞ p -Selmer groups and rational points on CM elliptic curves 333 ± − with F T free of rank one over O, and the quotient F T affording an unramiﬁed f ,χ f ,χ w w action of G .Put ± ∓ ∨ F W = F T , ⊗ . f ,χ f ,χ O w w 0 Then the Selmer group S (W ) deﬁned by ord f ,χ 1 1 − S (W ) := ker H (K /K, W ) → H (K , F W ) ord f ,χ f ,χ w f ,χ w| p × H (K , W ) (3.4) w f ,χ w∈,w p satisﬁes S (W ) ⊗ lim Sel (A /K ) ⊗ , ord f ,χ O P f ,χ n O − → and so X ( f ,χ) := Hom (S (W ), Q /Z ) ⊗ X(A /K ). (3.5) ord cts ord f ,χ p p O f ,χ ∞ Letting T := T ⊗ with G -action via ρ ⊗ , and deﬁning S (T ) ⊂ f ,χ f ,χ O 0 K f ord f ,χ H (K /K, T ) in the same manner as in (3.4), we similarly have f ,χ S ( f ,χ) := S (T ) ⊗ S(A /K ) (3.6) ord ord f ,χ O f ,χ ∞ (see e.g. [17, §4]). 3.2 Selmer groups of characters We keep the hypothesis that the imaginary quadratic ﬁeld K satisﬁes (spl), and let ξ be a Hecke character of K of conductor f .Let F be a number ﬁeld containing the values of ξ . Let be the completion of F at the prime P of F above p induced by ı ,and let O be the ring of integers of . Denote by T the free O-module of rank one on which G acts via ξ K −1 ξ , and consider the G -module W := T ⊗ , ξ ξ O ∨ −1 where as before the Galois action on is given by the character . Deﬁnition 3.2 Let be a ﬁnite set of places of K containing ∞ and the primes dividing p or f .The Selmer group S (W ) is deﬁned by v ξ 1 1 1 X (W ) := ker H (K /K, W ) → H (K , W ) × H (K , W ) . v ξ ξ v ξ w ξ w∈,w p We also set X (ξ ) = X (W ) ⊗ . v v ξ O Remark 3.3 Suppose ξ has inﬁnity type (−1, 0), and denote by ξ the composition of ξ with the non-trivial automorphism of K/Q,so ξ has inﬁnity type (0, −1). Then from e.g. [1, §1.1] we see that X (ξ ) corresponds to the Bloch–Kato Selmer group of ξ over K /K, v ∞ whereas X (ξ ) corresponds to the Selmer group obtained by reversing the local conditions at the primes above p in the corresponding Bloch–Kato Selmer group of ξ . 123 334 A. Burungale et al. 3.3 Decomposition We now specialize the set-up in Sect. 3.1 to the case where f = θ is the theta series of a Hecke character ψ of K of inﬁnity type (−1, 0).Then f has level N = D · N(f ) and nebentypus ε = η · ω ,where η is the quadratic character associated to K/Q. f K ψ K One easily checks (see [6, Lem. 3.14]) that if f is a cyclic ideal of norm N(f ) prime to ψ ψ D ,then K satisﬁes the Heegner hypothesis (Heeg) relative to N , and one may take N = d · f , where d := ( −D ). (3.7) K K K In the following, we assume that f satisﬁes the above condition, and take N as in (3.7). −1 Fixaninteger c > 0 prime to Np,and let χ be a ﬁnite order character such that χ N ∈ (1) (c, N,ε ). cc f The following decomposition will play an important role later. Proposition 3.4 Let ψ and χ be as above. There is a -module isomorphism ∗ ∗ ∗ X (θ ,χ) X (ψ χ ) ⊕ X (ψ χ ). v ψ v v Proof Put f = θ , and note that there is a G -module decomposition ψ K V V ∗ ∗ ⊕ V ∗ . (3.8) f ,χ ψ χ ψχ Since the module X ( f ,χ) ⊂ H (K /K, W ) ⊗ does not depend on the lattice v f ,χ O T ⊂ V chosen to deﬁne W ,by(3.8) we may assume that T T ∗ ∗ ⊕ T ∗ as f ,χ f ,χ f ,χ f ,χ ψ χ ψχ G -modules, and so W W ∗ ∗ ⊕ W ∗ f ,χ ψ χ ψχ as G -modules. The result thus follows immediately by comparing the deﬁning local con- ditions of the three Selmer groups involved at all places. 4 p-Adic L-functions In this section we introduce the two p-adic L-functions needed for our arguments, and prove Proposition 4.5 relating the two. 4.1 The BDP p-adic L-function Let f = a ( f )q ∈ S ( (N )) be an eigenform with p N and nebentypus ε ,let n 2 1 f n=1 K be an imaginary quadratic ﬁeld satisfying (Heeg)and (spl), and ﬁx an ideal N ⊂ O with cyclic quotient of order N.Let c be a positive integer prime to Np,and let χ be a ﬁnite order (1) −1 Hecke character of K such that χ N ∈ (c, N,ε ). cc Let F be a number ﬁeld containing K, the Fourier coefﬁcients of f , and the values of χ,and let be the completion of F at the prime of F above p induced by ı , with ring of integers O.Let and be the anticyclotomic Iwasawa algebras as in (2.3), and set ur ur ur ur ur := ⊗ Z O , := ⊗ , 0 Z O 0 p p 0 ur where Z is the completion of the ring of integers of the maximal unramiﬁed extension of Q . 123 ∞ p -Selmer groups and rational points on CM elliptic curves 335 The p-adic L-function in the next theorem was ﬁrst constructed in [7] as a continuous ur function on characters of . Its realization as a measure in was given in [19] following an approach introduced in [9]. As it will sufﬁce for our purposes, we describe below a multiple of that p-adic L-function by an element in . As in [19, §2.3], deﬁne ϑ ∈ K by D + −D D if 2 D , K K K ϑ := , where D = D /2else, and let and be CM periods attached to K as in [op. cit., §2.5]. p K ur Theorem 4.1 There exists an element L ( f ,χ) ∈ such that for every character ξ of crystalline at both v and v ¯ and corresponding to a Hecke character of K of inﬁnity type (n, −n) with n ∈ Z , we have ≥1 4n −1 (n) (n + 1)ξ(N ) 2 −1 L ( f ,χ) (ξ ) = · · 1 − a ( f )χ ξ(v)p v p 4n 2n+1 2n−1 4(2π) (Im ϑ) 2 −1 +ε (p)χ ξ(v) p · L( f ,χξ, 1). Proof Let η be an anticyclotomic Hecke character of K of inﬁnity type (1, −1) and conductor ur dividing cO , and deﬁne L ( f ,χ) ∈ by K v,η −1 L ( f , χ )(φ) = ηχ (a)N(a) η (φ|[a]) dμ v,η v × a [a]∈Pic(O ) for all continuous characters φ : → Q ,where: • f = a ( f )q is the p-depletion of f , pn • μ is the measure on Z corresponding (under the Amice transform) to the power series −1 N(a)c −D K ur f (t ) ∈ O t − 1 a a with t the Serre–Tate coordinate of the reduction of the point x on the Igusa tower of a a tame level N constructed in [19, (2.5)], × ab • η (x ) := η(rec (x )) with rec : Q = K → G the local reciprocity map at v, v v v p v K × −1 • φ|[a]: Z → Q is deﬁned by (φ|[a])(x ) = φ(rec (x )σ ) with σ the Artin symbol v a p p a of a. ur The same calculation as in [19, Prop. 3.8] then shows that the element L ( f ,χ) ∈ deﬁned by −1 L ( f , χ )(ξ ) := L ( f , χ )(η ξ) v v,η has, in view of the explicit Waldspurger formula in [29, Thm. 3.14], the stated interpolation property up to ﬁxed element in . The result follows. Remark 4.2 We our later use, we note that the complex period ∈ C in Theorem 4.1 (which also agrees with that in [7, (5.1.16)]) is different from the complex period ∈ C deﬁned in [23, p. 66] and [30, (4.4b)]. In fact, one has = 2πi · . ∞ K 123 336 A. Burungale et al. In terms of , the interpolation formula in Theorem 7.1 reads 4n −1 (n) (n + 1)ξ(N ) 2 −1 L ( f ,χ) (ξ ) = · · 1 − a ( f )χ ξ(v)p v p 4n 1−2n 2n−1 4(2π) (Im ϑ) 2 −1 +ε (p)χ ξ(v) p · L( f ,χξ, 1). Specialized to the range of critical values for the representation V , the Iwasawa– f ,χ Greenberg main conjecture [26] predicts the following. Conjecture 4.3 The module X ( f ,χ) is -torsion, and char (X ( f ,χ)) = (L ( f ,χ) ). v v In Theorem 5.2, we will explain the close link between Conjectures 2.2 and 4.3. 4.2 Katz p-adic L-functions We continue to assume that K satisﬁes (spl). Let c ⊂ O be an ideal prime to p,and let ∞ ∞ K(c p ) be the ray class ﬁeld of K of conductor c p . We say that a Hecke character φ of K is self-dual if it satisﬁes φφ = N . Note that the inﬁnity type of such φ is necessarily of the form (−1 + j , − j ) for some j ∈ Z. The p-adic L-function in the next theorem follows from the work of Katz [32], as extended by Hida–Tilouine [30](seealso[23]). Here we shall use the construction in [28], and similarly as in Theorem 4.1, it will sufﬁce for our purposes to describe a ﬁxed -multiple of the integral measure constructed in op. cit.. For any Hecke character ξ of K,wedenoteby L (ξ, s) the Hecke L-function L(ξ, s) with the Euler factors at the primes l|c removed. Theorem 4.4 Let φ be a character of Gal(K(c p )/K) corresponding to a self-dual Hecke character of inﬁnity type (−1 + j , − j ), with j ∈ Z . Then there exists an element L (φ) ∈ ≥0 v ur such that for every character ξ of crystalline at both v and v ¯ and corresponding to a Hecke character of inﬁnity type (n, −n) with n ≥ j, we have 2n−2 j +1 n− j (2π) −1 2 −1 L (φ)(ξ ) = · (n + 1 − j ) · · (1 − φ ξ(v)) · L (φ ξ, 0). v c 2n−2 j +1 n− j (Im ϑ) Proof Let L be the integral p-adic measure on Gal(K(cp )/K) constructed in [28, §4.8], so for every character χ of Gal(K(c p )/K) corresponding to a Hecke character of K of inﬁnity type (k + , −) with k > ≥ 0we have k+2 (2π) −1 −1 L (χ ) = · (k + ) · · (1 − χ (v)p )(1 − χ(v) ¯ ) · L (χ , 0). v c k+2 (Im ϑ) Setting −1 ∗ L (φ)(ξ ) := L (φ · π ξ) v v for all characters ξ of ,where π ξ is the pullback of ξ under the projection π : Gal(K(cp )/K) → , the result follows immediately from [28, Prop. 4.9], noting that 123 ∞ p -Selmer groups and rational points on CM elliptic curves 337 −1 the condition n ≥ j assures that the inﬁnity type of φ ξ , namely (1 + n − j , j − n),isof the form (1 + , −) with ≥ 0, and the p-adic multiplier that appears is −1 −1 −1 −1 2 (1 − φξ (v)p )(1 − φ ξ(v) ¯ ) = (1 − φ ξ(v) ¯ ) , since φ is self-dual and ξ is anticyclotomic. 4.3 Factorization As in Sect. 3.3, we now specialize to the case where f = θ for a Hecke character ψ of K of inﬁnity type (−1, 0) and conductor f with cyclic quotient of norm prime to D ,sothat K satisﬁes hypothesis (Heeg) relative to N = D · N(f ). K ψ Fix an integer c > 0 prime to Np,and let χ be a ﬁnite order Hecke character of K such −1 (1) that χ N ∈ (c, N,ε ).Thenwehavea G -module decomposition cc f K V V ∗ ∗ ⊕ V ∗ , (4.1) f ,χ ψ χ ψχ where V is as in (3.1). Note that each of the characters ψχ and ψ χ are self-dual (see [6, f ,χ Rem. 3.7]). For the rest of this paper, we shall write L (φ) for the p-adic L-function in Theorem 4.4 constructed with the auxiliary tame conductor c = cN used in the proof. The following result is a manifestation of the Artin formalism arising from the decompo- sition (4.1). A similar result in shown in [6,Thm.3.17].Asweshall seeinSect. 7,thisisa counterpart on the analytic side of the Selmer group decomposition in Proposition 3.4. Proposition 4.5 Suppose that f = θ and χ are as above. Then 2 ∗ ∗ ∗ L ( f ,χ) = u · L (ψ χ ) · L (ψ χ ), v v v ur × where u is a unit in ( ) . Proof This will follow by comparing the values interpolated by each side of the desired ur equality, using that an element in is uniquely determined by its values at inﬁnitely many characters. Let ξ be a character of of inﬁnity type (n, −n) with n ∈ Z as in the statement of ≥1 Theorem 4.1. The decomposition (4.1) yields −1 ∗ −1 L( f ,χξ, 1) = L(ψ χ ξ N , 0) · L(ψ χξ N , 0) K K ∗ ∗ −1 ∗ −1 = L((ψ χ ) ξ, 0) · L((ψ χ ) ξ, 0), (4.2) ∗ ∗ ∗ using that ψχ and ψ χ are self-dual. The factors in (4.2) are interpolated by L (ψ χ )(ξ ) and L (ψ χ )(ξ ), respectively. Noting that ∗ ∗ −1 ∗ −1 −1 2 −1 (1 − (ψ χ ) ξ(v)) · (1 − (ψ χ ) ξ(v)) = 1 − a ( f )χ ξ(v)p + ε (p)χ ξ(v) p , p f ∗ ∗ ∗ in light of Theorem 4.4 for L (ψ χ ) and L (ψ χ ) (with j = 1and j = 0, respectively), v v we thus ﬁnd 2n−1 2n+1 n−1 n (2π) (2π) p p L (ψ χ )(ξ ) · L (ψ χ )(ξ ) = · · (n) (n + 1) · · v v 2n−1 2n+1 n−1 n (Im ϑ) (Im ϑ) ∞ ∞ −1 2 −1 × 1 − a ( f )χ ξ(v)p + ε (p)χ ξ(v) p · L( f ,χξ, 1). p f The result now follows from Theorem 4.1 and Remark 4.2. 123 338 A. Burungale et al. Remark 4.6 Note that the trivial character is in the range of interpolation for L (ψ χ ),but ∗ ∗ lies outside the range of interpolation for both L (ψ χ ) and L ( f ,χ). v v 5 Explicit reciprocity law In this section we explain a variant of the explicit reciprocity law proved in [19] relating the -adic Heegner class z to the p-adic L-function L ( f ,χ) via a Perrin-Riou big logarithm f ,χ v map, and record a key consequence. We let f = θ and χ be as in Sect. 3.3. 1 + 1 For every w| p, the natural map H (K , F T ) → H (K , T ) induced by (3.3)is w f ,χ w f ,χ 0 − injective, since its kernel is H (K , F T ) = 0. Therefore, in view of (3.6)the imageof w f ,χ z under the restriction map f ,χ 1 1 loc : H (K, T ) → H (K , T ) w f ,χ w f ,χ 1 + ur ur is naturally contained in H (K , F T ).Let the compositum of and Q . w f ,χ w p ur 1 + ur ur Theorem 5.1 There is a -linear isomorphism Log : H (K , F T ) ⊗ → v v f ,χ such that Log loc (z ) = c · L ( f ,χ) v f ,χ v ur × for some c ∈ ( ) . ur ur Proof The existence of the map Log (with coefﬁcients in , rather than ) follows v 0 from the two-variable extension by Loefﬂer–Zerbes [33] of Perrin-Riou’s big logarithm map [41], and the proof of the explicit reciprocity law (integrally) is given in [19, §5.3]. That ur the -linear map Log is injective follows from [33, Prop. 4.11], and so it becomes an ur ur isomorphism after extending scalars to = ⊗ . Similarlyasobservedin[15, 51], the equivalence between Conjectures 2.2 and 4.3 can be deduced from Theorem 5.1 using Poitou–Tate global duality. Theorem 5.2 Assume that the class z is not -torsion. Then the following are equivalent: f ,χ (a) rank S ( f ,χ) = rank X ( f ,χ) = 1, ord ord (a’) X ( f ,χ) is -torsion; and the following are equivalent: (b) char X ( f ,χ) ⊂ char S ( f ,χ)/ · z , ord -tors ord f ,χ (b’) char X ( f ,χ) ⊂ L ( f ,χ) , v v and similarly for the opposite divisibilities. In particular, Conjectures 2.2 and 4.3 are equiv- alent. Remark 5.3 Note that for the last claim in the theorem we are using the isomorphisms (3.5) and (3.6). Proof of Theorem 5.2 This can be extracted from the arguments in [16, App. A], but since our setting is slightly different (in particular, E (K )[p] is reducible) we provide the necessary details for the convenience of the reader. We explain the implications (a) ⇒ (a ) and (b ) ⇒ (b) (the only implication we will need later), and note that the other implication follows from the same ideas. 123 ∞ p -Selmer groups and rational points on CM elliptic curves 339 Following [16, §2.1], below we denote by S ( f ,χ) (resp. S ( f ,χ), etc.) the str,rel ord,rel Selmer group deﬁned as in Sect. 3.1 but with the strict at v and relaxed at v (resp. ordinary at v and relaxed at v, etc.) local conditions, so in particular S ( f ,χ) = S ( f ,χ) by ord,ord ord deﬁnition. Assume (a), and consider the exact sequence from global duality loc 1 + 0 → S ( f ,χ) → S ( f ,χ) − − → H (K , F T ) → X ( f ,χ) str,ord ord v f ,χ rel,ord → X ( f ,χ) → 0. (5.1) ord Since z is not -torsion by hypothesis, by Theorem 5.1 it follows from (5.4)that f ,χ X ( f ,χ) has -rank one and S ( f ,χ) is -torsion. Since rel,ord str,ord rank X ( f ,χ) = 1 + rank X ( f ,χ) rel,ord ord,str (cf. [16, Lem. 2.3]), we conclude that X ( f ,χ) is -torsion, and from the exact sequence ord,str loc 1 + 0 → S ( f ,χ) → S ( f ,χ) − − → H (K , F T ) → X ( f ,χ) str,rel ord,rel v f ,χ rel,str → X ( f ,χ) →0(5.2) ord,str we conclude that X ( f ,χ) = X ( f ,χ) is -torsion, i.e., (a’) holds. rel,str v Now, in addition to (a), assume (b’). Then S ( f ,χ) is -torsion (since so is X ( f ,χ), str,rel v as we just showed), and since H (K /K , T ) is -torsion free as a consequence of (3.8) f ,χ and [1, Prop. 1.1.6], it follows that in fact S ( f ,χ) = 0. (5.3) str,rel Thus (5.2) reduces to the exact sequence loc 1 + 0 → S ( f ,χ) − − → H (K , F T ) → X ( f ,χ) → X ( f ,χ) → 0. (5.4) ord,rel v f ,χ v ord,str 1 + Since H (K , F T ) has -rank one, the assumption that z is not -torsion together v f ,χ f ,χ with Theorem 5.1 implies that S ( f ,χ) has -rank one. Since z ∈ S ( f ,χ) ⊂ ord,rel f ,χ ord S ( f ,χ), it follows that S ( f ,χ) also has -rank one, and by [16, Lem. 2.3(1)] so ord,rel ord does X ( f ,χ). ord Hence the quotient S ( f ,χ)/S ( f ,χ) is -torsion, and since it injects in ord,rel ord H (K , F T ) which is -torsion-free, this shows the equality S ( f ,χ) = S ( f ,χ). v f ,χ ord ord,rel Therefore the ﬁrst two terms in the exact sequence (5.4) agree with the ﬁrst two terms in the exact sequence loc 1 + 0 → S ( f ,χ) − − → H (K , F T ) → X ( f ,χ) → X ( f ,χ) → 0 (5.5) ord v f ,χ rel,ord ord (note that S ( f ,χ) as a consequence of (5.3)), and this yields str,ord 1 + S ( f ,χ) loc H (K , F T ) ord v v f ,χ 0 → − − → → coker(loc ) → 0. · z · loc (z ) f ,χ v f ,χ In view of Theorem 5.1, it follows that S ( f ,χ) ord ur char · char coker(loc ) = (L ( f , χ )). (5.6) v v · z f ,χ Next, from (5.4)and (5.5) we can extract the short exact sequences 0 → coker(loc ) → X ( f ,χ) → X ( f ,χ) → 0, v v ord,str 0 → coker(loc ) → X ( f ,χ) → X ( f ,χ) → 0, v rel,ord ord 123 340 A. Burungale et al. from which we readily obtain (taking -torsion in the ﬁrst exact sequence and using a straightforward variant of [16, Lem. 2.3]) the relations char X ( f ,χ) = char X ( f ,χ) · char coker(loc ) v ord,str v = char X ( f ,χ) · char coker(loc ) rel,ord -tors v = char X ( f ,χ) · char (coker(loc )) . ord -tors v Combined with (5.6), we thus obtain S ( f ,χ) ord ur 2 char X ( f ,χ) · char = char X ( f ,χ) · L ( f ,χ) . v ord -tors v · z f ,χ The result follows. 6 Twisted anticyclotomic main conjectures for K Let K be an imaginary quadratic ﬁeld satisfying (spl). The Iwasawa main conjecture for K wasprovedbyRubin[43] under some restrictions on p (including p O ) that were removed in subsequent work by Johnson-Leung–Kings [31] and Oukhaba–Viguié [39]. In this section we record a consequence of these results for the anticyclotomic Z -extension. Note that if ξ is a self-dual Hecke character in the sense of Sect. 4.2, then the Hecke −1 L-function L(ξ , s) is self-dual, with a functional equation relating its values at s and −s. In the following, by the sign of ξ we refer to the sign appearing in the functional equation −1 for L(ξ , s). Theorem 6.1 Let ψ be a Hecke character of K of inﬁnity type (−1, 0), and let χ be a ﬁnite order of character of such that the product ψχ is self-dual. Assume that ψ χ has sign +1. Then: (i) X (ψ χ ) is -torsion and the following equality holds: ∗ ∗ char X (ψ χ ) = L (ψ χ ) . v v (ii) The following divisibility holds: ∗ ∗ ∗ ∗ char X (ψ χ ) ⊂ L (ψ χ ) . v v Proof As noted in Remark 3.3, the Iwasawa module X (ψ χ ) recovers the Bloch–Kato Selmer group for ψχ over the anticyclotomic Z -extension K /K, and so the result of p ∞ part (i) follows from [1, Thm. 2.4.17], as extended in [2, Thm. 3.9]. (In these references, the hypothesis p > 3 arises from their appearance in [43], but as already mentioned this restriction can be removed thanks to [31, 39].) ∞ ∗ ∗ For (ii), put = OGal(K(c p )/K), = ⊗ , and deﬁne X (ψ χ ) and 0 0 O v ∗ ∗ ∗ ∗ X (ψ χ ) similarly as X (ψ χ ) and X (ψ χ ) in Sect. 3.2 but with in place of .By v v v 0 0 the Iwasawa main conjecture for K, the module X (ψ χ ) is -torsion, with v 0 ∗ ∗ char X (ψ χ ) = L (ψ χ ) , (6.1) v v where L is the integral p-adic measure L appearing in the proof of Theorem 4.4,and v v ∗ ∗ ∗ ∗ L (ψ χ) denotes its twist by ψ χ. Noting that X (ψ χ ) is the twist (in the sense of [46, v v 123 ∞ p -Selmer groups and rational points on CM elliptic curves 341 ∗ ∗−1 §6.1]) of X (ψ χ ) by ψψ , from Corollary 6.2.2 and Lemma 6.1.2 in loc. cit. we deduce ∗ ∗ from (6.1) that the module X (ψ χ ) is -torsion, with v 0 ∗ ∗ ∗ ∗ char X (ψ χ ) = L (ψ χ ) . (6.2) v v The divisibility in (ii) now follows from (6.2) after descent. 7 The main results Recall that K is an imaginary quadratic ﬁeld of discriminant −D < 0 satisfying (spl), with v the prime of K above p induced by our ﬁxed embedding ı . Theorem 7.1 Let ψ be a Hecke character of K of inﬁnity type (−1, 0) and conductor f with cyclic quotient of norm prime to D , and set f = θ , N = D · N(f ), N = d · f . K ψ K ψ Let c be a positive integer prime to N p, and let χ be a ﬁnite order character such that −1 (1) χ N ∈ (c, N,ε ). Assume that ψχ has sign −1.Then: cc f (i) The class z is not -torsion. f ,χ (ii) The module X ( f ,χ) is -torsion, and ur 2 char X ( f ,χ) ⊂ L ( f ,χ) . v v Proof Part (i) follows from [5, Thm. 1.1], so we focus on (ii). By the Gross–Zagier formula, the non-triviality of z implies that for all but ﬁnitely many ﬁnite order characters ξ : → f ,χ μ we have ord L( f ,χξ, s) = 1. (7.1) s=1 Fix any such ξ , and note that L( f ,χξ, s) factors as L( f ,χξ, s) = L(ψ χ ξ, s) · L(ψ χξ, s) (7.2) and has sign −1, since (Heeg) holds in our setting (see Sect. 4.3). By our sign assumption on ψχ, it follows that L(ψ χ, s) has sign +1 and from (7.1)and (7.2) we conclude ord L(ψ χ ξ, s) = 1, L(ψ χξ, 1) = 0. s=1 By an application of the Gross-Zagier formula [53]and [36,Thm.3.2]wehave ∗ ∗ ∗ ord L(ψ χ ξ, s) = 1 ⇒ corank H (K, W ) = 1, s=1 O ψ χ ξ and by [37,Thm.B]wehave ∗ 1 ∗ ∗ L(ψ χξ, 1) = 0 ⇒ corank H (K, W ) = 0, O ψχ ξ 1 1 ∗ ∗ ∗ where H (K, W ∗ ∗ ∗ ) and H (K, W ∗ ∗ ) are the Bloch–Kato Selmer groups for ψ χ ξ ψ χ ξ ψχ ξ f f ∗ ∗ and ψχ ξ , respectively, whose deﬁnition is recalled in Sect. 8 below. By the analogue of the decomposition (8.1) below, it follows that H (K , W ) has O- f ,χ ξ corank one, and therefore so does Sel (A /K). Varying ξ , by a variant of Mazur’s f ,χ ξ control theorem it follows that X(A /K ) (or equivalently, S ( f ,χ) and X ( f ,χ)) f ,χ ∞ ord ord has -rank one, and so by Theorem 5.2 we conclude that X ( f ,χ) is -torsion. Finally, by the decomposition in Proposition 3.4 and the factorization in Proposition 4.5, the divisibility in part (ii) of the theorem follows from Theorem 6.1. 123 342 A. Burungale et al. Corollary 7.2 Let f = θ and χ be an in Theorem 7.1, and assume that ψχ has sign −1. Then the modules S(A /K ) and X(A /K ) have both -rank one, and f ,χ ∞ f ,χ ∞ char (X(A /K ) ) ⊂ char S(A /K )/ · z . f ,χ ∞ -tors f ,χ ∞ f ,χ Proof That S(A /K ) and X(A /K ) have both -rank one has been shown in the f ,χ ∞ f ,χ ∞ course of the proof of Theorem 7.1, and the divisibility in the statement of Corollary 7.2 follows from Theorem 5.2 and the divisibility in part (ii) of Theorem 7.1. 8The p-converse In this section we deduce from our main results the proof of Theorem A in the Introduction. Let λ be a self-dual Hecke character of inﬁnity type (−1, 0) and conductor f , and suppose that: (a) λ has sign −1; (b) λ has central character ω = η ; λ K (c) d f . Note that f is divisible by d = ( −D ) by condition (b). Since λ is self-dual, f is K K λ λ invariant under complex conjugation, so by condition (c) we can write f = (c)d for a unique c > 0. We shall apply Corollary 7.2 for a pair (ψ, χ ) which is good for λ in the following sense: (G1) ψ has inﬁnity type (−1, 0) and conductor f with cyclic quotient of norm prime to pD ; −1 (1) (G2) χ is a ﬁnite order character such that χ N ∈ (c, N,ε ),where f = θ and cc f ψ N = f d ; (G3) ψχ = λ; ∗−1 −1 (G4) L(ψ χ , 0) = 0. The existence of good pairs for λ is shown in [6, Lem. 3.29] building on the non-vanishing results of Greenberg [25] and Rohrlich [42]. Fix a good pair (ψ, χ ) for λ,and let F be a number ﬁeld of containing the values of ψ and χ.Let P be the prime of F above p induced by our ﬁxed embedding ı ,let be the completion of F at P,and let O be the ring of integers of . Similarly as in (3.2), for any Hecke character ξ put W := T ⊗ D ξ ξ O where D = /O.Let a ﬁnite set of places of K containing ∞, p,and theprimesof K dividing the conductor of λ. Denote by H (K, W ∗ ∗ ) the Bloch–Kato Selmer group for ψ χ ∗ ∗ ψ χ : H (K , W ∗ ∗ ) v ψ χ 1 1 H (K, W ∗ ∗ ) = ker H (K /K, W ∗ ∗ ) → ψ χ ψ χ ∗ ∗ H (K , W ) v ψ χ div 1 1 ur ×H (K , W ∗ ∗ ) × H (K , W ∗ ∗ ) , v ψ χ ψ χ w∈,w p 123 ∞ p -Selmer groups and rational points on CM elliptic curves 343 1 1 ur ∗ ∗ ∗ ∗ where H (K , W ) ⊂ H (K , W ) is the maximal divisible submodule and K v ψ χ div v ψ χ denotes the maximal unramiﬁed extension of K . Similarly, let H (K , W ∗ ) v ψχ 1 1 1 ∗ ∗ ∗ H (K, W ) = ker H (K /K, W ) → H (K , W ) × ψχ ψχ v ¯ ψχ H (K , W ) v ψχ div 1 ur × H (K , W ∗ ) ψχ w∈,w p be the Bloch–Kato Selmer group for ψχ (see also [1, §1.1]). Finally, let V be as in (3.1). f ,χ Lemma 8.1 In the above setting, we have 1 1 corank Sel ∞ (A /K) = corank H (K, W ∗ ∗ ) + corank H (K, W ∗ ). O P f ,χ O ψ χ O ψχ f f Proof It is a standard fact (see e.g. [8]), Sel ∞ (A /K) agrees with the Bloch–Kato Selmer P f ,χ group 1 1 H (K, W ) ⊂ H (K, W ), f ,χ f ,χ where W := T ⊗ D for the G -stable O-lattice T ⊂ V coming from T A . f ,χ f ,χ O K f ,χ f ,χ P f In turn (see e.g. [27, Prop. 2.2]), the local conditions deﬁning H (K, W ) at the primes w f ,χ of K above p can be described in terms of the ﬁltration (3.3), namely: H (K , W ) w f ,χ 1 1 H (K, W ) = ker H (K /K, W ) → f ,χ f ,χ H (K , F W ) w w f ,χ div w| p 1 ur × H (K , W ), f ,χ w∈,w p + + where F W := F T ⊗ D. Note that since f = θ we have f ,χ f ,χ O ψ w w 1 + 1 1 1 H (K , F W ) = H (K , W ∗ ), H (K , F W ) = H (K , W ∗ ∗ ). v f ,χ v ψχ v f ,χ v ψ χ Since different lattices T give rise to Selmer groups H (K, W ) having the same f ,χ f ,χ O-corank, taking T so that W W ∗ ∗ ⊕ W ∗ , comparing the local conditions we f ,χ f ,χ ψ χ ψχ thus ﬁnd 1 1 1 ∗ ∗ ∗ H (K, W ) H (K, W ) ⊕ H (K, W ) (8.1) f ,χ ψ χ ψχ f f f and the result follows. The following recovers Theorem A in the introduction as a special case. Theorem 8.2 Let λ be a self-dual Hecke character of K of inﬁnity type (−1, 0) with central character ω = η and whose conductor f satisﬁes d f .Then λ K K λ λ corank Sel (B /K) = 1 ⇒ ord L(λ, s) = 1. O λ s=1 Proof By the p-parity conjecture [35], if corank Sel (B /K) = 1then λ has sign −1. O λ Let (ψ, χ ) be a good pair for λ, i.e., satisfying conditions (G1)–(G4) above, so in particular ∗−1 −1 L(ψ χ , 0) = 0. (8.2) 123 344 A. Burungale et al. By Theorem 4.4, the nonvanishing (8.2) implies that the p-adic L-function L (ψ χ ) does not vanish at trivial character, so by Theorem 4.4 it follows that ∗ ∗ # X (ψ χ )/(γ − 1)X (ψ χ ) < ∞, v v where γ ∈ is any topological generator. Since X (ψ χ ) corresponds to the Bloch–Kato ∗ 1 Selmer group for ψχ over K /K (see Remark 3.3), it follows that corank H (K, W ∗ ) = ∞ O ψχ ∞ ∗ ∗ 0. Since Sel (B /K) H (K, W ), from Lemma 8.1 we thus obtain λ ψ χ ∞ ∞ corank Sel (B /K) = 1 ⇒ corank Sel (A /K) = 1. (8.3) O λ O f ,χ P P Now, Corollary 7.2 together with a variant of Mazur’s control theorem immediately yields the implication corank Sel (A /K) = 1 ⇒ z = 0 ∈ S (A /K) ⊗ . (8.4) O P f ,χ f ,χ ,0 P f ,χ O where z is the image of z under the specialization map S(A /K ) → H (K, V ) f ,χ ,0 f ,χ f ,χ ∞ f ,χ at the trivial character. By deﬁnition, the class z is a nonzero multiple of f ,χ ,0 χ(σ ) · y , f ,c σ ∈Gal(K )/K where y is the Heegner class introduced in Sect. 2.2,and so f ,c z = 0 ⇐⇒ ord L( f ,χ, s) = 1 (8.5) f ,χ ,0 s=1 by virtue of the general Gross–Zagier formula [20, 53]. Finally, we note once more that (3.8) yields the factorization L( f ,χ, s) = L(ψ χ , s) · L(ψ χ, s). Combining (8.3), (8.4), and (8.5) we thus obtain corank Sel (B /K) = 1 ⇒ ord L( f ,χ, s) = 1 O λ s=1 ⇒ ord L(ψ χ , s) = 1, s=1 using (8.2) and the above factorization for the last implication. Since ψχ = λ, this concludes the proof. Acknowledgements We thank Matthias Flach, Jacob Ressler and Qiyao Yu for helpful discussions. We also thank Henri Darmon and Antonio Lei for giving us the opportunity to contribute to this special issue, and the anonymous referee for a detailed reading. During the preparation of this paper, A.B. was partially supported by the NSF grant DMS-2001409; F.C. was partially supported by the NSF grants DMS-1946136 and DMS- 2101458; C.S. was partially supported by the Simons Investigator Grant #376203 from the Simons Foundation and by the NSF Grant DMS-1901985; Y.T. was partially supported by the NSFC grants #11688101 and #11531008. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 123 ∞ p -Selmer groups and rational points on CM elliptic curves 345 References 1. Adebisi Agboola and Benjamin Howard. Anticyclotomic Iwasawa theory of CM elliptic curves. Ann. Inst. Fourier (Grenoble), 56(4):1001–1048, 2006. 2. Trevor Arnold. Anticyclotomic main conjectures for CM modular forms. J. Reine Angew. Math., 606:41– 78, 2007. 3. Ashay Burungale, Francesc Castella, and Chan-Ho Kim. A proof of Perrin-Riou’s Heegner point main conjecture. Algebra Number Theory, 15(7):1627–1653, 2021. 4. Ashay A. Burungale, Francesc Castella, Christopher Skinner, and Ye Tian. p-converse to a theorem of Gross–Zagier and Kolyvagin: CM elliptic curves over totally real ﬁelds. 2021. preprint. 5. Ashay A. Burungale and Daniel Disegni. On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions. Ann. Inst. Fourier (Grenoble), 70(5):2077–2101, 2020. 6. Massimo Bertolini, Henri Darmon, and Kartik Prasanna. p-adic Rankin L-series and rational points on CM elliptic curves. Paciﬁc J. Math., 260(2):261–303, 2012. 7. Massimo Bertolini, Henri Darmon, and Kartik Prasanna. Generalized Heegner cycles and p-adic Rankin L-series. Duke Math. J., 162(6):1033–1148, 2013. With an appendix by Brian Conrad. 8. Spencer Bloch and Kazuya Kato. L-functions and Tamagawa numbers of motives. In The Grothendieck Festschrift, Vol. I, volume 86 of Progr. Math., pages 333–400. Birkhäuser Boston, Boston, MA, 1990. 9. Miljan Brakoce ˇ vic. ´ Anticyclotomic p-adic L-function of central critical Rankin-Selberg L-value. Int. Math. Res. Not. IMRN, (21):4967–5018, 2011. 10. Manjul Bhargava, Christopher Skinner, and Wei Zhang. A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture. 2015. preprint, arXiv:1505.08165. 11. Ashay A. Burungale and Ye Tian. A rank zero p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin. 2019. preprint. 12. Ashay A. Burungale and Ye Tian. p-converse to a theorem of Gross-Zagier, Kolyvagin and Rubin. Invent. Math., 220(1):211–253, 2020. 13. Ashay Burungale and Ye Tian. The even parity Goldfeld conjecture: congruent number elliptic curves. J. Number Theory, 230:161–195, 2022. 14. Ashay A. Burungale. On the μ-invariant of the cyclotomic derivative of a Katz p-adic L-function. J. Inst. Math. Jussieu, 14(1):131–148, 2015. 15. Francesc Castella. On the p-adic variation of Heegner points. 2013. Thesis (Ph.D.)–McGill University. 16. Francesc Castella. p-adic heights of Heegner points and Beilinson-Flach classes. J. Lond. Math. Soc., 96(1):156–180, 2017. 17. J. Coates and R. Greenberg. Kummer theory for abelian varieties over local ﬁelds. Invent. Math., 124(1- 3):129–174, 1996. 18. Francesc Castella, Giada Grossi, Jaehoon Lee, and Christopher Skinner. On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes. Invent. Math., 227(2):517–580, 2022. 19. Francesc Castella and Ming-Lun Hsieh. Heegner cycles and p-adic L-functions. Math. Ann., 370(1- 2):567–628, 2018. 20. Li Cai, Jie Shu, and Ye Tian. Explicit Gross-Zagier and Waldspurger formulae. Algebra Number Theory, 8(10):2523–2572, 2014. 21. Francesc Castella and Xin Wan. Perrin-Riou’s main conjecture for elliptic curves at supersingular primes. 2016. preprint, arXiv:1607.02019. 22. Daniel Disegni. The p-adic Gross-Zagier formula on Shimura curves. Compos. Math., 153(10):1987– 2074, 2017. 23. Ehud de Shalit. Iwasawa theory of elliptic curves with complex multiplication, volume 3 of Perspectives in Mathematics. Academic Press, Inc., Boston, MA, 1987. 24. Dorian Goldfeld. Conjectures on elliptic curves over quadratic ﬁelds. In Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), volume 751 of Lecture Notes in Math., pages 108–118. Springer, Berlin, 1979. 25. Ralph Greenberg. On the critical values of Hecke L-functions for imaginary quadratic ﬁelds. Invent. Math., 79(1):79–94, 1985. 26. Ralph Greenberg. Iwasawa theory and p-adic deformations of motives. In Motives (Seattle, WA, 1991), volume 55 of Proc. Sympos. Pure Math., pages 193–223. Amer. Math. Soc., Providence, RI, 1994. 27. Ralph Greenberg. Iwasawa theory for elliptic curves. In Arithmetic theory of elliptic curves (Cetraro, 1997), volume 1716 of Lecture Notes in Math., pages 51–144. Springer, Berlin, 1999. 28. Ming-Lun Hsieh. On the μ-invariant of anticyclotomic p-adic L-functions for CM ﬁelds. J. Reine Angew. Math., 688:67–100, 2014. 29. Ming-Lun Hsieh. Special values of anticyclotomic Rankin-Selberg L-functions. Doc. Math., 19:709–767, 123 346 A. Burungale et al. 30. H. Hida and J. Tilouine. Anti-cyclotomic Katz p-adic L-functions and congruence modules. Ann. Sci. École Norm. Sup., 26(2):189–259, 1993. 31. Jennifer Johnson-Leung and Guido Kings. On the equivariant main conjecture for imaginary quadratic ﬁelds. J. Reine Angew. Math., 653:75–114, 2011. 32. Nicholas M. Katz. p-adic L-functions for CM ﬁelds. Invent. Math., 49(3):199–297, 1978. 33. David Loefﬂer and Sarah Livia Zerbes. Iwasawa theory and p-adic L-functions over Z -extensions. Int. J. Number Theory, 10(8):2045–2095, 2014. 34. Yifeng Liu, Shouwu Zhang, and Wei Zhang. A p-adic Waldspurger formula. Duke Math. J., 167(4):743– 833, 2018. 35. Jan Nekováˇr. On the parity of ranks of Selmer groups. II. C. R. Acad. Sci. Paris Sér. I Math., 332(2):99–104, 36. Jan Nekováˇr. The Euler system method for CM points on Shimura curves. In L-functions and Galois representations, volume 320 of London Math. Soc. Lecture Note Ser., pages 471–547. Cambridge Univ. Press, Cambridge, 2007. 37. Jan Nekováˇr. Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two. Canad. J. Math., 64(3):588–668, 2012. 38. Loren D. Olson. Points of ﬁnite order on elliptic curves with complex multiplication. Manuscripta Math., 14:195–205, 1974. 39. Hassan Oukhaba and Stéphane Viguié. On the μ-invariant of Katz p-adic L-functions attached to imag- inary quadratic ﬁelds. Forum Math., 28(3):507–525, 2016. 40. Bernadette Perrin-Riou. Fonctions Lp-adiques, théorie d’Iwasawa et points de Heegner. Bull. Soc. Math. France, 115(4):399–456, 1987. 41. Bernadette Perrin-Riou. Théorie d’Iwasawa des représentations p-adiques sur un corps local. Invent. Math., 115(1):81–161, 1994. With an appendix by Jean-Marc Fontaine. 42. David E. Rohrlich. On L-functions of elliptic curves and anticyclotomic towers. Invent. Math., 75(3):383– 408, 1984. 43. Karl Rubin. The “main conjectures” of Iwasawa theory for imaginary quadratic ﬁelds. Invent. Math., 103(1):25–68, 1991. 44. Karl Rubin. p-adic L-functions and rational points on elliptic curves with complex multiplication. Invent. Math., 107(2):323–350, 1992. 45. Karl Rubin. p-adic variants of the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication. In p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), volume 165 of Contemp. Math., pages 71–80. Amer. Math. Soc., Providence, RI, 1994. 46. Karl Rubin. Euler systems, volume 147 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2000. Hermann Weyl Lectures. The Institute for Advanced Study. 47. Jacob Ressler and Qiyao Yu. p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin for small primes. 2021. in preparation. 48. Goro Shimura. On the zeta-function of an abelian variety with complex multiplication. Ann. of Math. (2), 94:504–533, 1971. 49. Christopher Skinner. A converse to a theorem of Gross, Zagier, and Kolyvagin. Ann. of Math., 191(2):329– 354, 2020. ∞ ∞ 50. Alexander Smith. 2 -Selmer groups, 2 -class groups, and Goldfeld’s conjecture. 2017. preprint, arXiv:1702.02325. 51. Xin Wan. Heegner Point Kolyvagin System and Iwasawa Main Conjecture. Acta Math. Sin., 37(1):104– 120, 2021. 52. Qiyao Yu. p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin for small primes. 2021. https:// resolver.caltech.edu/CaltechTHESIS:06022021-000654935 Caltech, Senior thesis. 53. Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang. The Gross-Zagier formula on Shimura curves, volume 184 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2013. 54. Wei Zhang. Selmer groups and the indivisibility of Heegner points. Camb. J. Math., 2(2):191–253, 2014. 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Annales mathématiques du Québec – Springer Journals
Published: Oct 1, 2022
Keywords: P-adic; L-functions; Selmen groups; Elliptic waves; Heegen points; Primary 11G05; Secondary 11G40
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