p-ADIC L-FUNCTIONS AND RATIONAL POINTS ON CM ELLIPTIC CURVES AT INERT PRIMES

Abstract Let K be an imaginary quadratic field and 
$p\geq 5$
 a rational prime inert in K. For a 
$\mathbb {Q}$
 -curve E with complex multiplication by 
$\mathcal {O}_K$
 and good reduction at p, K. Rubin introduced a p-adic L-function 
$\mathscr {L}_{E}$
 which interpolates special values of L-functions of E twisted by anticyclotomic characters of K. In this paper, we prove a formula which links certain values of 
$\mathscr {L}_{E}$
 outside its defining range of interpolation with rational points on E. Arithmetic consequences include p-converse to the Gross–Zagier and Kolyvagin theorem for E. A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic 
${\mathbb {Z}}_p$
 -extension 
$\Psi _\infty $
 of the unramified quadratic extension of 
${\mathbb {Q}}_p$
 . Along the way, we present a theory of local points over 
$\Psi _\infty $
 of the Lubin–Tate formal group of height 
$2$
 for the uniformizing parameter 
$-p$
 .


Introduction
Since the seminal work of Coates and Wiles, Iwasawa theory of CM elliptic curves influences general Iwasawa theory.It continues to have applications to classical Diophantine problems.The nature of prime p is inherent to Iwasawa theory.For primes split in the CM field, CM Iwasawa theory is well-developed.In contrast, for non-split primes, new phenomena abound and CM Iwasawa theory is still incipient.
Let K be an imaginary quadratic field and p ≥ 5 a rational prime which is inert in K. Let K n be the n-th layer of the anticyclotomic Z p -extension K ∞ of K. Let E be an elliptic curve defined over Q with complex multiplication by O K .In the early 1980s, R. Greenberg found the formula for root numbers, where ϕ denotes the Hecke character of E and χ an anticyclotomic finite character of K of order p n > 1.It led him to the formula for all n sufficiently large (cf.[24], [26], see also Corollary 3.11).Here, ε n is 0 or 2 and ε n = 2 if and only if W (ϕ) = (−1) n .So, new points of infinite order occur in the alternate anticyclotomic layers.This behavior of the Mordell-Weil rank is peculiar to the inert case.For example, for a split prime p, we have rank Z E(K ∞ ) < +∞ if W (ϕ) = +1.In the late 1980s K. Rubin envisioned an Iwasawa theory echoing such phenomena and made a fundamental conjecture on the structure of anticyclotomic local units (cf.[47]).Recently, we proved the conjecture [12].The resolution has unexpectedly led us to new developments in supersingular Iwasawa theory.This is the first of the series of papers of our study.
In [47], Rubin constructed an anticyclotomic p-adic L-function L E interpolating special values L(ϕχ,1) for finite anticyclotomic characters χ of K with W (ϕχ) = +1.If one expects a p-adic Birch and Swinnerton-Dyer conjecture for L E , the function should encode the rank behavior (1.1).The main result of this paper is a formula relating the value of L E at a finite anticyclotomic character χ of K with W (ϕχ) = −1 to the formal group logarithm of a rational point on E(K n ) χ behind the phenomenon (1.1) (see Theorems 1.1 and 1.2).It has an application for the Birch and Swinnerton-Dyer (BSD) conjecture, namely a p-converse to the Gross-Zagier and Kolyvagin theorem (see Theorem 1.5).

Main results.
Let p ≥ 5 be a prime.Let Q be an algebraic closure of Q. Fix embeddings ι ∞ : Q → C and ι p : Q → C p .
Let K be an imaginary quadratic field with p inert and O K the integer ring.Let O (resp.Φ) be the completion of O K (resp.K ) at p.In this introduction, we assume the class number h K of K equals 1; however, the main text only assumes p h K .Let K ∞ be the anticyclotomic Z p -extension of K and Γ = Gal(K ∞ /K).Let Ξ be the set of finite characters of Γ.Let Ξ + = {χ ∈ Ξ | cond r χ is an even power of p}, Ξ − = {χ ∈ Ξ | cond r χ is an odd power of p}.
Let E be an elliptic curve defined over Q with complex multiplication by O K with good reduction at p (note that E has supersingular reduction at p since p is inert).Let T be the p-adic Tate module of E, which is an O-module of rank 1, and put T ⊗−1 = Hom O (T,O).Fix a minimal Weierstrass model of E over Z (p) and let ω be the associated Néron differential form.Let Ω ∈ C × be a CM period so that ΩO K is the period lattice.Let ϕ be the associated Hecke character of K.In particular, Let W (ϕ) be the root number of the Hecke L-function L(ϕ,s).

Rubin p-adic L-function.
Here, we introduce Rubin's theory in terms of Galois cohomology.The relation to Rubin's original formulation is explained in Section 2.
For χ ∈ Ξ −ε , note that L(ϕχ,1) = 0 by the functional equation and L E (χ) is not related to L(ϕχ,1) directly.In light of the BSD conjecture, it is natural to seek: links between L E (χ) for χ ∈ Ξ −ε and rational points in E(K ∞ ) χ . (Q) This question is due to Rubin [47, p. 421].
Theorem 1.1.Let E /Q be a CM elliptic curve with root number −1 and K the CM field.Let p ≥ 5 be a prime of good supersingular reduction for E /Q and L E the Rubin p-adic L-function as in (1.4).Then, there exists a rational point P ∈ E(Q) with the following properties.
(a) We have for , ∞ the Néron-Tate height pairing.
See also Theorem 4.8 in a more general setting.
The formula is the principal result of this paper.It gives a p-adic criterion for E to have analytic rank one.For such curves, the p-adic L-value in turn leads to a p-adic construction of a rational point of infinite order which is independent of the choice of v − (cf.Corollary 4.9).
Our second result is an interpolation of the Rubin p-adic L-function at higher order characters in Ξ −ε .Theorem 1.2.Let E /Q be a CM elliptic curve and K the CM field.Let ϕ be the associated Hecke character and ε the sign of the root number of ϕ.Let p ≥ 5 be a prime of good supersingular reduction for E /Q and L E the Rubin p-adic L-function as in (1.4).Let χ ∈ Ξ −ε be a Hecke character with conductor p n+1 .Let z χ ∈ H 1 (K n ,T p E) χ be the image of a system of elliptic units of E (cf.§3.1.2).Then, , and it has the following properties.
(a) We have See also Theorem 3.16 in a more general setting.Note that ord s=1 L(ϕχ,s) = 1 for all but finitely many χ ∈ Ξ −ε (cf.[45]).So, in view of Theorem 1.2 (b), the Rubin p-adic L-function leads to a construction of new points of infinite order in the alternate anticyclotomic layers (cf.(1.1)).
An evidence appears in [12,Thm. 2.4].It seems interesting to compare Theorems 1.1 and 1.2 with the exceptional zero conjecture of Mazur, Tate and Teitelbaum [39].
As a corollary of the above theorems, we obtain a refined (non-asymptotic) version of (1.1).
Corollary 1.4.Let E /Q be a CM elliptic curve and K the CM field.
(ii) Suppose that ord s=1 L(E /Q ,s) = 1 and there exists a rational point P ∈ E(Q) whose image generates the free where ε n = 0 (resp.2) for n even (resp.odd).
In particular, if X(E /Kn ) is finite, then A key to the proof of main results is a theory of local points, similarly as [31, §8] underlies the cyclotomic signed Iwasawa theory [31] (cf.Section 2.3.)In the case of cyclotomic deformation, such a theory is the core of Perrin-Riou theory.However, Perrin-Riou theory for the anticyclotomic Z p -extension is not yet developed sufficiently to be applicable to our case.Instead, we use Rubin's conjecture to construct local points.It may give some insight towards a Perrin-Riou theory for the anticyclotomic Z p -extension.

p-converse to a theorem of Gross-Zagier and Kolyvagin.
Theorem 1.5.Let E /Q be a CM elliptic curve with good supersingular reduction at p ≥ 5.
See also Theorem 4.18 in a more general setting.Just as the Bertolini-Darmon-Prasanna formula is employed in the proof of Skinner's p-converse [52], our approach is based on Theorem 1.1.

Remark 1.6.
(i) The first results towards the p-converse were due to Rubin [51], which treated CM elliptic curves and ordinary primes p.The first general results for non-CM curves were independently due to Skinner [52] and Zhang [56] a few years back.

Background
An impetus to Theorems 1.1 and 1.2 is a formula of Rubin.For primes p split in an imaginary quadratic field K, Rubin proved an influential formula [50] which links certain values of the Katz p-adic L-function of K to the formal group logarithm of rational points on elliptic curves with CM by K (cf.[42], [44]).The last decade has led to a revival of Rubin's formula.For an arbitrary elliptic curve E /Q and K an imaginary quadratic field satisfying Heegner hypothesis for E with p split, the Bertolini-Darmon-Prasanna (BDP) formula relates certain values of a Rankin-Selberg p-adic L-function L Gr E of E /K with the formal group logarithm of Heegner points on E (cf.[6], [34]).Since its advent, the BDP formula has influenced the arithmetic of elliptic curves and inspired progress towards the BSD conjecture, with an instance being p-converse to the Gross-Zagier and Kolyvagin theorem due to Skinner (cf.[52]), which is a p-adic criterion for E /Q to have both algebraic and analytic rank one.The p-converse is based on the BDP formula and an Iwasawa theory of L Gr E .Subsequently, Liu-Zhang-Zhang interpreted the BDP formula as a p-adic Waldspurger formula and generalised it to modular elliptic curves over totally real fields (cf.[37]).
An emerging search is the analogue of the BDP formula2 over imaginary quadratic fields with p non-split, and a pertinent Iwasawa main conjecture (the conjectural backdrop of Iwasawa theory excludes such a non-split setting; cf.[25], [44], [31]).The ensuing CM case is perhaps the first instance, whose investigation we plan to continue (cf.[13], [14]).

Plan
Section 2 presents the local theory.In Section 3, certain global aspects appear, including (1.1) and Theorem 1.2.Then Section 4 treats Theorems 1.1 and 1.5.
The proof of Theorem 1.1 is based on the appendices to which the reader may refer prior to Section 4. Appendix A describes a variant of the p-adic Gross-Zagier formula [32] in which the p-adic logarithm of Heegner points appears (see Theorem A.6). Appendix B exhibits another consequence: the Perrin-Riou conjecture [42] for GL 2 -type abelian varieties at primes of good non-ordinary reduction (see Theorem B.3).

The set-up
We introduce the module of anticyclotomic local units as well as its signed submodules following [47], [12].

Notation.
Let p ≥ 5 be a prime.Let Φ be the unramified quadratic extension of Q p and O the integer ring.We fix a Lubin-Tate formal group F over O for the uniformizing parameter π := −p.Let λ denote the logarithm of F .
For n ≥ 0, write Φ n = Φ(F [π n+1 ]), the extension of Φ in C p generated by the π n+1torsion points of F , and put be the natural isomorphism induced by the Galois action on the π-adic Tate module T π F =: T. Let Θ n be the subfield of Φ n with [Θ n : Φ] = p 2n and Θ ∞ = ∪ n≥1 Θ n the Z 2 p -extension of Φ.Let Ψ ∞ be the anticyclotomic Z p -extension of Φ and Ψ n the n-th layer.We put Let U n be the group of principal units in Φ n , that is, the group of elements in O × Φn congruent to one modulo the maximal ideal.Let where the superscript Δ means the Δ-invariants.Define the Iwasawa algebras It is known that U * ∞ is a free Λ 2 -module of rank 2 (cf.[54]).A primary object is the anticyclotomic projection Φ n be the Coates-Wiles homomorphism as in [12, §2].For a finite character χ of Gal(Φ ∞ /Φ) of conductor dividing p n+1 and u ∈ U * ∞ , let 2.1.2.Rubin's conjecture.Let Ξ be the set of finite characters of G − .Let Ξ + = {χ ∈ Ξ | cond r χ is an even power of p}, Rubin showed that V * ,± ∞ is a free Λ-module of rank one (cf.[47,Prop. 8.1]).The following is central to the construction of local points.
Theorem 2.1.(Rubin's conjecture) We have Here, the first isomorphism is the Kummer map and the third is a consequence of the Gal(Φ ∞ /Φ n )-action on O(1) ⊗ T ⊗−1 /π n being trivial.The Δ-invariants of (2.3) give an isomorphism be the dual exponential map which arises from the identification of Fil 0 D dR (T commutes, where the upper horizontal map is (2.4).The anticyclotomic projection induces an isomorphism as well as a commutative diagram where δ ac n := Tr Φn/Ψn • δ n .Hence, for a character χ of Gal( Therefore, we may naturally identify V * ,± ∞ with the module H 1 ± introduced in (1.2), and Theorem 2.1 implies the decomposition (1.3) of lim (1) The quotient by the ideal (γ p n − 1)Λ induces an isomorphism . In light of the inflation-restriction sequence and [47,Prop. 4 The local duality thus implies (2.10). ( Hence, by the commutative diagram (2.8), the proof concludes.
In the following, (2.10) will be often treated as an identification.

An optimal basis.
We introduce a basis of the submodule of signed anticyclotomic local units and duality pairings, which will be used in the construction of local points.
We fix a Λ-basis v ± of V * ,± ∞ and regard it as an element of lim (2) ( , ) n is a perfect pairing.
) are orthogonal complements of each other under the pairing.
Proof.The first assertion is a simple consequence of Theorem 2.1.
Note that where T τ denotes conjugation of T by the complex conjugate.Then, by [47,Prop. 4 free, and we have natural identifications Hence, the local duality induces where the isomorphism arises from the perfect pairing ( , ) n is also an isomorphism, and hence, ( , ) n is perfect.
The assertion (3) then follows from the fact that )⊗ Q p under the base change of ( , ) n , and vice versa.By Lemma 2.3, we have a perfect pairing ( , (2.12) Note that v ⊥ ±,n depends on the choice of v ∓ but is independent of v ± .

Local points
We introduce an optimal system of local points, which generate the signed submodules of the underlying Lubin-Tate group. For Define 1≤k≤n, k:odd for Φ p k (X) the p k -th cyclotomic polynomial, and we also put

Definition 2.4 (local points)
. For v ± and γ as above, let Proof.It suffices to show that for a finite character χ of Gal(Ψ n /Φ), the image lies in the finite part, and it gives rise to a generator of ) with respect to the local duality, it thus follows that χ(c ± n ) lies in the finite part.
By Lemma 2.5, we may naturally regard c ± n as an element in F (m n ).In particular, Salient features of the local points are given by the following.
(a) As Λ n -modules, we have Proof.By definition, Thus, we may choose h ∓ as above to be 0.
Remark 2.8.The local points are also elemental to Iwasawa theory of the Z panticyclotomic deformation of a non-CM elliptic curve over imaginary quadratic fields with p inert (cf.[9]).

Rubin p-adic L-function and global points
The main results are Theorems 3.9 and 3.16.

Notation
Let Q be an algebraic closure of For a number field L, let G L = Gal(Q/L).For a finite dimensional Q p -vector space V endowed with a continuous G L -action and v a prime of L, the Bloch-Kato subgroup is given by If M denotes V or a Z p -lattice in V, then the Bloch-Kato Selmer group is defined as .
For an extension N/L of number fields, let Ind N L (•) denote the induction Ind GN GL (•).For an abelian variety A, let T p (A) denote the p-adic Tate module and put (inr) Let K ∞ be the anticyclotomic Z p -extension of K and K n the n-th layer.Let G − also denote Gal(K ∞ /K).
Let ϕ be a Hecke character over K of infinity type (1,0) such that the Hecke character ϕ • N H/K is associated to a Q-curve E over H which has good reduction at each prime of H above p.In particular, E satisfies the Shimura condition.
In this section, let O denote the integer ring of a finite extension of Φ which contains the Hecke field K(ϕ( K × )) for K := K ⊗ lim ← −m Z/mZ.Let f be the conductor of ϕ.Let T be the p-adic Galois representation of G K associated to ϕ, which is an O-module free of rank one so that its restriction to as formal groups over O Φ , the results in §2 may be utilized by replacing F with Ê and identifying We put T ⊗−1 = Hom O (T,O) and note that T ⊗−1 (1) is identified with the complex conjugation of T as follows.Let τ be the complex conjugation.We have a natural decomposition where ⊗ OΦ,τ is the tensor product with respect to the map O Φ → O induced by τ and the natural inclusion.This decomposition and the base change of the Weil pairing over O induce a perfect O-bilinear pairing Thus, we may naturally identify where ι : Λ → Λ denotes the involution induced by g → g −1 for g ∈ G − and for a Λ-module M, we put M ι = M ⊗ Λ,ι Λ.

Construction of Selmer elements.
Based on elliptic units, we associate a Selmer element to a Hecke character.

Proposition 3.1.
There exists an elliptic unit associated to b E such that for a character χ of Gal(K n /K), we have Here, is the localization as w varies over the places of K n above p.
We put Definition 3.2 (Selmer element).For a character χ of Gal(K n /K), let z χ ∈ H 1 (K n ,T ⊗−1 (1)) χ −1 denote the image of the elliptic unit z n under the composite Here, the second and third maps are corestriction and restriction, respectively.
Note that Proof.By definition and [30, Prop.15.9], z χ lies in the image of where K(fp m ) denotes the ray class field of K of conductor fp m .For a prime v p of K n and a prime w | v of ∪ m≥1 K(fp m ), note that the completion of ∪ m K(fp m ) at w contains the maximal pro-p unramified extension of K n,v , and so ).Since L(ϕχ −1 ,1) = 0, by the explicit reciprocity law (3.3) and (3.5), ) coincides with the kernel of exp * Kn , the proof concludes.

Global points
3.2.1.Mordell-Weil groups over Q.In this subsection, for sufficiently large n with (−1) n+1 = −W (ϕ) and χ an anticyclotomic character of conductor p n+1 , the Selmer element z χ is shown to arise from a rational point.
Let χ be a finite character of Gal(K ∞ /K) and ) the theta series attached to ϕχ −1 , where c χ denotes the conductor of χ.In particular, L(f χ ,s) = L(ϕχ −1 ,s).Let F χ denote the Hecke field.Fix an abelian variety In this subsection, O is enlarged to also contain the image of χ, and m denotes the maximal ideal.We begin with a preliminary.
As for the irreducibility, in light of the proof of [30,Lem. 15.20], it suffices to show that for a finite character χ of Gal(K ∞ /K), there exists an integral ideal b of K relatively prime to pf such that via ι p and notice an abstract isomorphism which follows from considering the action of Frobenius elements.By Lemma 3.4, there exists an isomorphism Ind -modules, and so we have an identification and the Tate-Shafarevich group X(A χ/Q ) is finite.In particular, if r = 1, we have Proof.Since ord s=1 L(f χ ,s) = r, by the main result of [8], there exists an imaginary quadratic field L such that (ii) the pair (f χ ,L) satisfies the Heegner hypothesis.
Then, the Gross-Zagier formula [28], [55] implies that the Heegner point y L ∈ A χ (L) is non-torsion, and so the assertion is due to Kolyvagin [35] (see also [40]).As for the "in particular" part, note that A χ (Q) is p-torsion-free by Lemma 3.4.

Anticyclotomic Mordell-Weil groups.
This independent subsection presents an anticyclotomic variation of the Mordell-Weil groups.
(a) We have Proof.Let B n denote the Weil restriction Res Kn/K (A /Kn ) of A over K n .By considering the Galois action on valued points of B n , note that the Galois group Gal(K n /K) embeds into End B n , which in turn implies as algebras (cf.[23,Thm. 3]).
In light of the decomposition and factorisation of the underlying L-functions, we have an isogeny of abelian varieties over K. Here, A n is the abelian variety defined as a product of copies of ).Note that the set of K -rational points is given by Now, we consider Gal(K n /K)-action which leads to and In light of the Gross-Zagier formula [28], [55] and Proposition 3.5, we have and so Hence, in conjunction with (3.10), it follows that where χ 1 ranges over the conjugates of χ.
(a) Recall the short exact sequence Now, as T p (X(A /Kn )) is p-torsion-free, (3.13) readily implies the second asserted equality of part (a).Since p is unramified in So, in view of (3.9) and (3.12), it follows that Hence, the evaluation at χ yields the first asserted equality of part (a).
Proof.This is a simple consequence of Theorem 3.9 (a) and Remark 3.8.
The corollary implies that new points of infinite order appear in the alternate anticyclotomic layers.As shown in Corollary 3.18 below, these points correspond to the Selmer elements y χ .Remark 3.12.
(ii) An analogue of Corollary 3.11 for Selmer groups appears in [2, Thm.A].

Rubin p-adic L-function and global points
The section presents a Rubin type special value formula for the Rubin p-adic L-function, which is a result towards the question (Q).
Assume that Then, the Galois group Gal(K ∞ /K) is naturally identified with Gal(Ψ ∞ /Φ).For n ≥ 0, let p denote the prime of K n above p.

A Rubin type formula. The subsection explores
As in (3.1), we identify by which an element v of H 1 (Ψ n ,T ⊗−1 (1)) will be regarded as an element of H 1 (Ψ n ,T ).
In view of the identifications the pairings ( , ) Λm in (2.11) induce a perfect pairing which is Λ-bilinear (as the pairing (2.11) is sesquilinear).In the following, we regard ( , ) n and ( , ) Λn as pairings on Lemma 3.15.We have Proof.As for the first assertion, by Nakayama's lemma, it suffices to show that In view of Theorem 2.1, and so by (3.16), ) is a maximal isotropic subgroup.Hence, (3.16) implies that (v +,0 ,v −,0 ) 0 ∈ O × .We now consider the second assertion.For any χ ∈ Ξ ∓ , note that δ χ (v ± ) = 0, i.e. the image of v ± under lim From now, we fix v + ,v − so that and then v ⊥ ±,n in (2.12) is identified with v ±,n via (3.15).The main result of this subsection is the following.Theorem 3.16.Let K be an imaginary quadratic field and p ≥ 5 a prime satisfying (inr) and (cp).Let E be a Q-curve with complex multiplication by O K with good reduction at p, ϕ the associated Hecke character of K and ε the sign of the root number.Let L be the Rubin p-adic L-function as in (3.14).
(a) Let χ ∈ Ξ −ε be a Hecke character with conductor p n+1 .Let be the image of a system of elliptic units of E (cf.§3.1.2).Then, where τ (χ,α) : Proof.(a) By Definition 3.13, note that where z n also denotes loc p (z n ).
(b) This just follows by letting n = 0 and χ = 1 in the above argument.
up to an element in O × .
(ii) One may seek a Coleman integration approach to Theorem 3.16.The preliminary study of a p-adic Eisenstein series in [5] maybe relevant.
(iii) A natural problem is to investigate a special value formula L (χ) for anticyclotomic characters χ of infinity type (j, − j) with j > 0. It will be investigated in a forthcoming paper.
(ii) We have for a non-zero Proof.The following is based on Iwasawa main conjectures [2], [12], to which we refer for notation.
up to an element in O × .

Rubin p-adic L-function and rational points
The main results are Theorems 4.8 and 4.18, and Proposition 4.14.

Notation
Let the setup be as in §3.3.In particular, f ∈ S 2 (Γ 0 (N )) denotes the theta series associated to the Hecke character ϕ.Let F ⊂ C denote the Hecke field of f.Let A /Q be a GL 2 -type abelian variety so that ).Let L denote the subfield of C generated by ϕ( K × ) over K, a finite extension of K containing F. As in §3, let O be the integer ring of the completion L p at the prime p compatible with the embedding ι p .Let λ f : A(Q p ) → F p be a formal group logarithm arising from the differential attached to the newform f as in §A.3.
Our normalisation differs from [30]; namely, our Replacing A by an isogeny, we may assume that Theorem 4.1.Let A /Q be a GL 2 -type CM abelian variety.Let K be the corresponding imaginary quadratic field and F the Hecke field.Suppose that the root number of the associated CM newform is −1.Let p ≥ 5 be a prime of good non-ordinary reduction for A /Q with p h K and L the Rubin p-adic L-function as in (3.14).Then, there exists a rational point P ∈ A(Q) with the following properties.

p-adic
(a) We have for some c ∈ L × .

Tools of the proof.
We outline the strategy.
Elliptic units and Beilinson-Kato elements.The following link between zeta elements is a key.
) be the elliptic unit as in (3.2) under the identification (3.1) and z f ∈ H 1 (Q,V (f )) a Beilinson-Kato element associated to the newform f.Then, under the identification (4.2), we have Proof.This is [30, (15.16.1)].
Beilinson-Kato elements and rational points.The following connects Beilinson-Kato elements with Heegner points.
for some c P ∈ F × and a p (f ) the p-th Fourier coefficient of f.
(b) P is non-torsion if and only if ord s=1 L(f,s) = 1.
(c) If the equivalent conditions in (b)hold, then This is an evidence towards a conjecture of Perrin-Riou [42].We refer to Appendix B for details (cf.Theorem B.3).
p-adic Gross-Zagier formula.Theorem 4.3 is based on the following interrelation between p-adic logarithm of a Heegner point and central derivative L p,γ (f,1) of the cyclotomic p-adic L-function L p,γ (f,s) for γ ∈ {α,β} a root of the Hecke polynomial at p. Theorem 4.4.Suppose that the root number of L(f,s) is −1.Then, there exists a point P ∈ A(Q) and a non-zero constant c P ∈ Q such that Moreover, P is non-torsion if and only if ord s=1 L(f,s) = 1, and This is a variant of the p-adic Gross-Zagier formula [32] (cf.Appendix A).In combination with Proposition B.4, it yields Theorem 4.3.

Proof of Theorem 4.1.
The approach is based on Theorem 3.16 (b) and a link between elliptic units and Heegner points (cf.Theorems 4.2 and 4.3).

Proof. Fix an isomorphism Ind
be the induced identification.Let be a Beilinson-Kato element as in [30, Thm.12.5], which depends on a choice of an element in Since the root number of f is −1, L(f,1) = 0, and so loc p (z f ) ∈ H 1 f (Q p ,V (f )) by Kato's reciprocity law [30,Thm. 12.5 (1)].Now in view of Theorems 3.16 and 4.2, it follows that up to an element in L × .Hence, Theorem 4.3 concludes the proof.
Remark 4.5.Theorem 4.1 concerns an anticyclotomic p-adic L-value, yet its proof relies on central derivative of cyclotomic p-adic L-functions.

p-adic Beilinson formula: a refined form
The main result is Theorem 4.8.
To consider a refinement of Theorem 4.1, we first specify an abelian variety A in the associated isogeny class (cf.§4.2.1), leading to an explicit form of Theorem 4.2 (cf.Proposition 4.12).

A CM abelian variety.
We begin with a preliminary (cf.[27, §5.1]).Lemma 4.6.Let E be a CM elliptic curve as in §3.1.1 and j ∈ H denote its j-invariant.Then the following holds.
(2) Q(j) has at least one real place and Suppose that E is defined over Q(j).
We now describe some structures on A arising from E.

The canonical identification
is compatible with the Hodge filtration, via which the Néron differential ω of E gives an element ω A of coLie(A /Q ).Since H = Q(j)K, we have So, the one-dimensional L-vector space coLie(A /Q ) ⊗ F L leads to a one-dimensional subspace S(ϕ) of coLie(E /H ) ⊗ K L, namely its ϕ-part.This induces an identification which is the same as the identification [30, (15.11.2)] (recall that our V Lp (f ) is isomorphic to the L p -linear dual of that of [30]).In turn, λ E :

Main result and applications.
Theorem 4.8.Let A /Q be a GL 2 -type abelian variety associated to a CM newform f as in §4.2.1.Let K be the CM field and F the Hecke field.Suppose that (rt) holds and the root number of f equals −1.Let p ≥ 5 be a prime of good non-ordinary reduction for A /Q with p h K and L the Rubin p-adic L-function as in (3.14).Then there exists a rational point P ∈ A(Q) with the following properties.
(a) We have Part (a) leads to the following p-adic construction of a rational point of infinite order.
Corollary 4.9.Let E /Q be a CM elliptic curve with root number −1.Let p ≥ 5 be a prime of good supersingular reduction for E /Q and L E the Rubin p-adic L-function as in (3.14).Suppose that ord s=1 L(E,s) = 1, and the Birch-Swinnerton-Dyer formula is true for is a rational point of infinite order, where c denotes the Tamagawa number at .
(i) The BSD formula is known to be true up to an element in Z[ [32,Cor. 1.4]).
(ii) The rational point is independent of the choices involved, besides that of the square root.
(iii) Rubin initiated p-adic construction of rational points of infinite order (cf.[50,Thm. 10.4].) Another application is the following variant of Corollary 3.11.
Corollary 4.11.Let E /Q be a CM elliptic curve and K the CM field.Let p ≥ 5 be a prime of good supersingular reduction for E /Q and K n the n th -layer of the anticyclotomic Z p -extension of K.
(ii) Suppose that ord s=1 L(E /Q ,s) = 1 and there exists a rational point P ∈ E(Q) whose image generates the Z p -module where ε n = 0 (resp.2) for n even (resp.odd).
In particular, if X(E /Kn ) is finite, then Proof.We first consider the case (i).
Hence, the image of z in S rel is a Λ q -basis for S rel,q up to tensoring with Q p , where S rel denotes the relaxed compact Selmer group and q the prime ideal of Λ corresponding to the χ −1 -specialization.In turn, X str,q is finite by [2,Prop. 3.3] and then so is . Hence, Tate's Euler characteristic formula implies The assertion follows from this.
The case (ii) is similarly proven by using Theorem 4.8.
) be the elliptic unit as in (3.2) under the identification (3.1) and z f,0 ∈ H 1 (Q,V (f )) the Beilinson-Kato element associated to the newform f as in (4.8).Then, under the identification (4.7), we have Proof.By [30,Lem. 15.11 (2)], there is a unique isomorphism of L[Gal(C/R)]-modules such that the following diagram commutes.Here, H 1 (A(C),Q) is regarded as a Gal(C/R)-module via the complex conjugation on A(C), per ϕ is the period map induced by that of E (cf.[30, §15.8]) and the right vertical map is the base change of (4.5 where f denotes the conductor of ϕ and K(fp n ) the ray class field of K of conductor fp n (cf.[30, p. 254]).Let Q n be the n-th layer of the cyclotomic Here, the first map is induced by the corestriction maps with respect to K ⊗ Q n ⊆ K(fp n ), and the equality is a consequence of (4.4) and Shapiro's lemma.Note that b E = Ω −1 per ϕ (ω) maps to Ω −1 per f (ω A ) under (4.5) and z ell is associated to b E .Thus, in light of (4.6) and [30, (15.16.1)], it follows that z ell coincides with the system of Beilinson-Kato elements associated to Since Q ∞ is a totally real field, the last assertion in [30, Thm.12.5 (1)] implies that z ell also coincides with the system of Beilinson-Kato elements associated to and that z f coincides with the one associated to )), where ι denotes the involution induced by the complex conjugation on A(C).Therefore, we have , the proposition follows.

Proof of Theorem 4.8.
We proceed as in the proof of Theorem 4.1 (cf.§4.1.3).The additional ingredient is Proposition 4.12.
Proof.By (4.4) and Shapiro's lemma, we have an identification Let T (f ) be a Galois stable O Fp -lattice of V (f ).Let be the Beilinson-Kato element associated to b A as in [30, Thm.12.5 ( 1)] (since our V Lp (f ) is a Tate twist of that in [30], z f is the corresponding twist of z (p) bA as in [30]).In particular, z f satisfies the explicit reciprocity law (B.4).

Proof. We have
(4.12) To begin, Sel str (K,V /T ) is finite by (4.12).Put Since z 0 ∈ H 1 (K,T ) is non-torsion, pick an integer m such that p −m+a z 0 ∈ H 1 (K,V ) does not lie in the image of H 1 (K,T ).Suppose that loc p (z 0 ) = 0 ∈ H 1 (K p ,V ).Then, as z 0 is unramified outside p, the image w m of p −m z 0 in H 1 (K,V /T ) lies in Sel str (K,V /T ) and so does p a w m .However, This contradiction yields (4.13).
Finally, we have the following p-converse.

Corollary
In particular, if the Hecke polynomial at p is irreducible over the p-adic completion of K f , the height pairing , c Q ,Nα is non-trivial.Proof.This follows from Theorem A.2 since λ f is non-trivial.The pairings , c Q ,Nα and , c Q ,N β are conjugate if α and β are.Corollary A.5.Let p be a non-ordinary (good) prime for f.Suppose that ord s=1 L(f,s) = 1 and the Iwasawa main conjecture for f is true for p.Then, the p-part of the full Birch and Swinnerton-Dyer conjecture (Bloch-Kato's tamagawa number conjecture) is true for f.
We now return to Theorem B.3. (i) A recent progress towards Conjecture B.1 appears in [7], [17], [19].The key tools are (variants of) the Beilinson-Flach element and the BDP formula.In the nonordinary case, these results assume additional hypotheses such as p odd, while our independent approach treats the general non-ordinary case.
(ii) Theorem B.3 is a tool in the proof of yet another CM p-converse (cf.[15]), and in turn, a result towards the cube sum problem (cf.[1]).

3. 1 .
Elements of Selmer groups 3.1.1.The set-up.Let K be an imaginary quadratic field of discriminant −D K < 0 and H the Hilbert class field.Suppose p is inert in K.
Fix a minimal Weierstrass model of E over O H,p ∩ H = O Φ ∩ H for p | p the prime of H arising via ι p and let ω be the Néron differential.Pick a non-zero b and so ϕ(b)γ b − ϕ( b)γb ∈ Λ × , from which (3.6) follows.(Here, γ a ∈ G − denotes the element which corresponds via the Artin map to an integral ideal a of K relatively prime to pf.) i.e. part (b) holds.Remark 3.10.The above argument is a variation of Rubin's argument for[3, Prop.A.8].

Theorem 4 . 3 .
If L(f,1) = 0, then there exists a rational point P ∈ A(Q) with the following properties.

. 9 ) 3 . 4 . 13 .
Hence, Theorem 4.8 is a consequence of Theorems 3.16 (b) and 4.Remark For a given b E or ω, note that λ f (v −,0 )Ω • L (1) is independent of the choices of Ω and v − .Moreover, the right-hand side of Theorem 4.8 (a) is independent of the choice of b A .

Corollary A. 4 .
The p-adic Gross-Zagier formula of f holds for inert primes if f is nonordinary at p. (cf.[33, Theorem 3])Proof.We first show that [ϕω f ,ω f ] = 0. We have a strongly divisible lattice D in M f by the Fontaine-Laffaille theory.Suppose that [ϕω f ,ω f ] = 0.Then, Fil 1 D is stable by ϕ.Hence, ϕ(Fil 1 D) ⊂ Fil 1 D ∩ pD = pFil 1 D. This implies that one of the eigenvalues α, β is divisible by p, which contradicts the non-ordinary assumption.By Corollary A.3, choose α for which , c Q ,Nα is non-trivial.Then, see a remark after[33, Theorem 3].

2.2. Local cohomology 2.2.1. Kummer theory.
We recast the modules of anticyclotomic local units in terms of the local Iwasawa cohomology.Define a natural isomorphism of O Theorem 4.18.Let A /Q be a GL 2 -type CM abelian variety.Let K be the CM field and F the Hecke field.Suppose that (rt) holds and O the cyclotomic height pairing (here, Tr F/Q λ d,Nα,N β (a) := σ:F →Cp λ d σ ,Nα,N β (a σ )).We put A(F ) f for the f -part of A(F ) ⊗ L be the Hecke action.Let λ f : A(F ) f → C p be the logarithm associated to ω f .Assume that [ϕω f ,ω f ] = 0.Then, we have F → EndA.Let p ≥ 5 be a prime of good non-ordinary reduction for A /Q with p h K , and p a prime of F above p.Suppose either of the following.for