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Abstract Let $$E$$ be an elliptic curve defined over $${\mathbb{Q}}$$ of conductor $$N$$, $$p$$ an odd prime of good ordinary reduction such that $E[p]$ is an irreducible Galois module, and $$K$$ an imaginary quadratic field with all primes dividing $Np$ split. We prove Iwasawa main conjectures for the $${\mathbb{Z}}_{p}$$-cyclotomic and $${\mathbb{Z}}_{p}$$-anticyclotomic deformations of $$E$$ over $${\mathbb{Q}}$$ and $K,$ respectively, dispensing with any of the ramification hypotheses on $E[p]$ in previous works. The strategy employs base change and the two-variable zeta element associated to $$E$$ over $$K$$, via which the sought after main conjectures are deduced from Wan’s divisibility towards a three-variable main conjecture for $$E$$ over a quartic CM field containing $$K$$ and certain Euler system divisibilities. As an application, we prove cases of the two-variable main conjecture for $$E$$ over $$K$$. The aforementioned one-variable main conjectures imply the $$p$$-part of the conjectural Birch and Swinnerton-Dyer formula for $$E$$ if $$\operatorname{ord}_{s=1}L(E,s)\leq 1$$. They are also an ingredient in the proof of Kolyvagin’s conjecture and its cyclotomic variant in our joint work with Grossi [1]. 

Abstract LetKbe an imaginary quadratic field and$$p\geq 5$$a rational prime inert inK. For a$$\mathbb {Q}$$-curveEwith complex multiplication by$$\mathcal {O}_K$$and good reduction atp, K. Rubin introduced ap-adicL-function$$\mathscr {L}_{E}$$which interpolates special values ofL-functions ofEtwisted by anticyclotomic characters ofK. In this paper, we prove a formula which links certain values of$$\mathscr {L}_{E}$$outside its defining range of interpolation with rational points onE. Arithmetic consequences includep-converse to the Gross–Zagier and Kolyvagin theorem forE. 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$$-p$$. 

We show that for certain non-CM elliptic curves E / Q E_{/\mathbb {Q}} such that 3 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E ψ E_{\psi } of E E have Mordell–Weil rank one and the 3 3 -adic height pairing on E ψ ( Q ) E_{\psi }(\mathbb {Q}) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p p -adic height. It is not known – though expected – that the archimedian height of these higher-codimensional cycles is non-zero. 

Résumé Soit $$E/{\mathbb {Q}}$$ E / Q une courbe elliptique à multiplication complexe et p un nombre premier de bonne réduction ordinaire pour E . Nous montrons que si $${\mathrm{corank}}_{{\mathbb {Z}}_p}{\mathrm{Sel}}_{p^\infty }(E/{\mathbb {Q}})=1$$ corank Z p Sel p ∞ ( E / Q ) = 1 , alors E a un point d’ordre infini. Le point de non-torsion provient d’un point de Heegner, et donc $${{{\mathrm{ord}}}}_{s=1}L(E,s)=1$$ ord s = 1 L ( E , s ) = 1 , ce qui donne une réciproque à un théorème de Gross–Zagier, Kolyvagin, et Rubin dans l’esprit de [49, 54]. Pour $$p>3$$ 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].