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The rationality of Stark-Heegner points over genus fields of real quadratic fields. (English) Zbl 1203.11045
Let $$E$$ be an elliptic curve over $$\mathbb Q$$ of conductor $$N=pM$$, where $$p$$ is an odd prime not dividing $$M$$, and let $$f$$ be the normalised cusp form of weight 2 on $$\Gamma_0(N)$$ attached to $$E$$. Let $$K$$ be a real quadratic field such that $$p$$ is inert in $$K$$ and all prime factors of $$M$$ are split. As an analogue to the classical theory of Heegner points, the second author [Ann. Math. (2) 154, No. 3, 589–639 (2001; Zbl 1035.11027)] constructed so-called Stark–Heegner points $$P_\tau$$, points on $$E$$ over the completion $$K_p$$, determined by $$f$$ and by an element $$\tau$$ of $$K$$ belonging to the $$p$$-adic upper half plane $${\mathbb C}_p - {\mathbb Q}_p$$. He predicted that some integral multiple of $$P_\tau$$ is defined over a ring class field of $$K$$ depending on $$\tau$$. The main result of the present article gives some evidence for this prediction.
To state the result, let us explain the situation. A genus character $$\chi$$ of $$K$$ determines an extension $$H_\chi={\mathbb Q}(\sqrt{D_1},\,\sqrt{D_2})$$ with $$D_1D_2=D$$, the discriminant of $$K$$. Let $$G_D$$ denote the group of $$\text{SL}_2({\mathbb Z})$$-equivalence classes of primitive integral binary quadratic forms of discriminant $$D$$, so that, by class field theory, there is an isomorphism $$\text{{rec}}:\,G_D \longrightarrow \text{{Gal}}(H_D/K)$$. Here $$H_D$$ denotes the narrow ring class field attached to $${\mathcal O}_D$$. Define $$P_\chi = \sum_{g\in G_D} \chi(g)P_{\tau^g} \in E(K_p)$$. Conjectures by the second author [op. cit.] predict that $$P_{\tau^g}= \text{{rec}}(g)^{-1}(P_\tau)$$ for all $$g\in G_D$$, and thus an integral multiple of $$P_\chi$$ belongs to $$E(H_\chi)^\chi$$, and moreover that the point $$P_\chi$$ is of infinite order if and only if the twisted Hasse–Weil $$L$$-series $$L(E/K,\chi,s)$$ has nonzero derivative at $$s=1$$.
Define log$$_E$$ on $$E(K_p)$$ by $$\text{{log}}_E(P) = \text{{log}}_q(\Phi^{-1}(P))$$, where $$q\in p{\mathbb Z}_p$$ is the Tate period attached to $$E$$ and $$\Phi$$ denotes the Tate uniformisation. Let $$\chi_k$$ be the Dirichlet character associated to $${\mathbb Q}(\sqrt{D_k})$$ ($$k=1,2$$). Denote by $$w_M$$ the sign of the Fricke involution at $$M$$ acting on $$f$$. Suppose that $$E$$ has at least two primes of multiplicative reduction and that $$\chi_1(-M)=-w_M$$. The main result asserts that (1) there is a global point $${\mathbb P}_\chi\in E(H_\chi)^\chi$$ and a nonzero rational number $$t$$ such that $$\text{{log}}_E(P_\chi)= t\text{{log}}_E(\mathbb P_\chi)$$, and (2) the point $${\mathbb P}_\chi$$ is of infinite order if and only if $$L'(E/K,\chi,1)\neq 0$$. In particular, this implies that a suitable integral multiple of $$P_\chi$$ belongs to the natural image of $$E(H_\chi)^\chi$$ in $$E(K_p)$$.
One key step in the proof is the result that $$2\text{{log}}_E^2(P_\chi)$$ equals the second derivative of the Hida $$p$$-adic $$L$$-function $$L_p(f_\infty/K,\chi,k)$$ at $$k=2$$. (For the Hida family $$f_\infty$$, see the authors’ article [Invent. Math. 168, No. 2, 371–431 (2007; Zbl 1129.11025)]).

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 11G05 Elliptic curves over global fields 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11R11 Quadratic extensions
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