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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)]).

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
Full Text: DOI Link
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