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Stark-Heegner points on modular Jacobians. (English) Zbl 1173.11334
Summary: We present a construction which lifts Darmon’s Stark-Heegner points from elliptic curves to certain modular Jacobians. Let \(N\) be a positive integer and let \(p\) be a prime not dividing \(N\). Our essential idea is to replace the modular symbol attached to an elliptic curve \(E\) of conductor \(Np\) with the universal modular symbol for \(\Gamma_0(Np)\). We then construct a certain torus \(T\) over \(Q_p\) and lattice \(L\subset T\), and prove that the quotient \(T/L\) is isogenous to the maximal toric quotient \(J_0(N_p)^{p\text{-new}}\) of the Jacobian of \(X_0(Np)\). This theorem generalizes a conjecture of B. Mazur, J. Tate and J. Teitelbaum [Invent. Math. 84, 1–48 (1996; Zbl 0699.14028)] on the \(p\)-adic periods of elliptic curves, which was proven by R. Greenberg and G. Stevens [Invent. Math. 111, 407–447 (1993; Zbl 0778.11034)]. As a by-product of our theorem, we obtain an efficient method of calculating the \(p\)-adic periods of \(J_0(Np)^{p\text{-new}}\).

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
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