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Heegner points on Mumford-Tate curves. (English) Zbl 0882.11034
Let \(E/\mathbb{Q}\) be a modular elliptic curve of conductor \(N\). Fix a prime number \(p\). Let \(K\) be a quadratic imaginary field, and \(K_\infty\) be the compositum of all the ring class fields \(K_n\) of conductor \(p^n\) which contain the anticyclotomic \(\mathbb{Z}_p\)-extension of \(K\). The authors define a certain Heegner distribution on \(G_\infty=\text{Gal}(K_\infty/K)\) with values in \(\text{Pic}(X)\) (Prop. 2.7). (Here \(X\) is a Shimura curve corresponding to a quaternion algebra, whose definition depends on the data \((N,K,p)\).) To such a Heegner distribution they associate a certain anticyclotomic \(p\)-adic \(L\)-function (§2.7). They show that the so-called exceptional zero phenomena also occur in this context: they express (conjecturally) the \(p\)-adic periods of \(E\) in terms of the derivatives of the anticyclotomic \(p\)-adic \(L\)-function (section 4). The split exceptional case resembles the exceptional zero case studied by B. Mazur, J. Tate and J. Teitelbaum [Invent. Math. 84, 1-48 (1986; Zbl 0699.14028)]. Section 5 presents some numerical evidence for their conjectures.

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11R23 Iwasawa theory
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G20 Local ground fields in algebraic geometry
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