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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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