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Casazza, Daniele; Rotger, Victor

On the elliptic Stark conjecture at primes of multiplicative reduction. (English) Zbl 1471.11192

Indiana Univ. Math. J. 68, No. 4, 1233-1253 (2019).
Summary: In [Forum Math. Pi 3, Paper No. e8, 95 p. (2015; Zbl 1392.11034)], H. Darmon et al. formulated a \(p\)-adic elliptic Stark conjecture for the twist of an elliptic curve \(E/\mathbb{Q}\) by the self-dual tensor product \(\rho_1\otimes\rho_2\) of two odd and two-dimensional Artin representations. These authors provided abundant numerical evidence and proved the conjecture in the special setting where \(p\) is a prime of good reduction for \(E\) and \(\rho_1\) and \(\rho_2\) are induced from finite-order characters \(\psi_g, \psi_h\) of the same imaginary quadratic field. The key step in their proof is a factorization of one-variable \(p\)-adic \(L\)-functions, where \(\psi_g\) varies in a \(p\)-adic family of Hecke characters.
The main goal of this article is to prove a new case of the conjecture, placing ourselves in the setting where \(p\) is a prime of multiplicative reduction for \(E\). In order to achieve our theorem, we need to work with {\em two-variable} \(p\)-adic \(L\)-functions, where the weight \(2\) cusp form associated with \(E\) also moves independently along a Hida family. Our main result then follows from a factorization of \(p\)-adic \(L\)-series extending to two variables the one obtained in [loc. cit.]. On the way we also generalize to our setting the results obtained in [the authors, “Stark points and \(p\)-adic iterated integrals”, Ramanujan J. (to appear)].

Cited in 2 Documents

MSC:

11G16 Elliptic and modular units
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F11 Holomorphic modular forms of integral weight

Keywords:

elliptic Stark conjecture

Citations:

Zbl 1392.11034
PDFBibTeX XMLCite
\textit{D. Casazza} and \textit{V. Rotger}, Indiana Univ. Math. J. 68, No. 4, 1233--1253 (2019; Zbl 1471.11192)
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References:

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