Congruences for critical values of higher derivatives of twisted Hasse–Weil $L$-functions, II
Volume 195 / 2020
Acta Arithmetica 195 (2020), 327-365
MSC: Primary 11G40; Secondary 11G35, 11R34.
DOI: 10.4064/aa181101-4-12
Published online: 27 May 2020
Abstract
Let $A$ be an abelian variety defined over a number field $k$ and let $F$ be a finite Galois extension of $k$ with Galois group $G$. We discuss the formulation of ‘higher’ analogues of the ‘refined conjectures of Birch and Swinnerton-Dyer type’ of Mazur and Tate. These include, in particular, integral congruences for ‘higher’ analogues of modular elements, interpolating values of higher derivatives of Hasse–Weil–Artin $L$-functions of $A$ at $s=1$, that involve natural $G^{\rm ab}$-valued height pairings.
