Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

Streszczenie

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.