The epsilon constant conjecture for higher dimensional unramified twists of ${\mathbb Z}_p^r$(1)

Bley, Werner ; Cobbe, Alessandro

Canadian journal of mathematics, Tome 74 (2022) no. 5, pp. 1405-1449

Résumé

Let $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$, which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$. In this paper, we consider the $\mathrm {Gal}(N/K)$-equivariant local $\varepsilon $-conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$. For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$. This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

Détail

DOI : 10.4153/S0008414X2100033X

Mots-clés : epsilon constant conjecture, equivariant Tamagawa number conjecture, weakly ramified extensions