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

Part of: Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  29 June 2021

Werner Bley  and
Alessandro Cobbe
Werner Bley
Affiliation:
Mathematisches Institut der Ludwig-Maximilians-Universität München, München, Germany e-mail: [email protected]
Alessandro Cobbe*
Affiliation:
Institut für Theoretische Informatik, Mathematik und Operations Research, Universität der Bundeswehr München, Neubiberg, Germany
*
e-mail: [email protected]
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Abstract

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.

Keywords

epsilon constant conjectureequivariant Tamagawa number conjectureweakly ramified extensions

MSC classification

Primary: 11S23: Integral representations 11S25: Galois cohomology
Secondary: 11S40: Zeta functions and $L$-functions
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 5 , October 2022 , pp. 1405 - 1449
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Copyright
© Canadian Mathematical Society 2021

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