Tamagawa Torsors of an Abelian Variety

TL;DR

This paper investigates the arithmetic properties of Tamagawa torsors associated with abelian varieties over number fields, focusing on their behavior at primes and the structure of these torsors over local fields.

Contribution

It introduces the concept of Tamagawa torsors for abelian varieties and analyzes their arithmetic properties at various primes, providing new insights into their structure.

Findings

Characterization of Tamagawa torsors over local fields

Analysis of splitting behavior at unramified extensions

Insights into the arithmetic structure of torsors

Abstract

For an abelian variety $A$ over a number field $K$, we define the set of Tamagawa torsors of $A$ at a prime $v$ of $K$ to be the set of principal homogeneous spaces of $A$ over the completion $K_v$ of $K$ at $v$ that are split by an unramified extension of $K_v$. In this paper, we study the arithmetic properties of the Tamagawa torsors.