The Semisimplicity Conjecture for A-Motives

TL;DR

This paper proves the semisimplicity conjecture for A-motives over finitely generated fields, establishing the semisimplicity of their rational Tate modules and analyzing the structure of associated algebraic monodromy groups.

Contribution

It confirms the semisimplicity conjecture for A-motives and explores implications for their algebraic monodromy groups, extending known results in the field.

Findings

Rational Tate modules of semisimple A-motives are semisimple as Galois representations.

The algebraic monodromy group G_p(M) equals the Zariski closure of the Galois image.

Connected component of G_p(M) is reductive when M is semisimple with separable endomorphism algebra.

Abstract

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Akio Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G_p(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G_p(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V_p(M). The second states that the connected component of G_p(M) is reductive if M is semisimple and has a separable endomorphism algebra.