Construction of Abelian varieties with given monodromy

TL;DR

This paper constructs families of Abelian varieties over algebraic curves in positive characteristic with prescribed monodromy groups and representations, advancing the understanding of their algebraic and geometric properties.

Contribution

It introduces a deformation method to realize Abelian varieties with specific monodromy groups and representations over fields of positive characteristic.

Findings

Existence of Abelian varieties with prescribed monodromy groups

Construction of families over algebraic curves in positive characteristic

Monodromy representations contain given faithful representations

Abstract

Let $\rho$ be a finite-dimensional faithful representation of a semisimple algebraic group $G$. By means of a deformation argument, we show that there exists a family of Abelian varieties over a smooth and projective curve over the algebraic closure of a prime field of positive characteristic, such that its $\ell$-adic monodromy group covers $G$ and its $\ell$-adic monodromy representation contains $\rho$.