Vari\'et\'es ab\'eliennes CM et grosse monodromie finie sauvage

TL;DR

This paper investigates the wild finite monodromy groups of CM abelian varieties over number fields, providing new bounds and constructions for maximal wild monodromy, especially addressing the odd and even prime cases.

Contribution

It introduces solutions to Grunwald problems for specific groups and establishes new bounds on the 2-part of monodromy groups for CM abelian varieties.

Findings

Constructed CM abelian varieties with maximal wild monodromy for odd primes.

Proved a new bound on the 2-part of the monodromy group for even primes.

Solved Grunwald problems for groups of the form Z/pZ wr S_n over number fields.

Abstract

In this paper we study the wild part of the finite monodromy groups of abelian varieties over number fields. We solve Grunwald problems for groups of the form $\mathbf{Z}/p\mathbf{Z}\wr \mathfrak{S}_n$ over number fields to build CM abelian varieties with maximal wild finite monodromy in the odd prime case. For the even prime case we prove a new bound on the 2-part of the order of the finite monodromy group for CM abelian varieties and build varieties that reach it.