Repr\'esentations galoisiennes di\'edrales et formes \`a multiplication complexe

TL;DR

This paper proves that under certain conditions, dihedral Galois representations in characteristic l correspond to CM newforms with explicitly controlled weight, level, and character, establishing a congruence between their associated l-adic representations.

Contribution

It demonstrates the existence of CM newforms associated to dihedral Galois representations with explicit control over key invariants, under specific hypotheses.

Findings

Existence of CM newforms congruent to given dihedral Galois representations.

Control over weight, level, and Nebentypus of the newforms.

Conditions under which the congruence holds.

Abstract

Pour une repr\'esentation galoisienne di\'edrale en caract\'eristique l on \'etablit (sous certaines hypoth\`eses) l'existence d'une newform \`a multiplication complexe, dont on contr\^ole le poids, le niveau et le caract\`ere, telle que la repr\'esentation l-adique associ\'ee est congrue modulo l \`a celle de d\'epart. Given a dihedral Galois representation in characteristic l, we establish (under some assumption) the existence of a CM newform, whose weight, level and Nebentypus we pin down, such that its l-adic representation is congruent modulo l to the one we started with.