Un crit\`ere d'\'epointage des sections $l$-adiques

TL;DR

This paper investigates a criterion for identifying the cuspidalization of sections in $l$-adic homology, contributing to the understanding of Grothendieck's section conjecture through Galois representations and modular curves.

Contribution

It provides a criterion for cuspidalization in $l$-adic homology and offers concrete examples involving modular curves, advancing the study of the cuspidalization conjecture.

Findings

Established a criterion for cuspidalization of sections in $l$-adic homology.

Identified conditions under which $l$-adic homology is a pure Galois representation.

Presented examples of modular curves where cuspidalization is achieved at the $l$-adic level.

Abstract

The cuspidalization conjecture emerged as an approach of Grothendieck's famous section conjecture. We address a weak form of it by using a mild generalization of a theorem of Uwe Jannsen which describes exactly when the $l$-adic homology of an open curve is a pure Galois representation. We also give some concrete examples of modular curves for which the cuspidalization is possible at the $l$-adic level.