A Splitting Criterion for Galois Representations Associated to Exceptional Modular Forms (mod p)

TL;DR

This paper develops a criterion to determine when certain Galois representations attached to exceptional modular forms split at p, extending understanding from nonexceptional cases and providing a framework for future generalizations.

Contribution

It introduces a splitting criterion for Galois representations associated with exceptional modular forms, addressing a previously less understood case.

Findings

Established a criterion for splitting of Galois representations at p

Extended the understanding of Galois representations for exceptional modular forms

Provided a framework potentially applicable to more general cases

Abstract

This paper continues the study of certain two-dimensional Galois representations attached to modular forms (mod p) via a construction due to Deligne. In particular, we prove a criterion for determining when the representation is split when restricted to a decomposition group at p.While this problem was already well understood for "nonexceptional" modular forms, here we specifically address the "exceptional" case. Although the problem is not solved in all generality, it is reasonable to hope that this proof will in fact provide the framework of a proof in the more general case.