Mod-$2$ Hecke algebras of level $3$ and $5$

TL;DR

This paper investigates the structure of mod-2 Hecke algebras at levels 3 and 5 using deformation theory, establishing isomorphisms with universal deformation rings and describing their detailed structure.

Contribution

It proves isomorphisms between big Hecke algebras and universal deformation rings for levels 3 and 5, and characterizes their structure and grading.

Findings

Maximal reduced quotients of big Hecke algebras are isomorphic to those of universal deformation rings.

Complete determination of the structure of the big Hecke algebra at levels 3 and 5.

Introduction of a natural grading on mod-$p$ Hecke algebras.

Abstract

We use deformation theory to study the big Hecke algebra acting on mod-2 modular forms of prime level $N$ and all weights, especially its local component at the trivial representation. For $N = 3, 5$, we prove that the maximal reduced quotient of this big Hecke algebra is isomorphic to the maximal reduced quotient of the corresponding universal deformation ring. Then we completely determine the structure of this big Hecke algebra. We also describe a natural grading on mod-$p$ Hecke algebras.