Abstract

In this posthumously published work, Bellaïche continues the study of modular forms modulo $2$ of level $1$ begun by Nicolas, Serre, and himself in 2012. After a careful analysis of the Galois group in question — the Galois group of the maximal pro-$2$ extension of $\mathbb{Q}$ unramified outside $2$ — Bellaïche, in collaboration with Serre, gives explicit matrices realizing the representation of this Galois group on the big Hecke algebra whose trace is the universal in the sense of Chenevier and Mazur. Bellaïche then analyzes the ideals of the Hecke algebra corresponding to certain special forms mod $2$ — those whose prime Fourier coefficients depend on Frobenius conjugacy classes in finite abelian or dihedral extensions of $\mathbb{Q}$ — and gives a basis of these forms.

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