Galois Representations Associated to $p$-adic Families of Modular Forms of Finite Slope

Abstract

We define a pro-$p$ Abelian sheaf on a modular curve of a fixed level $N\ge 5$ divisible by a prime number $p\ne 2$. Every $p$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ associated to an eigenform is obtained as a quotient of its étale cohomology. For any compact ${\mathbb{Z}}_p[[1+N{\mathbb{Z}}_p]]$-algebra ${\Lambda }_1$ satisfying certain suitable conditions, we construct a representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over ${\Lambda }_1$ associated to a ${\Lambda }_1$-adic cuspidal eigenform of finite slope as a scalar extension of a quotient of the étale cohomology.

96 pages

References in corpus (1)

• Revêtements étales et groupe fondamental (SGA 1)