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

TL;DR

This paper constructs Galois representations associated to $p$-adic families of modular forms of finite slope, using étale cohomology on modular curves and extending to $oldsymbol{ ext{Lambda}}_1$-adic eigenforms.

Contribution

It introduces a new method to realize Galois representations for finite slope $p$-adic families via quotients of étale cohomology on modular curves.

Findings

Galois representations are obtained as quotients of étale cohomology.

Construction works over certain $oldsymbol{ ext{Lambda}}_1$-algebras.

Provides a framework for $p$-adic families of modular forms of finite slope.

Abstract

We define a pro-$p$ Abelian sheaf on a modular curve of a fixed level $N \geq 5$ divisible by a prime number $p \neq 2$. Every $p$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ associated to an eigenform is obtained as a quotient of its \'etale 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 \'etale cohomology.