Hilbert modular forms and Galois representations

TL;DR

The paper surveys Hilbert modular forms and Galois representations, clarifying what it means to attach a compatible system $\{\rho_{\lambda}\}$ to a Hilbert cuspidal eigenform $f$ within both classical and adelic frameworks and highlighting the role of Hecke algebras and $L$-functions. It reviews the historical development and foundational results that connect automorphic forms on $\mathrm{GL}_2$ over totally real fields to $2$-dimensional Galois representations, including the use of congruence methods to construct such representations. The main focus is the existence of a continuous representation $\rho_{\lambda}: \mathrm{Gal}(\overline{F}/F) \to \mathrm{GL}_2(\mathcal{O}_{K,\lambda})$, unramified outside $\mathfrak{n}l$, with $\mathrm{tr} \rho_{\lambda}(\mathrm{Frob}_{\mathfrak{q}}) = \theta_f(T_{\mathfrak{q}})$ and $\mathrm{det} \rho_{\lambda}(\mathrm{Frob}_{\mathfrak{q}}) = \theta_f(S_{\mathfrak{q}})N(\mathfrak{q})$, forming a strictly compatible system. It synthesizes results of Deligne, Carayol, Taylor, Rogawski–Tunnell, Ohta, and Jarvis, and discusses the adelic Hilbert modular form framework that underpins attaching Galois representations to Hilbert modular forms. The work has significance for modularity over totally real fields and provides tools for arithmetic applications, including connections to generalized Fermat equations and Darmon's program toward associating Hilbert modular forms with GL(2)-type abelian varieties.

Abstract

In this expository article, we present a brief introduction to the theory of Hilbert modular forms and Galois representations, and describe what it means to attach a compatible system of Galois representations to a Hilbert modular form.