Compatibility between base change and Hecke orbits of Hilbert newforms

TL;DR

This paper investigates how Galois actions influence Hecke orbits of Hilbert newforms and their geometric counterparts, providing new insights into Langlands functoriality and illustrating these concepts with a specific example involving Shimura curves and Galois extensions.

Contribution

It establishes the compatibility between Galois actions on Hecke orbits of Hilbert newforms and their geometric analogs, linking to Langlands functoriality and providing explicit examples.

Findings

Galois action on Hecke orbits relates to abelian varieties of $ ext{GL}_2$-type.

Explicit example with a Galois extension unramified outside 2.

Determination of the 2-torsion field of Jacobians of certain Shimura curves.

Abstract

Let $F/E$ be a Galois extension of totally real number fields, with Galois group $\mathrm{Gal}(F/E)$. Let $\mathfrak{N}$ be an integral ideal which is $\mathrm{Gal}(F/E)$-invariant, and $k \ge 2$ an integer. In this note, we study the action of $\mathrm{Gal}(F/E)$ on the Hecke orbits of Hilbert newforms of level $\mathfrak{N}$ and weight $k$. We also discuss the geometric counterpart to this action, which is closely related to the notion of abelian varieties potentially of $\mathrm{GL}_2$-type. The two actions have some consequences in relation with Langlands Functoriality. We conclude with an example over the maximal totally real subfield $F = \mathbf{Q}(\zeta_{32})^+$ of the cyclotomic field of 32nd root of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and $7$ real places, and $X_0^D(1)$ the Shimura curve attached to $D$. Among other things, our example shows that the field of $2$-torsion of the Jacobian of the curve $X_0^D(1)$ (and its Atkin-Lehner quotient) is the unique Galois extension $N/\mathbf{Q}$ unramified outside $2$, with Galois group the Frobenius group $F_{17} = \mathbf{Z}/17\mathbf{Z} \rtimes (\mathbf{Z}/17\mathbf{Z})^\times$. This completes Noam Elkies' answer~\cite{elk15} to a question posed by Jeremy Rouse on \verb|mathoverflow.net|.