Hilbert modular forms with prescribed ramification

Let $K$ be a totally real field. In this article we present an asymptotic formula for the number of Hilbert modular cusp forms $f$ with given ramification at every place $v$ of $K$. When $v$ is an infinite place, this means specifying the weight of $f$ at $k$, and when $v$ is finite, this means specifying the restriction to inertia of the local Weil-Deligne representation attached to $f$ at $v$. Our formula shows that with essentially finitely many exceptions, the cusp forms of $K$ exhibit every possible sort of ramification behavior, thus generalizing a theorem of Khare and Prasad. From this fact we compute the minimal field over which a modular Jacobian becomes semi-stable.Comment: 30 pages, published versio

Higher modularity of elliptic curves over function fields

We investigate a notion of "higher modularity" for elliptic curves over function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we say that $E$ is $r$-modular when there is an algebraic correspondence between a stack of $r$-legged shtukas, and the $r$-fold product of $E$ considered as an elliptic surface. The (known) case $r=1$ is analogous to the notion of modularity for elliptic curves over $\mathbf{Q}$. Our main theorem is that if $E/\mathbf{F}_q(t)$ is a nonisotrivial elliptic curve whose conductor has degree 4, then $E$ is 2-modular. Ultimately, the proof uses properties of K3 surfaces. Along the way we prove a result of independent interest: A K3 surface admits a finite morphism to a Kummer surface attached to a product of elliptic curves if and only if its Picard lattice is rationally isometric to the Picard lattice of such a Kummer surface.Comment: Contains an appendix by Masato Kuwat