English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We investigate a notion of "higher modularity" for elliptic curves overfunction fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, wesay that $E$ is $r$-modular when there is an algebraic correspondence between astack of $r$-legged shtukas, and the $r$-fold product of $E$ considered as anelliptic surface. The (known) case $r=1$ is analogous to the notion ofmodularity for elliptic curves over $\mathbf{Q}$. Our main theorem is that if$E/\mathbf{F}_q(t)$ is a nonisotrivial elliptic curve whose conductor hasdegree 4, then $E$ is 2-modular. Ultimately, the proof uses properties of K3surfaces. Along the way we prove a result of independent interest: A K3 surfaceadmits a finite morphism to a Kummer surface attached to a product of ellipticcurves if and only if its Picard lattice is rationally isometric to the Picardlattice of such a Kummer surface.

Higher modularity of elliptic curves over function fields