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Number Theory Seminar

Thursday, December 1, 2022
2:00pm to 3:00pm
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Linde Hall 387
Higher modularity for elliptic curves over function fields
Jared Weinstein, Department of Mathematics and Statistics, Boston University,

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 the rationals. Our main theorem is that if $E/\F_q(t)$ 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.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].