Higher modularity of elliptic curves over function fields
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913089540063232 |
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| author | Logan, Adam Weinstein, Jared |
| author_facet | Logan, Adam Weinstein, Jared |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11149 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Higher modularity of elliptic curves over function fields Logan, Adam Weinstein, Jared Number Theory Algebraic Geometry 11G05, 14J28, 11R39 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. |
| title | Higher modularity of elliptic curves over function fields |
| topic | Number Theory Algebraic Geometry 11G05, 14J28, 11R39 |
| url | https://arxiv.org/abs/2211.11149 |