On pairs of 17-congruent elliptic curves

Tom Fisher

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arXiv · 2106.02033

On pairs of 17-congruent elliptic curves

Tom Fisher

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Abstract

We compute explicit equations for the surfaces Z(17,1) and Z(17,3) parametrising pairs of $17$-congruent elliptic curves. We find that each is a double cover of the same elliptic K3-surface. We use these equations to exhibit the first non-trivial example of a pair of symplectically 17-congruent elliptic curves over the rationals. We also compute the corresponding genus 2 curve whose Jacobian has a $(17,17)$-splitting.

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Tom Fisher. 2021-06-03. On pairs of 17-congruent elliptic curves. https://arxiv.org/abs/2106.02033

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