An exceptional isomorphism between level 13 modular curves via Torelli’s Theorem

The Jacobians of the modular curves $X_{\mathrm{ns}}(13)$ and $X_{\mathrm{s}}(13)$ respectively associated with the normalizers of non-split and split Cartan subgroups of level $13$ are isogenous over $\mathbb{Q}$. In this note, we construct a $\mathbb{Q}$-isomorphism between these Jacobians which respects their canonical principal polarizations. In particular, we obtain a $\mathbb{Q}$-isomorphism between $X_{\mathrm{ns}}(13)$ and $X_{\mathrm{s}}(13)$; this has no known “modular” explanation.

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Published 9 December 2014