Mathematical Research Letters

Volume 21 (2014)

Number 5

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

Pages: 919 – 936



Burcu Baran (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


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.

Full Text (PDF format)