## Conference "Arithmetic, Geometry and Coding Theory"## March 14 - 18, 2011## CIRM, Marseille, France |

**Organisers:** Yves Aubry *(Université de la Méditerranée, IML)*, Christophe Ritzenthaler *(Université de la Méditerranée, IML)*, Alexey Zykin *(Mathematical Department of HSE, Laboratoire Poncelet, IITP)*

For any n > 0, let X_ns(n) denote the modular curve over Q associated to the normalizer of a non-split Cartan subgroup of level n. The integral points and the rational points of X_ns(n) are crucial in two interesting problems: the class number one problem and the Serre's uniformity problem. In this talk we focus on the genus 3 curve X_ns(13). It has no Q-rational cusp (as for any level n > 2), so to compute an equation for this curve as a quartic in P^2(Q) we use representation theory. Our explicit description of X_ns(13) yields a surprising exceptional Q-isomorphism to another modular curve. We also compute the j-function on X_ns(13); evaluating it at the known Q-rational points, we obtain the expected CM values.