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.