On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane)

Abstract

Riemann showed that a period matrix of a compact Riemann surface of genusg≧1 satisfies certain relations. We give a further simple combinatorial property, related to the length of the shortest non-zero lattice vector, satisfied by such a period matrix, see (1.13). In particular, it is shown that for large genus the entire locus of Jacobians lies in a very small neighborhood of the boundary of the space of principally polarized abelian varieties. We apply this to the problem of congruence subgroups of arithmetic lattices in SL2(ℝ). We show that, with the exception of a finite number of arithmetic lattices in SL2(ℝ), every such lattice has a subgroup of index at most 2 which is noncongruence. A notable exception is the modular groupSL 2(ℤ).