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

Riemann showed that a period matrix of a compact Riemann surface of genus g≧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 group SL₂(ℤ).