### Abstract

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 SL_{2}(ℝ). We show that, with the exception of a finite number of arithmetic lattices in SL_{2}(ℝ), every such lattice has a subgroup of index at most 2 which is noncongruence. A notable exception is the modular group SL_{2}(ℤ).

Original language | English (US) |
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Pages (from-to) | 27-56 |

Number of pages | 30 |

Journal | Inventiones Mathematicae |

Volume | 117 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 1994 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)