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On the period matrix of a Riemann surface of large genus (with an appendix by J. H. Conway and N. J. A. Sloane). (English) Zbl 0814.14033

Let \(H\) be a polarization on an abelian variety \(X = \mathbb{C}^ g/\Lambda\), considered as a hermitian form on the vector space \(\mathbb{C}^ g\). The number \(m = m(X,H)\), defined as the minimum of all values \(H(\lambda,\lambda)\) over all \(0 \neq \lambda \in \Lambda\) is an invariant of the polarized abelian variety \((X,H)\). Geometrically \(m\) is the square of the length of the shortest closed geodesic on the corresponding flat real torus. By a well-known compactness theorem of Mahler \(m\) may be thought of as a distance function to the boundary of the corresponding moduli space. Using results of the geometry of numbers the authors prove some explicit inequalities for the invariant \(m\). The surprising fact is that for large genus the whole Schottky locus lies in a very small neighborhood of the boundary of the moduli space of principally polarized abelian varieties. – This is applied to give examples of families of period matrices which are not Jacobians. – There are some applications concerning arithmetic lattices. It is shown that with the exception of a finite number every arithmetic lattice in \(\text{SL}_ 2(\mathbb{R})\) has a subgroup of index at most 2 which is noncongruence.
This paper contains several appendices: Appendix 0: Polarisations and quadratic forms (p. 50-51). – Appendix 1: Some explicit examples (p. 51- 53). – Appendix 2 (by J.H. Conway and N. J. A. Sloane): \(D_ 4\), \(E_ 8\), Leech and certain other lattices are symplectic (p. 53-55).
Reviewer: H.Lange (Erlangen)

MSC:

14H42 Theta functions and curves; Schottky problem
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
32G20 Period matrices, variation of Hodge structure; degenerations
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References:

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