## 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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