On the second successive minimum of the Jacobian of a Riemann surface

TL;DR

This paper investigates bounds on successive minima of Jacobians of Riemann surfaces, showing that for hyperelliptic surfaces these are bounded by a constant and for general surfaces, the second minimum grows logarithmically with genus.

Contribution

It establishes new bounds for the second successive minimum of Jacobians, especially highlighting the case of hyperelliptic surfaces and surfaces with small geodesics.

Findings

Second successive minimum is bounded by a constant for hyperelliptic surfaces.

Second successive minimum grows logarithmically with genus for general surfaces.

Improved bounds for the k-th successive minimum when the surface has small geodesics.

Abstract

To a compact Riemann surface of genus g can be assigned a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Schottky problem is to discern the Jacobians among the PPAVs. Buser and Sarnak showed, that the square of the first successive minimum, the squared norm of the shortest non-zero vector in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. We show, that in the case of a hyperelliptic surface this geometric invariant is bounded from above by a constant and that for any surface of genus g the square of the second successive minimum is equally of order log(g). We obtain improved bounds for the k-th successive minimum of the Jacobian, if the surface contains small simple closed geodesics.