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

2010/08/12 by Bjoern Muetzel, Muetzel, Bjoern
Mathematics · #14H40 #14H42 #30F15 #30F45 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:14H40 #msc:14H42 #msc:30F15 #msc:30F45

paper · pdf · doi:10.48550/arxiv.1008.2233

arxiv created 2010/10/22 · arxiv updated 2010/10/25

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.

Related

• The asymptotic Schottky problem
• The Jacobian of a Riemann surface and the geometry of the cut locus of simple closed geodesics
• Effective Bounds on the Dimensions of Jacobians Covering Abelian Varieties
• The Schottky Problem
• Classification of real Riemann surfaces and their Jacobians in the critical case