  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.
