  Consider a polarized abelian variety $(A,L)$ over the field of complex
numbers. Following Demailly, one can associate to $(A,L)$ a real number
$\epsilon(A,L)$, its {\em Seshadri constant}, which in effect measures how much
of the positivity of $L$ can be concentrated at any given point of $A$. There
has been considerable recent interest in finding bounds on the Seshadri
constants of abelian varieties and on smooth projective varieties in general.
In the present paper we first generalize an approach of Buser and Sarnak to
give a lower bound on the minimal period length of $(A,L)$ in terms of the type
of the polarization, which by a recent result of Lazarsfeld leads to a lower
bound on the Seshadri constant of $(A,L)$. Secondly, we consider Prym varieties
and show that they have small Seshadri constants and therefore unusually small
periods. Finally, we obtain refined results for the case of abelian surfaces,
which imply in particular the surprising fact that Seshadri constants are
always rational in this case.
