  To any closed subvariety $Y$ of a complex abelian variety one can attach a
reductive algebraic group $G$ which is determined by the decomposition of the
convolution powers of $Y$ via a certain Tannakian formalism. For a theta
divisor $Y$ on a principally polarized abelian variety, this group $G$ provides
a new invariant that naturally endows the moduli space $A_g$ of principally
polarized abelian varieties of dimension $g$ with a finite constructible
stratification. We determine $G$ for a generic principally polarized abelian
variety, and for $g=4$ we show that the stratification detects the locus of
Jacobian varieties inside the moduli space of abelian varieties.
