Characterizations of Jacobians of curves with automorphisms

We obtain a characterization of theta functions of Jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties (p.p.a.v.). We first give a characterization in terms of finite dimensional orbits for a suitable action in the Sato Grassmannian. Secondly, the introduction of formal Baker-Akhiezer functions and formal $\tau$-functions attached to a p.p.a.v. (for the multipuncture case) allows us to characterize, in terms of bilinear identities, those Baker-Akhiezer functions that are Baker-Akhiezer functions of Jacobians of curves with automorphisms. Further, in the case of automorphisms with fixed points, we rewrite the previous result as a hierarchy of partial differential equations for the $\tau$-function of a p.p.a.v. Finally, since Baker-Akhiezer and $\tau$ functions are written in terms of theta functions, these results give rise to characterizations of p.p.a.v. in terms of their theta functions.