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[EN] We obtain a characterization of theta functions of Jacobian vari
eties of curves with automorphisms among theta functions of principally po
larized 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 τ
functions attached to a p.p.a.v. (for the multipuncture case) allows us to char
acterize, 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 τ-function of a p.p.a.v.
Finally, since Baker-Akhiezer and τ 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.
