Deformations of abelian varieties with an automorphism group

Given a polarized abelian variety with an automorphism group \(G\), we prove that the associated local moduli functor is pro-representable, the algebra that pro-represents it is formally smooth, and compute the dimension of this algebra as a function of the analytic action of the group. We present the explicit computations in the case of the action of the symmetric group \(S_3\) on the factors of the product \(E\times E\times E\) of an elliptic curve.