Hyperelliptic curves with many automorphisms

Abstract

A complex algebraic curve is said to have many automorphisms if it cannot be deformed nontrivially together with its automorphism group. Oort asked whether the jacobian of any such curve has complex multiplication. We answer this question completely when the curve is hyperelliptic. For this, we first classify all hyperelliptic curves with many automorphisms, finding 3 infinite sequences and [Formula: see text] separate isomorphism classes. Of these, all members of the infinite sequences and [Formula: see text] of the isolated curves have a jacobian with complex multiplication, but the remaining [Formula: see text] curves do not. The methods used include a criterion of Wolfart, a representation theoretic criterion of streit and the non-integrality of the [Formula: see text]-invariants of certain quotient curves of genus [Formula: see text]. For the hardest cases, we use an ad hoc developed computational criterion depending on the splitting fields of the characteristic polynomials of Frobenius acting on the Tate module of the Jacobian.