Hyperelliptic curves with many automorphisms

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 has multiplication. We answer this question completely when hyperelliptic. For this, we first classify all hyperelliptic curves automorphisms, finding 3 infinite sequences and [Formula: see text] separate isomorphism classes. Of these, members isolated a multiplication, but remaining do not. The methods used include criterion Wolfart, representation theoretic streit non-integrality text]-invariants certain quotient genus text]. hardest cases, use an ad hoc developed computational depending on splitting fields characteristic polynomials Frobenius acting Tate module Jacobian.