We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli that can not be defined over is also given.
Given a finite -group acting on a smooth projective curve over an algebraically closed field of characteristic , the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of acting on the space of global holomorphic quadratic differentials on . We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when is cyclic or when the action of on is weakly...
We determine all modular curves X(N) (with N ≥ 7) that are hyperelliptic or bielliptic. We also give a proof that the automorphism group of X(N) is PSL₂(ℤ/Nℤ), whence it coincides with the normalizer of Γ(N) in PSL₂(ℝ) modulo ±Γ(N).
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