On the de-Rham cohomology of hyperelliptic curves
Abstract
For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of Cech cohomology. We use this to produce a family of curves in characteristic p>2 for which the Hodge-de-Rham short exact sequence does not split equivariantly; this generalises a result of Hortsch. Further, we use our basis to show that the hyperelliptic involution acts on the first de-Rham cohomology by multiplication by -1, i.e., acts as the identity when p=2.
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