Moduli of Hyperelliptic Curves and Invariants of Binary Forms

For families of elliptic and genus 2 hyperelliptic curves over an algebraically closed field k of characteristic p ≠ 2, we derive smooth local normal forms, parameterized by families of binary n-forms over k with distinct linear factors. We also prove that the GL₂(k) invariant theory of binary n-forms in positive characteristic is isomorphic to the invariant theory of the permutation group S_n acting on an affine k-variety. We show that this ring of S_n-invariants is purely inseparable over the tensor product of k with a polynomial ring defined over $Z$. Next we show that the GL₂(k) invariant theory of binary sextics is purely inseparable over a ring given by classical invariant theory of binary sextic forms taken modulo p for all but finitely many primes p. When char. k = p ≠ 2 these results imply that for all but finitely many primes, p, a power of the Frobenius endomorphism induces a morphism between the coarse moduli scheme of genus 2 curves and the spectrum of a ring constructed using the classical invariant theory of binary sextics taken modulo p. Moreover, this morphism is a bijection at the level of closed points.;We also give, via a combinatorial argument, explicit formulae for the Picard-Fuchs equations satisfied by the periods of hyperelliptic curves of arbitrary genera over $C$. This allows us to conclude directly that the Picard-Fuchs equations are regular-singular.