UCLA

For families of elliptic and genus 2 hyper-elliptic curves over an algebraically closed field k of characteristic not equal to 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 invariant theory of the general linear group of two-by-two matrices acting on binary n-forms is isomorphic to the invariant theory of the permutation group of n elements acting on an affine k-variety. We show that this ring of permutation group invariants is purely inseparable over the tensor product of k with a polynomial ring defined over the integers. Next we show that the GL_2(k) invariant theory of binary sextic forms in positive characteristic is purely inseparable over a ring given by the classical invariant theory of binary sextic forms taken modulo p for all but finitely many primes p. When the characteristic of k is not 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-Fuch equations satisfied by the periods of hyperelliptic curves of arbitrary genera defined over the complex numbers. This allows us to conclude directly that the Picard-Fuchs equations are regular singular.