Thetanulls of cyclic curves of genus 4

Abstract

Let be an irreducible smooth projective cyclic curve of genus defined over the complex field . These are by definition compact Riemann surfaces of genus (unless we allow singular points) admitting an automorphism such that and generates a normal subgroup of the automorphism group of When the curve is hyperelliptic, then the curve has extra automorphisms, in particular is not the hyperelliptic involution. The condition implies to having an equation for the curve, where is an affine coordinate on and has order . The branch points of together with the signature of the cover provide algebraic coordinates for the curve in moduli. Choosing a symplectic homology basis for a given curve of genus such that the intersection products and where is the Kronecker delta and a basis for the space of holomorphic 1- forms such that we can define the period matrix of It can be shown that is an element of the Siegel upper-half space . For any and any the Riemann’s theta function is defined as Any point where is the Jacobian of the curve can be written uniquely as , where For any the theta function with rational characteristics is defined as When the entries of column vectors are from the set , then the corresponding theta functions with rational characteristics are known as theta characteristics. A scalar obtained by evaluating a theta characteristics at is called a thetanull. The problem of expressing branch points in terms of transcendentals (period matrix, thetanulls, etc.,) is classical. This is an old problem that goes back to Riemann, Jacobi, Picard and Rosenhein. We do not aim here at a complete account of the classical or contemporary work on these problems. We determine the curves of genus 4 in terms of thetanulls and further study relations among the classical thetanulls of cyclic curves (of genus 4) with an automorphisms. In our work we use formulas for small genus curves introduced by Rosenhein, Thomae’s formulas for hyperelliptic curves, some recent results of Hurwitz space theory, and symbolic manipulation. Inverting the period map has an application in fast genus two curves arithmetic incryptography. We determine similar formulas for genus 4 hyperelliptic curves as the one used in cryptography using genus 2 algebraic curves.