Theta functions and algebraic curves with automorphisms

Let X be an irreducible, smooth, projective curve of genus g ≥ 2 defined over the complex field C . Then there is a covering pi : X&rarrr;P1 , where P1 denotes the projective line. The problem of expressing branch points of the covering pi in terms of the transcendentals (period matrix, thetanulls, e.g.) is classical. It goes back to Riemann, Jacobi, Picard and Rosenhein. Many mathematicians, including Picard and Thomae, have offered partial treatments for this problem. In this work, we address the problem for cyclic curves of genus 2, 3, and 4 and find relations among theta functions for curves with automorphisms. We consider curves of genus g > 1 admitting an automorphism sigma such that Xs has genus zero and sigma generates a normal subgroup of the automorphism group Aut( X ) of X . To characterize the locus of cyclic curves by analytic conditions on its Abelian coordinates, in other words, theta functions, we use some classical formulas, recent results of Hurwitz spaces, and symbolic computations, especially for genera 2 and 3. For hyperelliptic curves, we use Thomae's formula to invert the period map and discover relations among the classical thetanulls of cyclic curves. For non hyperelliptic curves, we write the equations in terms of thetanulls. Fast genus 2 curve arithmetic in the Jacobian of the curve is used in cryptography and is based on inverting the moduli map for genus 2 curves and on some other relations on theta functions. We determine similar formulas and relations for genus 3 hyperelliptic curves and offer an algorithm for how this can be done for higher genus curves. It is still to be determined whether our formulas for g = 3 can be used in cryptographic applications as in g = 2.;In the second part of this thesis we study the use of theta functions in coding theory. Let ℓ > 0 be a square free integer and OK be the ring of integers of the imaginary quadratic field K = Q( -ℓ ). Codes C over K determine lattices Λ ℓ(C) over rings OK/pOK via construction A. If p ∤ ℓ then the ring R:=OK/pO K is isomorphic to Fp2 or FpxF p . Given a code C over R , theta functions on the corresponding lattices are defined. These theta series qLℓ (C) can be written in terms of the complete weight enumerator of C. We prove that for any two ℓ < ℓ' the first ℓ+14 terms of their corresponding theta functions are the identical. Moreover, we prove that when p = 2: (i) for ℓ ≥ 2n+1 n+2n-1,QL ℓ (C) determines the code C uniquely, (ii) for ℓ < 2n+1 n+2n-1 there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to QLℓ (C).;When p is a prime other than 2, we conjecture that for ℓ > pn+1 n+22 there is a unique complete weight enumerator corresponding to a given theta function. We verify the conjecture for ℓ ≤ 59 and primes p < 7.