[Paper Review] Theta functions and algebraic curves with automorphisms

G. S. Wijesiri, Wijesiri, G. S.|arXiv (Cornell University)|Oct 5, 2012

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper derives explicit formulas expressing branch points of cyclic algebraic curves of genus 2, 3, and 4 with automorphisms as ratios of theta constants (thetanulls), using Thomae’s formula for hyperelliptic curves and a general method based on Riemann’s theta functions and period integrals. The key contribution is a systematic inversion of the period map for such curves, enabling analytic characterization of their moduli via theta functions.

ABSTRACT

Let $\X$ be an irreducible, smooth, projective curve of genus $g \geq 2$ defined over the complex field $\C.$ Then there is a covering $π: \X \longrightarrow ¶^1,$ where $¶^1$ denotes the projective line. The problem of expressing branch points of the covering $π$ 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 $σ$ such that $\X^σ$ has genus zero and $σ$ 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 $g=2.$

Motivation & Objective

• To express the branch points of cyclic algebraic curves with automorphisms in terms of theta constants (thetanulls).
• To provide analytic conditions on the period matrix and theta functions that characterize the locus of cyclic curves in moduli space.
• To extend the inversion of the period map from genus 2 to genus 3 and 4, particularly for curves with nontrivial automorphism groups.
• To develop a general method for computing branch points as ratios of theta functions using symbolic computation and classical formulas.
• To assess the potential cryptographic applicability of genus 3 formulas analogous to those used in genus 2 curve arithmetic.

Proposed method

• Uses Thomae’s formula to express branch points of hyperelliptic curves as ratios of theta constants.
• Applies Lemma 7 to relate divisor classes and theta functions via period integrals, enabling inversion of the period map.
• Employs symbolic computation in Maple 10 to evaluate complex integrals and derive explicit algebraic relations among thetanulls.
• Utilizes Riemann’s theta functions with rational characteristics and their quasi-periodicity to relate integrals to theta constants.
• Applies classical formulas from Hurwitz spaces and symplectic group actions on Siegel upper half-space to constrain moduli space loci.
• Derives explicit parametrizations of curves (e.g., $y^3 = (x^2-1)(x^4 - eta x^2 + 1)$) and expresses their invariants in terms of thetanulls.

Experimental results

Research questions

• RQ1 How can the branch points of cyclic curves of genus 2, 3, and 4 with automorphisms be expressed as ratios of theta constants?
• RQ2 What analytic conditions on the period matrix and theta functions characterize the moduli locus of such curves?
• RQ3 Can the method used for genus 2 be generalized to genus 3 and 4 curves with automorphisms to invert the period map?
• RQ4 What explicit algebraic relations exist between thetanulls and invariants of cyclic curves with automorphism groups $C_3$, $C_5$, $C_6 \times C_2$, etc.?
• RQ5 To what extent can the derived formulas for genus 3 curves be applied in cryptographic systems, as in genus 2?

Key findings

• For genus 2 hyperelliptic curves with automorphism group $C_3$, the branch points are expressed as rational functions of thetanulls via Thomae’s formula.
• For genus 3 hyperelliptic curves with $C_3$ or $C_5$ automorphisms, explicit formulas are derived relating the invariants $\alpha_1, \alpha_2, \alpha$ to ratios of theta constants.
• The curve $y^3 = (x^2 - 1)(x^4 - \alpha x^2 + 1)$ has its parameter $\alpha$ expressed as a product of ratios of theta functions evaluated at specific period integrals.
• The method of Lemma 7 enables the derivation of branch point expressions by comparing divisor classes and theta functions, though symbolic evaluation of integrals remains challenging.
• The paper provides a general algorithmic framework for inverting the period map for higher genus cyclic curves using theta functions and symbolic computation.
• Some results for genus 2 and 3 curves were previously published in [15], but this work extends them to genus 4 and provides new explicit formulas for non-hyperelliptic cases.