[Paper Review] Some addition formulae for Abelian functions for elliptic and hyperelliptic curves of cyclotomic type

J. C. Eilbeck, Shigeki Matsutani+1 more|arXiv (Cornell University)|Mar 27, 2008

Algebraic Geometry and Number Theory17 references4 citations

TL;DR

This paper presents novel multi-term addition formulae for Abelian functions on elliptic and hyperelliptic curves of cyclotomic type, where special moduli induce extra automorphisms via roots of unity. The key contribution is deriving explicit addition laws—such as for genus one equianharmonic and lemniscate curves, and genus two curves with 5th root of unity symmetries—that are only valid under these symmetric conditions, generalizing classical Weierstrass formulae to higher genus with cyclotomic structure.

ABSTRACT

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of genus one and two with many automorphisms. In the genus one case we find new addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves, including the Burnside curve.

Motivation & Objective

• To develop explicit addition formulae for Weierstrass $σ$ and $π$ functions on elliptic and hyperelliptic curves with enhanced automorphisms.
• To generalize classical addition laws—such as the two- and three-variable formulae—for genus one and two Abelian functions to cases where curve moduli are chosen to induce cyclotomic symmetries.
• To present these formulae in a way that makes the underlying automorphism groups and symmetries manifest, particularly those involving complex roots of unity.
• To initiate a systematic compendium of such formulae for curves with complex multiplication of cyclotomic type, filling a gap in the absence of comprehensive handbooks for higher genus Abelian functions.
• To explore the structure of multi-term addition formulae in higher genus (e.g., g=3) and for other special curves, such as trigonal curves with $3a$-th root of unity actions.

Proposed method

• Derive addition formulae by exploiting the extra automorphisms present when curve moduli are set to zero, leading to curves with symmetries related to roots of unity.
• Use the classical Weierstrass $σ$ and $π$ functions and their logarithmic derivatives in genus one, generalizing to multi-variable $σ$ and $π_{ij}$ functions in genus two.
• Apply known identities such as the two-variable formula $-\frac{\sigma(u+v)\sigma(u-v)}{\sigma(u)^2\sigma(v)^2} = \wp(u) - \wp(v)$ and extend them to multi-term forms involving $\sigma(u + \zeta^j v)$ for $\zeta$ a root of unity.
• Construct higher genus formulae by analyzing the structure of the space $\Gamma(J, \mathcal{O}(3\Theta^{[2]}))$ and expressing symmetric combinations of $\sigma$ functions as polynomials in $\wp_{ij}$, $\wp_{ijk}$, and their derivatives.
• Use the action of the Galois group of roots of unity to define automorphisms $[\zeta^j]$ on the Jacobian, which induce symmetric relations among $\sigma$-functions at transformed arguments.
• Compute explicit formulae by expanding symmetric rational functions of $\sigma$-functions and matching coefficients to known bases of symmetric functions in $\wp$-tensors and their derivatives, as illustrated in the $C_{30}$ term of the genus three case.

Experimental results

Research questions

• RQ1 How do addition formulae for Abelian functions change when the underlying curve has extra automorphisms induced by roots of unity?
• RQ2 What novel multi-term addition formulae emerge for genus one curves in the equianharmonic ($g_2=0$) and lemniscate ($g_3=0$) cases?
• RQ3 Can explicit addition formulae be derived for genus two hyperelliptic curves with 5th root of unity symmetries, such as $y^2 = x^5 + \mu_{10}$?
• RQ4 How do the symmetric structures of $\sigma$-functions transform under the action of $[\zeta^j]$, and what polynomial relations arise in the quotient $\sigma(u + \zeta^j v)/\sigma(u)^5\sigma(v)^5$?
• RQ5 What is the structure of multi-term addition formulae in higher genus, such as for a genus three curve with a 4th root of unity action, and how can they be decomposed into symmetric components?

Key findings

• For the equianharmonic case ($g_2=0$) in genus one, the paper derives a novel addition formula: $-\frac{\sigma(u\pm v)\sigma(u\pm\zeta v)\sigma(u\pm\zeta^2 v)}{\sigma(u)^3\sigma(v)^3} = \pm\frac{1}{2}(\wp'(u) \pm \wp'(v))$, valid only when $g_2=0$.
• For the genus two curve $y^2 = x^5 + \mu_{10}$, the paper presents a five-term addition formula: $\frac{\sigma(u+v)\sigma(u+[\zeta]v)\cdots\sigma(u+[\zeta^4]v)}{\sigma(u)^5\sigma(v)^5}$, which is a polynomial in $\wp_{ij}(u)$, $\wp_{ij}(v)$, $\wp_{ijk}(u)$, and $\wp_{ijk}(v)$, with $\zeta = \exp(2\pi i/5)$.
• In the genus three case with a 4th root of unity action, the paper derives a symmetric three-variable formula: $\frac{\sigma(u+v+w)\sigma(u+[\zeta]v+[\zeta^2]w)\cdots}{\sigma(u)^3\sigma(v)^3\sigma(w)^3} = \sum_{i,j,k} c_{ijk} U_i(u)V_j(v)W_k(w)$, where the $U_i, V_j, W_k$ are basis functions in $\Gamma(J, \mathcal{O}(3\Theta^{[2]}))$.
• The leading term $C_{30}$ of the genus three formula is explicitly computed and found to involve combinations of $\wp_{13}(u)\partial_3 Q_{1333}(v)\wp_{111}(w)$, $\wp_{11}(u)\wp_{11}(v)\wp_{11}(w)$, and other tensor products, with full symmetry under permutation of $u,v,w$.
• The formulae are shown to be non-trivial extensions of classical results, as they only hold under the specific cyclotomic symmetry conditions and do not reduce to standard addition laws in the general case.
• The paper demonstrates that the presence of extra automorphisms (e.g., from $\mu_{10} \neq 0$ in genus two) leads to new algebraic structures in the ring of Abelian functions, enabling new types of symmetric identities.