Addition formulae for Abelian functions associated with specialized curves

Area of Science:

• Algebraic Geometry
• Number Theory
• Complex Analysis

Background:

• Weierstrass functions are fundamental in the study of elliptic curves and related mathematical structures.
• Addition formulae for these functions are crucial for understanding the geometric and arithmetic properties of curves.
• Specialized curves with high degrees of symmetry (automorphisms) offer unique analytical challenges and opportunities.

Purpose of the Study:

• To derive and present a family of multi-term addition formulae for Weierstrass functions.
• To focus on curves of low genus, specifically genus 1 and 2, which possess numerous automorphisms.
• To extend the existing body of knowledge on addition formulae for these specialized mathematical objects.

Main Methods:

• The study employs techniques from algebraic geometry and complex analysis.
• Focus is placed on analyzing Weierstrass functions on curves with specific symmetry properties.
• Derivations involve detailed calculations for genus 1 and genus 2 cases.

Main Results:

• The paper provides addition formulae for Weierstrass functions in the equianharmonic and lemniscate cases for genus 1 curves.
• Novel addition formulae are established for several types of genus 2 curves.
• The findings contribute to the understanding of function addition on highly symmetric algebraic curves.

Conclusions:

• The derived addition formulae offer new tools for studying Weierstrass functions on specialized low-genus curves.
• The results highlight the importance of automorphisms in simplifying or structuring these formulae.
• This work advances the study of addition theorems in algebraic geometry and complex analysis.