On addition theorems related to elliptic integrals

TL;DR

This paper derives explicit addition formulas for elliptic integrals involving quartic polynomials, connecting classical elliptic integral theory with complex elliptic genera and extending Euler's addition theorem.

Contribution

It introduces new explicit formulas for addition theorems of elliptic integrals associated with degree 4 polynomials, enhancing understanding of their algebraic and geometric properties.

Findings

Explicit addition formulas for elliptic integrals with quartic polynomials

Connections established between elliptic integrals and complex elliptic genera

Extension of Euler's addition theorem to broader classes of elliptic integrals

Abstract

This paper provides some explicit formulas related to addition theorems for elliptic integrals $\int_0^x dt/R(t)$, where $R(t)$ is the square root from a polynomial of degree 4. These integrals are related to complex elliptic genera and are motivated by Euler's addition theorem for elliptic integrals of the first kind.