Transformations of hypergeometric elliptic integrals

The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1\mathrm{/}2,1\mathrm{/}4,1\mathrm{/}4)$, $(1\mathrm{/}2,1\mathrm{/}3,1\mathrm{/}6)$ and $(1\mathrm{/}3,1\mathrm{/}3,1\mathrm{/}3)$. These form a special class of algebraic transformations of Gauss hypergeometric functions, of arbitrary high degree. The Gauss hypergeometric functions can be identified as elliptic integrals on the genus 1 curves $y=x^3-x$ or $y=x^3-1$. Especially interesting are algebraic transformations of the hypergeometric functions into themselves; these transformations come from isogenies of the respective elliptic curves.