# Relation between elliptic curves and hypergeometric functions

If we are given an elliptic curve of the form $y^2 = (x-e_1)(x-e_2)(x-e_3)$, where $e_1 > e_2 > e_3$ and all $e_i$ are real, then we can evaluate the period and the dual period, which will be Gauss hypergeometric functions.

My question is: if we are given a hypergeometric function $_2 F_1 (a,b,c;z)$, is it possible to directly get the elliptic function whose periods are this hypergeometric function? More precisely, is there a way in which, given $(a,b,c)$, we can determine $(e_1,e_2,e_3)$?

• In my experience, one usually writes the real period of an elliptic curve in terms of $_{2}F_{1}(1/2,1/2,1; z)$ for some $z \in \mathbb{C}$. The corresponding curve is then $y^{2} = x(x-1)(x-z)$. In other words, I don't think you can determine $e_{1}$, $e_{2}$ and $e_{3}$ from $(a,b,c)$, because one basically always has $(e_{1},e_{2},e_{3}) = (1/2,1/2,1)$. – Jeremy Rouse Aug 10 '17 at 12:46