# Is there a precise mathematical connection between hypergeometric functions and modular forms

I've been playing around with Gauss' hypergeometric series $F(a,b,c,z)$ these days and I was wondering if there is some relation with the theory of modular forms.

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The answer to this question is yes. For instance there is a correspondence between the Gaussian hypergeometric function $_2F_1$ and the family of elliptic curves $_2E_1(\lambda)$ defined by $$y^2 = x(x- 1) (x-\lambda)$$ for $\Bbb{Q}\backslash\{0,1\}$. Through the modularity theorem which was proved by Breuil, Conrad, Diamond, Taylor, and Wiles, an elliptic curve has a corresponding weight $2$ cusp form. So we get a connection between Gaussian hypergeometric functions and modular forms.