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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.

See this paper of Ono's for more detail. Also this paper proves a congruence, using Gaussian hypergeometric functions, that allows one to compute coefficients of modular forms using only multinomial coefficients.

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I'm not sure this is what is meant in the question, as it asks about Gauss's function, rather than Greene's functions. I guess you would also have to add a step explaining how these two, one over complex number, the other "over" finite fields, are related. Note that both cited papers don't do this, not to mention almost not mentioning Gauss's hypergeometric functions at all (Ono's paper includes a remark by D. Stanton that some things look similar to other things). – simplequestions Feb 19 '13 at 22:42

A modular form $f$ of weight $1$ with respect to some discrete subgroup $\Gamma$ of $\operatorname{SL}(2,\mathbb{R})$ fulfills a linear second-order differential equation, if the independent variable of that equation is chosen to be a suitable modular function with respect to $\Gamma$. That is, if we express $f$ locally not as a function of $\tau\in\mathbb{H}$ but as a function of $t$ where $t$ is a modular function of $\tau$, then $f$, $\frac{\mathrm{d}f}{\mathrm{d}t}$, $\frac{\mathrm{d}^2f}{\mathrm{d}t^2}$ are linearly dependent with coefficients that are algebraic in $t$.

In some cases, the differential equation turns out to be the hypergeometric one, and if the initial conditions match, $f$ can be expressed directly as $f = {}_2F_1(a,b;c;t)$.

Of course, this only works locally because $t$ is invariant under $\Gamma$ whereas $f$ is not, but the linearity and familiarity of the resulting differential equation compensates for that. As a corollary, this approach yields values of modular forms at certain points as a power product of several Gamma function values.

This subject is treated in section 5.4 around proposition 21 in

  1. Don Zagier: Elliptic modular forms and their applications. In: Kristian Ranestad (ed.): The 1-2-3 of modular forms. Springer 2008, DOI: 10.1007/978-3-540-74119-0.

with three different and interesting proofs. Two examples are given there as well, they result in $$\begin{align} \varTheta_{00}^2 &= {}_2F_1\left(\tfrac{1}{2},\tfrac{1}{2};1;k^2\right) \\ \sqrt[4]{\operatorname{E}_4} &= {}_2F_1\left(\tfrac{1}{12},\tfrac{5}{12};1;\tfrac{1728}{j}\right) \end{align}$$ where $\varTheta_{00}$ is a Jacobi thetanull function, $k^2 = \frac{\varTheta_{10}^4}{\varTheta_{00}^4}$ is the parameter of the standard elliptic integrals, $\operatorname{E}_4$ is a normalized Eisenstein series, and $j$ is Klein's absolute invariant. All these entities are considered functions of the period ratio $\tau\in\mathbb{H}$ here, and as such you might know $k^2$ better as the elliptic lambda function $\lambda$.

Note that the exponent of the left-hand side is the reciprocal of the underlying modular form's weight, so the result has (formally) weight $1$. The main argument of the ${}_2F_1$ on the right-hand side is the chosen modular function $t$; for $f=\varTheta_{00}^2$, it is $t=k^2=\lambda$, and for $f=\sqrt[4]{\operatorname{E}_4}$, it is $t=\tfrac{1728}{j}$.

In both examples, the initial values have been set to match as $t\to0$, i.e. $\Im\tau\to\infty$; and the periodicity (in $\tau$) of both sides matches as well, so the identities hold as long as $\Im\tau$ is large enough. I am deliberately vague here because the precise set of admissible $\tau\in\mathbb{H}$ depends on the example, and I am going to give additional examples: $$\begin{align} \\ \varTheta_{00}^2 &= {}_2F_1\left(\tfrac{1}{4},\tfrac{1}{4};1; \tfrac{64}{\mathfrak{f}^{24}}\right) \\ \varTheta_{01}^2 &= {}_2F_1\left(\tfrac{1}{2},\tfrac{1}{2};1; -\tfrac{k^2}{k'^2}\right) = {}_2F_1\left(\tfrac{1}{4},\tfrac{1}{4};1; -\tfrac{64}{\mathfrak{f}_1^{24}}\right) \\ \sqrt[6]{\operatorname{E}_6} &= {}_2F_1\left(\tfrac{1}{12},\tfrac{7}{12};1; \tfrac{1728}{1728-j}\right) \end{align}$$ where $\varTheta_{01}$ is another Jacobi thetanull function; $k'^2=1-k^2$; furthermore, $\mathfrak{f}$, $\mathfrak{f}_1$ are Weber functions; and $\operatorname{E}_6$ is the normalized Eisenstein series of weight $6$ with respect to the full modular group.

From the above, it is easy to derive representations of related functions such as $$\begin{align} \varTheta_{10}^2 &= k\,\varTheta_{00}^2 = k\,{}_2F_1\left(\tfrac{1}{2},\tfrac{1}{2};1;k^2\right) \\ \eta^2 &= \sqrt[3]{\frac{\varTheta_{00}^2\varTheta_{01}^2\varTheta_{10}^2}{4}} = \sqrt[3]{\frac{k'k}{4}}\ \varTheta_{00}^2 = \sqrt[3]{\frac{k'k}{4}} \ {}_2F_1\left(\tfrac{1}{2},\tfrac{1}{2};1;k^2\right) \\ &= \frac{\varTheta_{00}^2}{\mathfrak{f}^4} = \frac{1}{\mathfrak{f}^4}{}_2F_1\left(\tfrac{1}{4},\tfrac{1}{4};1; \tfrac{64}{\mathfrak{f}^{24}}\right) \\ &= \frac{\varTheta_{01}^2}{\mathfrak{f}_1^4} = \frac{1}{\mathfrak{f}_1^4}{}_2F_1\left(\tfrac{1}{4},\tfrac{1}{4};1; -\tfrac{64}{\mathfrak{f}_1^{24}}\right) \\ &= \sqrt[4]{\frac{\operatorname{E}_4}{\gamma_2}} = \frac{1}{\sqrt[4]{\gamma_2}} {}_2F_1\left(\tfrac{1}{12},\tfrac{5}{12};1; \tfrac{12^3}{\gamma_2^3}\right) \\ &= \sqrt[6]{\frac{\operatorname{E}_6}{\gamma_3}} = \frac{1}{\sqrt[6]{\gamma_3}} {}_2F_1\left(\tfrac{1}{12},\tfrac{7}{12};1; -\tfrac{12^3}{\gamma_3^2}\right) \end{align}$$ Here $\varTheta_{10}$ is the remaining member of the triplet of standard Jacobi thetanull functions, $\eta$ is the Dedekind eta function, and $\gamma_2$, $\gamma_3$ are another couple of Weber functions. Note that in these "derived" examples, the appropriate choice of root branch depends on $\tau$.

Further occurences of ${}_2F_1$ can be found when inverting Klein's $j$-invariant and in the context of other modular functions. The latter example shows an aspect that I have not covered thus far: What if you want to express a modular function $f$ (weight zero) locally in terms of another modular function $t$, supposing their corresponding automorphy groups are commensurable? Then $f$ and $t$ are algebraically related, and in some cases those algebraic relations can again be solved with hypergeometric functions.

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