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

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.

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.