Abstract: There are four values of $s$ for which the hypergeometric function ${}_2F_1(\frac12-s,\frac12+s;1;\cdot)$ can be parametrized in terms of modular forms; namely, $s=0$, $\frac13$, $\frac14$, $\frac16$. For the classical $s=0$ case, the parametrization is in terms of the Jacobian theta functions $\theta_3(q)$, $\theta_4(q)$ and is related to the arithmetic-geometric mean iteration of Gauss and Legendre. Analogues of the arithmetic-geometric mean are given for the remaining cases. The case $s=\frac16$ and its relationship to the work of Ramanujan is highlighted. The work presented includes various pieces of joint work with combinations of the following: B. Berndt, S. Bhargava, J. Borwein, P. Borwein and M. Hirschhorn.