Are there any padic techniques that can be applied to the 2F1 hypergeometric function?
For e.g. I'm interested in which values this function converges padically.
Are there any padic techniques that can be applied to the 2F1 hypergeometric function? For e.g. I'm interested in which values this function converges padically. 


The Gauss hypergeometric function is the main example in the theory of padic differential equations. See K. S. Kedlaya, pAdic Differential Equations, Cambridge University Press, 2010, for the general theory. There were also two books by Dwork, almost completely devoted to ${}_2F_1$ (its padic theory is much more complicated than the classical one): B. Dwork, Generalized hypergeometric functions. Oxford: Clarendon Press, 1990. B. Dwork, Lectures on padic differential equations, Springer, 1982. It is easy to check local padic convergence for the hypergeometric series, but to study and even correctly define its analytic continuation properties one needs subtle analytic and algebraic techniques. 

