The Gauss hypergeometric function $_2F_1(a,b;c;z)$ is a special function expressed by the hypergeometric series that includes many other special functions as specific or limiting cases. It satisfies a second order linear ODE, denoted $DE(a, b; c)$, with three regular singularities at $0,1,\infty$. The solutions to $DE(a, b; c)$ form a rank-2 sheaf on $\mathbb{P}^1(\mathbb{C})\backslash\{0, 1,\infty\}$, admitting the action of the fundamental group $\pi_1(\mathbb{P}^1(\mathbb{C})\backslash\{0,1,\infty\}, *)$, called the monodromy representation associated to $_2F_1(a, b; c; z)$.

The above generalizes from $_2F_1$ to $_nF_{n-1}$ when the set of parameters $\{a,b;c\}$ is extended to a hypergeometric datum $HD =\{a_1, ..., a_n; b_1 = 1, b_2, ..., b_n\}$ with $a_i,b_j$ nonzero rationals. A parallel algebraic setting was introduced by Katz who defined a hypergeometric sheaf $\mathcal{F}(HD)$ on the multiplicative (algebraic) group $\mathbf{G}_m$ together with a representation of the absolute Galois group of a cyclotomic field acting on $\mathcal{F}(HD)$.

A well-known Whipple identity, proved analytically, relates $_7F_6$ to $_4F_3$. We shall reinterpret it in terms of the associated Katz representations. Moreover, these Galois representations, arising geometrically, are shown to be automorphic, as predicted by the Langlands program. Finally, we shall discuss the connection between special values of hypergeometric functions and periods. This is a joint work with Ling Long and Fang-Ting Tu.

## Various aspects of hypergeometric functions

2023/05/29 (Mon.) 14:00~15:00