國立清華大學 數學系

Tea time：15:30, Room 707
Abstract:
The ordinary hypergeometric function 2F1(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 singular points at 0, 1,∞. They were systematically studied by Gauss and characterized by Riemann. There are thousands of identities among the hypergeometric functions. The solutions to DE(a, b; c) form a sheaf on which the fundamental group π1(P1(C)\{0, 1,∞}, ∗) acts, called the monodromy representation associated to 2F1(a, b; c; z).
The above generalizes from 2F1 to nFn−1 when the set of parameters {a, b; c} is extended to a hypergeometric datum HD = {a1, ..., an; b1 = 1, b2, ..., bn} with ai, bj nonzero rationals. A parallel algebraic setting was introduced by Katz who defined a hypergeometric sheaf on the multiplicative (algebraic) group Gm on which the absolute Galois group of a cyclotomic field acts via ρ(HD). We shall reinterpret a Whipple identity relating 7F6 to 4F3, proved analytically, in terms of associated Galois representations ρ(HD). Moreover, these ρ(HD), 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.