Milner 317

Wednesdays, 12:30-1:30 PM

Texas A&M University

**Wednesday, February 21
Milner 317, 12:30 PM**

**Title:** Transcendence of values of hypergeometric functions

**Abstract:**
One of the recurrent themes in the theory of transcendental numbers is the
problem of determining the set of algebraic numbers at which a given
transcendental function assumes algebraic values. This set has come to be
known as the exceptional set of the function. The classical work of Hermite
(1973), Lindemann (1882) and Weierstrass (1885) established that the
exceptional set of the exponential function exp(x) consists only of x=0.
This implies for example that pi is transcendental. C.L. Siegel (1929)
suggested studying the exceptional set of the classical (Gauss)
hypergeometric function of one complex variable F(a,b,c;x) when a,b,c are
rational numbers. We recall the work of Wolfart and Cohen-Wustholz on this
problem. More recently, with Tretkoff and Desrousseaux, we have studied the
exceptional set of two different generalizations of the classical
hypergeometric function, namely the Appell-Lauricella hypergeometric
functions of several complex variables and the Tissot-Pochhammer functions
of one complex variable satisfying higher order differential equations. We
show how transcendence techniques relate these problems to the Andre--Oort
Conjecture on Zariski-density of complex multiplication points in
subvarieties of Shimura varieties and to generalizations of this conjecture
by R. Pink. This is joint work with Marvin Tretkoff and Pierre-Antoine Desrousseaux.