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Paula Tretkoff

Texas A&M

Transcendendence of values of hypergeometric functions

October 24, 2007
106 AH


(Joint work with Marvin Tretkoff and Pierre-Antoine Desrousseaux) 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 both e and pi are 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 myself with Wustholz, on this problem. More recently, with Tretkoff and Desrousseaux, we have studied the exceptional set of the Appell-Lauricella hypergeometric functions of several complex variables. 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. The talk will be accessible to a general audience.

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