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Paula Tretkoff
Texas A&M
Transcendendence of values of hypergeometric functions
October 24, 2007 106 AH
Abstract


(Joint work with Marvin Tretkoff and PierreAntoine 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 AppellLauricella hypergeometric
functions of several complex variables. We show how transcendence techniques relate these problems to the AndreOort Conjecture on Zariskidensity 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.







