**Number Theory Home**
| **Number
Theory Seminar**
| **ArithmeTexas**
| **Department Home**

# Texas A&M Number Theory Seminar

##
Department of Mathematics

Milner 317

Wednesdays, 12:30-1:30 PM

### Paula Tretkoff

Texas A&M

#### Wednesday, October 17

#### Milner 317, 12:30PM

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

(Joint work with Marvin Tretkoff and Pierre-Antoine Desrousseaux)

**Abstract:**
One of the recurrent themes in the theory of transcendental
numbers is the problem of determining the set of algebraic arguments 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 Pink.
The talk will be accessible to a general audience.