General Mathematics Colloquium
Speaker: Arno Kret
Title: Integral points on Hilbert-Blumenthal varieties
(joint work with Rafael von Kaenel)
A Diophantine equation X is a polynomial equation in finitely many unknowns x_1, x_2, ... x_r, where the coefficients of the polynomial are integers. A famous example is Fermat's equation x_1^n + x_2^n = x_3^n (n > 3).
The problem is to find all integer solutions of the Diophantine equation X.
Let's consider only those X for which the set of all solutions X(Z) over the integers Z is finite. To determine the set X(Z), an important first step is to bound the absolute values |x_1|, |x_2|, ... of a possible solution x_1, x_2, ... x_r. Then, one knows that there exists an algorithm to determine X(Z), in finite, and bounded time.
In more modern language, we consider these Diophantine equations as special examples of algebraic schemes X over the integers Z. Giving bounds for the values |x_1|, |x_2|, ... , means giving height bounds on the solutions.
We explain a new method to bound the integral points on a certain class of very particular varieties X (Shimura varieties of Hilbert-Blumenthal type). We give an effective bound on the size of the set X(Z[1/m[) and we give an effective bound on the height of the solutions in terms of the integer m.
Location: KdVI meeting room, Science Park 105-107, F3.20