Local-global principle for the existence of solutions for diophantine equations with hopefully many solutions
Mathematics Colloquium
Date
Wednesday, June 3, 2026 15:30 - 16:30
Speaker
Jean-Louis Colliot-Thélène (Université Paris-Saclay)
Location
Raiffeisen Lecture Hall, Central Building
Series
Colloquium
Tags
Colloquium
Host
Tim Browning
Contact
A hundred years ago, extending earlier results of Fermat, Legendre, Hilbert, Minkowski, Helmut Hasse etablished a "local-global principle" for the existence of rational solutions of quadratic polynomial equations with integral coefficients. The "local" conditions essentially require that no congruences prevent the existence of solutions. For polynomials of higher degree, but with a number of variables bigger than the degree, one might hope for a similar local-global principle. I shall survey the proto-history and history of this problem and the many techniques developed to prove or disprove this "principle" for specific classes of polynomials.