Séminaire Combinatoire et Théorie des Nombres ICJ

Unconditional discriminant lower bounds exploiting violations of GRH

by Eduardo Friedman (Universidad de Chile)

Europe/Paris
Bât. Braconnier, salle 112 (ICJ, Université Lyon 1)

Bât. Braconnier, salle 112

ICJ, Université Lyon 1

Description

I shall first explain Odlyzko's method for proving lower bounds for discriminants of number fields, and then describe how one can modify it to get new unconditional lower bounds.  The idea is that since good lower bounds are available if all the low-lying zeroes of the Dedekind zeta function are on the critical line (i.e., if the Riemann hypothesis holds close to the real axis), one should exploit the contribution of a hypothetical low zero positioned away from the critical line. This works reasonably well for fields of up to degree 9 or 10, but peters out by degree 13. 

This is joint work with Karim Belabas, Francisco Diaz y Diaz and Salvador Flores.