Tuesday, November 28, 2017 - 13:15
Humboldt-Universität zu Berlin, Institut für Mathematik
Rudower Chaussee 25, 12489 Berlin, 3.006
Forschungsseminar Arithmetische Geometrie
HU Berlin, Institut für Mathematik
Abstract: Fermat showed that every prime p=1 mod 4 is a sum of two squares: p=a^2+b^2, and hence such a prime gives rise to an angle whose tangent is the ratio b/a. Hecke proved that these Gaussian primes are equidistributed across sectors of the complex plane, by making use of (infinite) Hecke characters and their associated L-functions. In this talk I will present a conjecture, motivated by a random matrix model, for the variance of Gaussian primes across sectors, and discuss ongoing work about a more refined conjecture that picks up lower-order-terms. I will also introduce a function field model for this problem, which will yield an analogue to Hecke's equidistribution theorem. By applying a recent result of N. Katz concerning the equidistribution of "super even" characters (the function field analogues to Hecke characters), I will provide a result for the variance of function field Gaussian primes across sectors; a computation whose analogue in number fields is unknown beyond a trivial regime.
submitted by Marion Thomma (thomma@math.hu-berlin.de, 2093-5815)