Counting points mod $p$ with Chebotarev's

Specialeforsvar ved Jens Emil Østergaard Laursen

Titel: Counting points mod $p$ with Chebotarev's  Density Theorem

Abstract: The goal of this thesis is to prove a theorem of Serre [20], stated as Theorem 6.3.2 in ”Lectures on NX(p)”. This theorem tells us that if we are given two schemes of finite type over Z, such that the difference, in their respective number of Fp-points, is small for sufficiently many primes p, then this difference must exhibit an extraordinary amount of regularity. Along the way, we delve into a string of mathematical fields and examine how they are deployed in number theory. In the first part, the aim is Chebotarev’s density theorem, which will play a prominent role in the later discussion. While we will only prove a special case, several important and interesting applications will be pursued in depth.
In chapter 3, we develop the theory of infinite Galois extensions K{E where E is a number field. The purpose is to show that the Frobenius classes sp for p R S are dense in the conjugacy classes of GalpQS{Qq, where S is a finite set of rational primes and QS is the maximal extension which is unramified outside S. A consequence of this is that continuous representations of Gal(QS/(Q) are determined by their values on these classes.
In chapter 4, we will develop the theory of characters of finite dimensional representations of a group G over a field k of characteristic zero. To that end, we will define the representation ring RkpGq and the virtual characters it induces. The general approach in proving the theorems will be to reduce to the case where G is finite and then exploit the rich theory of representations of finite groups.
In chapter 5 we will state a couple of important theorems from l-adic group cohomology. In particular, we need a special case of the Grothendieck-Lefschetz formula.
In chapter 6 we apply the theorems of the previous chapters, to prove Theorem 6.3.2. Moreover, we will explore some possible examples and consequences. In particular, we will see that if two monic and irreducible polynomials over Z satisfy jNf (p) _ Ng(p) < 1 for a set of primes of upper density 1, where Nf (p) is the number of solutions to f mod p, then they must have the same splitting field, and thus the same Galois group. Even if their degrees are not equal.

 

Vejleder: Fabien Pazuki
Censor:  Niels Lauritzen, Aarhus Universitet