Lectures on N_X(p)

Lectures on N_X(p) deals with the question on how N_X(p), the number of solutions of mod p congruences, varies with p when the family (X) of polynomial equations is fixed. While such a general question cannot have a complete answer, it offers a good occasion for reviewing various techniques in l-adic cohomology and group representations, presented in a context that is appealing to specialists in number theory and algebraic geometry.

Along with covering open problems, the text examines the size and congruence properties of N_X(p) and describes the ways in which it is computed, by closed formulae and/or using efficient computers.

The first four chapters cover the preliminaries and contain almost no proofs. After an overview of the main theorems on N_X(p), the book offers simple, illustrative examples and discusses the Chebotarev density theorem, which is essential in studying frobenian functions and frobenian sets. It also reviews ℓ-adic cohomology.

The author goes on to present results on group representations that are often difficult to find in the literature, such as the technique of computing Haar measures in a compact ℓ-adic group by performing a similar computation in a real compact Lie group. These results are then used to discuss the possible relations between two different families of equations X and Y. The author also describes the Archimedean properties of N_X(p), a topic on which much less is known than in the ℓ-adic case. Following a chapter on the Sato-Tate conjecture and its concrete aspects, the book concludes with an account of the prime number theorem and the Chebotarev density theorem in higher dimensions.