Lectures on N_X(p)

<p><strong>Lectures on N<sub>X</strong></sub><strong>(p)</strong> deals with the question on how N<sub>X</sub>(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.</p> <p>Along with covering open problems the text examines the size and congruence properties of N<sub>X</sub>(p) and describes the ways in which it is computed by closed formulae and/or using efficient computers.</p> <p></p> <p>The first four chapters cover the preliminaries and contain almost no proofs. After an overview of the main theorems on N<sub>X</sub>(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.</p> <p></p> <p>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<sub>X</sub>(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. </p> <p> </p>