Description:
This book presents several basic techniques in algebraic geometry, group representations, number theory, ℓ-adic and standard cohomology, and modular forms. It explores how NX(p) varies with p when the family (X) of polynomial equations is fixed. The text examines the size and congruence properties of NX(p) and describes the ways in which it is computed. Along with covering open problems and offering simple, illustrative examples, the author presents various theorems, including the Chebotarev density theorem and the prime number theorem.
Review Quotes:
"The book is written by a master in the area. It puts the objects it treats into their natural conceptual framework. ... The book is highly recommended to anyone interested in the fundamental questions it treats. Those enjoying the mathematics created by Serre will also find pleasure and inspiration in this book."
--Gabor Wiese, Mathematical Reviews, April 2013
"This book may be regarded (and can be used) both as an introduction to the modern algebraic geometry, written by one of its creators, and as a research monograph, investigating in depth ..."
--B.Z. Moroz, Zentralblatt MATH 1238
"the mathematics is exquisite and the presentation is wonderful. ... The development of the background mathematics and methodology is crystal clear. ... this is another terrific book by Serre: it provides a splendid introduction to both a beautiful arithmetic (-geometric) theme and hugely important mathematical methods pertaining to the given theme. It should tantalize the reader and move him to go into these themes in greater depth, using Serre's exposition as a high-level road map."
--Michael Berg, MAA Reviews, June 2012