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Lectures on $$N_X(p)$$. (English) Zbl 1238.11001
Research Notes in Mathematics 11. Boca Raton, FL: CRC Press (ISBN 978-1-46650-192-8/hbk). ix, 163 p. (2012).
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 the properties of the function $N_X:\{p^e\mid e\in\mathbb N,\;p\in{\mathcal P}\}\to \mathbb Z$ for a scheme $$X$$ of finite type over $$\mathbb Z$$, where $$N_X(q):= |X(\mathbb F_q)|$$ and $${\mathcal P}$$, $$\mathbb N$$, and $$\mathbb F_q$$ stand respectively for the set of the rational primes, the set of the positive integers, and a finite field of $$q$$ elements. The author writes: “For a fixed $$X$$, one wants to understand how $$N_X(p)$$ varies with $$p$$: what is its size and its congruence properties? Can it be computed by closed formulae, by cohomology, and/or by efficient computer programs? What are the open problems? These questions offer a good opportunity for reviewing several basic techniques in algebraic geometry, group representations, number theory, cohomology (both $$l$$-adic and standard) and modular forms…”.
The book consists of nine chapters. Chapter 1 contains five sections: definition of $$N_X(p)$$ in the affine case, definition of $$N_X(p)$$ in the scheme setting, how large is $$N_X(p)$$ when $$p\to\infty$$, more properties of $$p\mapsto N_X(p)$$, the zeta point of view. The second chapter describes a few low-dimensional examples with $$\dim(X(\mathbb C))\in\{0,1,2\}$$. In the third chapter a few variants of the Chebotarev density theorem for a number field are stated and discussed; the author defines “Frobenian functions” and “Frobenian sets” (somewhat analogous to the multiplicative Frobenian functions defined by R. W. K. Odoni [Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 371–403 (1985; Zbl 0596.10040)]). Chapter 4, entitled “Review of $$l$$-adic cohomology”, summarises a few results from the SCA and Deligne’s works on the Weil conjectures, as well as some of the later improvements of the Deligne-Weil bounds for $$N_X(q)$$. Chapter 5 contains several results on the linear representations of a group $$G$$, applied in chapter 6 to $$G= \text{Gal}(\overline{\mathbb Q}\mid\mathbb Q)$$.
Let me cite a few results proved in chapter 6, entitled “The $$l$$-adic properties of $$N_X(p)$$”. Let $$d(X):= \dim(X\times_{\mathbb Z}\mathbb Q)$$. There is a finite subset $$S$$ of $${\mathcal P}$$ such that $N_X(p^e)= \sum_{0\leq i\leq 2d(X)} (-1)^i\, \text{Tr}(g^e_p\mid H^i(X,l))$ for $$p\in{\mathcal P}\setminus{\mathcal S}$$, where $$H^i(X,l)$$ is the $$i$$th $$l$$-adic cohomology group with proper support of the scheme $$X\times_{{\mathbb Z}}\mathbb Q$$ and $$g_p$$ is the “geometric Frobenius” (Theorem 6.1). Let $$X$$ and $$Y$$ be two schemes of finite type over $$\mathbb Z$$. Suppose that the density of the set $$\{p\mid N_X(p)= N_Y(p), p\in{\mathcal P}\}$$ in $${\mathcal P}$$ is equal to 1. There is then a real number $$R$$ such that $$N_X(p^e)= N_Y(p^e)$$ for all $$p\in{\mathcal P}$$ with $$p>R$$ and all $$e\in\mathbb N$$ (Theorem 6.6). Let $B:= \sum_{0\leq i\leq 2d(X)} \dim H^i(X,l)+ \sum_{0\leq i\leq 2d(Y)}\dim H^i(Y, l).$ There is a finite subset $$S$$ of $${\mathcal P}$$ such that if $$N_X(p)\neq N_Y(p)$$ for some $$p$$ in $${\mathcal P}\setminus{\mathcal S}$$, then the density of the set $$\{p\mid N_X(p)= N_Y(p)$$, $$p\in{\mathcal P}\}$$ in $${\mathcal P}$$ doesn’t exceed $$1-1/B^2$$ (Theorem 6.17).
In chapter 7, on the Archimedean properties of $$N_X(p)$$, one finds, in particular, the following information about the function $$N_X$$. Let $$d\in\mathbb Z$$, $$d\geq 0$$, then $\dim X(\mathbb C)\leq d\Leftrightarrow N_X(p)= O(p^d)$ as $$p\to\infty$$ and $\dim X(\mathbb C)\leq d\Leftrightarrow \limsup_{p\to\infty}\, (p^{-d} N_X(p))= r,$ where $$r$$ is the number of the $$\mathbb C$$-irreducible components of dimension $$d$$ of $$X(\mathbb C)$$ (Theorem 1.2). Let $$r_0$$ be the number of the $$\mathbb Q$$-irreducible components of dimension $$d$$, $$d\geq 0$$, of the scheme $$X\times_{\mathbb Z}\mathbb Q$$, there is then a positive constant $$c$$ such that $\sum_{p\leq x,\,p\in{\mathcal P}} N_X(p)= r_0\, li(x^{d+1})+ O(x^{d+1} \exp(- c(\log x)^{1/2}))$ as $$x\to\infty$$, where $$li(u):=\int^u_2(\log t)^{-1}\,dt$$ (Corollary 7.13). The Sato-Tate conjecture is discussed in chapter 8, subdivided into the five sections: equidistribution statements, the Sato-Tate correspondence, an $$l$$-adic construction of the Sato-Tate group, consequences of the Sato-Tate conjecture, examples. The final chapter 9 gives an account of (analogues of) the prime number theorem and the Chebotarev density theorem in higher dimension.
Reviewer: B. Z. Moroz (Bonn)

##### MSC:
 11-02 Research exposition (monographs, survey articles) pertaining to number theory 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 11G35 Varieties over global fields 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G25 Global ground fields in algebraic geometry 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)