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.

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) |