*(English)*Zbl 0958.11004

This book has its origin – as explicitly stated by the authors in its introduction – in the remarkable discovery by *A. M. Odlyzko* [The ${10}^{20}$th zero of the Riemann zeta-function and 70 millions of its neighbors, ATT Bell Laboratories, 1989; Math. Comput. 48, 273-308 (1987; Zbl 0615.10049)], carried out by means of numerical experiments, that the distribution of the spacings between successive (nontrivial) zeros of the Riemann zeta-function is empirically the same as the so-called GUE measure, which is a probability measure on $\mathbb{R}$ that arises in random matrix theory. Odlyzko’s numerical experiments had been, on its turn, inspired by work of *H. L. Montgomery* [Analytic Number Theory, Proc. Symp. Pure Math. 24, St. Louis Univ. Missouri 1972, 181-193 (1973; Zbl 0268.10023)], who had determined the pair correlation distribution between the zeros in a restricted range, and had already noted the compatibility of his results with the GUE prediction.

Recent new results have enforced the belief, explicitly conjectured in the book, that the distribution of the spacings between zeros of the Riemann zeta-function and also of quite general automorphic $L$-functions over $\mathbb{Q}$ are, in fact, given by the GUE measure, satisfying the by now called Montgomery-Odlyzko law. The authors recognize that proving this in such generality, for arbitrary number fields, seems well beyond the range of existing techniques.

In the book, they restrain this scope to the case of finite fields, namely the authors establish the Montgomery-Odlyzko law for wide classes of zeta and $L$-functions over finite fields.

To fix ideas, the authors start by considering, already at the introduction, a special case, namely that of a finite field, ${\mathbb{F}}_{q}$, and a proper, smooth, geometrically connected curve $C/{\mathbb{F}}_{q}$, of genus $g$ (the corresponding zeta function was introduced by E. Artin in his thesis). After reviewing in this example the defining of the normalized spacings between the zeros of the zeta function, they show that the spacing measure in this case is the probability measure which gives mass $1/2g$ to each of the $2g$ normalized spacings. Of course, the rest of the 420 page book is not that easy, but this example gives a clue to understand the essential phenomena, what is it all about, as the authors themselves like to remark. This pedagogical attitude is manifest throughout the book.

They go on by recalling the definition of the GUE measure on $\mathbb{R}$ (that is, the Wigner measure, for physicists): the limit for big $N$ of the probability measure $A\in U\left(N\right)$. Using the Kolmogorov-Smirnov discrepancy function, one is able to have a numerical measure of how close two probability measures in $\mathbb{R}$ are. The generalized Sato-Tate conjecture follows, as well as a number of conjectures, in particular, involving the low-lying zeros of $L$-functions of elliptic curves over $\mathbb{Q}$.

In Chapter 1 we find statements of the main result in the book. It deals with the measures attached to spacings of eigenvalues and with the expected values of spacing measures. Three main theorems on the existence, universality and discrepancy for limits of expected values of spacing measures are given there, with some applications, corollaries, and an appendix on the continuity properties of the $i$th eigenvalue as a function on $U\left(N\right)$.

Chapter 2 deals with a reformulation of the main results, under a different viewpoint, and provides several discussions on the combinatorics of spacings of finitely many points on a line. Chapter 3 deals with reduction steps in proving the main theorems, while the next chapters are devoted, respectively, to test functions (Ch. 4), the Haar measure (Ch. 5), tail estimates, a determinant-trace inequality, and multi-eigenvalue location measures (Ch. 6), large $N$ limits and Fredholm determinants (Ch. 7), the case of several variables, with corresponding large $N$ scaling limits (Ch. 8), equidistribution, with two versions of Deligne’s equidistribution theorem (Ch. 9), monodromy of families of curves and monodromy of some other families (Chs. 10, 11), GUE discrepancies in various families of curves, abelian varieties, hypersurfaces, and Kloosterman sums (Ch. 12), and finally, on the distribution of low-lying Frobenius eigenvalues in various families, of curves, abelian varieties, hypersurfaces, and Kloosterman sums, according to the measures corresponding to different groups of the $G\left(N\right)$, $USp$, and $SO$ types, with a passage to the large $N$ limit (Ch. 13).

The book finishes with two appendices, on densities and large $N$ limits, and on some graphs, how they were drawn and what they show, respectively. I’ve missed some more of these graphs, for different explicit examples, and also some specific applications (e.g. in physics, to honor Wigner’s pioneering discoveries) of the results obtained in the book, but this is just a personal thought.

To summarize, a very complete and useful reference on the subject, by two well known specialists in the field.

##### MSC:

11-02 | Research monographs (number theory) |

14-02 | Research monographs (algebraic geometry) |

11M06 | $\zeta \left(s\right)$ and $L(s,\chi )$ |

11Y35 | Analytic computations |

14D05 | Structure of families |

11G25 | Varieties over finite and local fields |

14G10 | Zeta-functions and related questions |

82B44 | Disordered systems (equilibrium statistical mechanics) |

60F99 | Limit theorems (probability) |