The Riemann hypothesis in characteristic \(p\) in historical perspective
The book deals with the history of the zeros of the zeta function of a global function field. Let \(k={\mathbb F}_q\) be the finite field of \(q=p^r\) elements and let \(K/k\) be a function field with exact field of constants \(k\). The zeta function of \(K\) is the function \[\zeta_K(s)= \prod_P \frac{1}{1-|P|^{-s}}=\sum_A |A|^{-s}\] where \(P\) ranges over the primes of \(K\) and \(A\) over the integer divisors of \(K\). Here \(|P|\) denotes the number of elements in the residue field corresponding to \(P\) and \(s\in {\mathbb C}\) satisfies \(\mathrm{Re}( s)>1\). The function \(\zeta_K(s)\) can be extended to a meromorphic function of \({\mathbb C}\) with poles of order \(1\) at \(s=0\) and \(s=1\). The Riemann hypothesis (RH) says that all zeros of \(\zeta_K(s)\) belong to the line with real part \(\mathrm{Re}( s)=\frac{1}{2}\). The RH was considered by Riemann for the field of rational numbers \({\mathbb Q}\) and the generalization of the zeta function to a general number field is known as the Dedekind zeta function of the field. The RH is an open problem in characteristic \(0\). In characteristic \(p>0\) the RH was finally settled by \textit{André Weil} in 1940--1941 [C. R. Acad. Sci., Paris 210, 592--594 (1940; Zbl 0023.29401); Proc. Natl. Acad. Sci. USA 27, 345--347 (1941; Zbl 0061.06406)]. Roquette's book is not just the history of Weil's proof of the RH. It is much more. It is a detailed and accurate history of the RH in characteristic \(p\) and it is also a description on how and when the main protagonists at the time made their corresponding contributions towards the proof of the RH. It is worth to point out that the author met most of the protagonists of the solution of the problem. The book is very pleasant to read and should be consulted by any one interested in history, in function fields or in general in the RH in any characteristic. The book can be used by specialists and by non-specialists as a brief but very interesting introduction to function fields including its relation with algebraic geometry. Along the manuscript, the author provides the source of the facts analyzed, principally the letters among the protagonists, of each of the main steps and contributions to the solution of the RH in function fields. At the end of each chapter, there is a summary of the topics discussed in the chapter. The summaries give a good abstract of the book. In the first chapter, the book describes how the story of the RH began at Göttingen in 1921 when \textit{Emil Artin} gave a talk in order to report on his PhD thesis [Math. Z. 19, 153--206, 207--246 (1924; JFM 50.0107.01)] and further work. However, David Hilbert interrupted Artin frequently and said that he had the impression that everything was trivial. The part of the Artin's talk interrupted by Hilbert pointed out in the direction of the RH. After this incident, Artin turned to other problems and never took up the RH. The RH was studied by Helmut Hasse beginning in 1932. Hasse provided the proof of the RH for elliptic function fields, that is, the genus one case. However, Hasse did not succeed for genus greater than one. The author gives a proof of the RH in this book, showing that all the elements for the proof were available at that time (approximately 1937). In Chapter 2 there is an introduction to the main facts in order to establish the RH. Chapter 3 describes Artin's thesis and can be used by non-specialists as an overview of global function fields, particularly quadratic extensions. The Artin-Hilbert affair is described in detail here. Chapter 4 tackles the contributions of Friedrich Karl Schmidt in the theory of function fields and in particular their zeta functions. Again, this chapter may well be used as an overview of the theory of function fields. Chapters 5 and 6 describe the interest of Hasse on the RH and his collaboration with Harold Davenport. The breakthrough for the proof of the RH were \textit{H. Hasse}'s two proofs in the elliptic case [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1933, 253--262 (1933; Zbl 0007.39702); J. Reine Angew. Math. 175, 55--62 (1936; Zbl 0014.14903); J. Reine Angew. Math. 175, 69--88 (1936; Zbl 0014.24901); J. Reine Angew. Math. 175, 193--208 (1936; Zbl 0014.24902)]. This achievement is narrated in Chapter 7. Further developments on elliptic function fields are described in Chapter 8 which may be used as an introduction to the endomorphism ring of elliptic functions fields, Jacobians, the Hasse's invariant, which controls the existence of unramified extensions of degree \(p\) and a brief description of complex multiplication and class field theory. Chapter 9 deals with the interrelation between Hasse and Weil trying to generalize Hasse's result for elliptic function fields to function fields of genus grater than one. Chapter 10 is the proof of the RH for general global function fields given by the author using what was already known in 1937. He calls it a ``virtual'' proof. Of course this is not a virtual proof but a nice algebraic proof that might have anticipated Weil's proof of the RH. Chapter 12 is the story of Weil's two proofs of the RH during 1940--1941, Weil's results were finally published in three books during 1946--1948 [\textit{A. Weil}, Foundations of algebraic geometry. New York: American Mathematical Society (1946; Zbl 0063.08198); 2nd ed. (1962; Zbl 0168.18701); Sur les courbes algébriques et les variétés qui s'en déduisent. Paris: Hermann \& Cie. (1948; Zbl 0036.16001); Variétés abéliennes et courbes algébriques. Paris: Hermann \& Cie. (1948; Zbl 0037.16202)]. In the appendix, \textit{Enrico Bombieri}'s proof of the RH is described [Lect. Notes Math. 383, 234--241 (1974; Zbl 0307.14011)] .
zeta function in characteristic \(p\), Emil Artin
scientific article; zbMATH DE number 6928403