Riemann Hypothesis in Characteristic p in Historical Perspective

After a successful mathematical career, Peter Roquette has now turned to the history of number theory. He has published some important original source material (for example, the correspondence between Hasse and Artin) and some very interesting studies, notably in Contributions to the History of Number Theory in the 20th Century. His most recent book is a mathematical history of the formulation and proof of the Riemann hypothesis for algebraic curves in characteristic \(p\).

The story begins with Emil Artin’s doctoral dissertation, in which he studied quadratic function fields in one variable over a finite field. That is, Artin started with the field \(K=\mathbb{F}_q(x)\) of rational functions with coefficients in a finite field \(\mathbb{F}_q\) with \(q\) elements and studied the field \(F=K(\sqrt{D})\) where \(D\) is a squarefree polynomial in \(\mathbb{F}_q[x]\). He showed that it was possible to study the field \(F\) in close analogy to the theory of quadratic number fields \(\mathbb{Q}(\sqrt{D})\). Just as there was a Dedekind zeta-function attached to the number field, there was a natural way to attach a zeta-function to \(F\). Artin calculated many examples and verified in those cases that something like the Riemann hypothesis was true.

Artin never published anything else on the subject. Roquette argues that this was largely due to David Hilbert’s reaction to a talk Artin gave in Göttingen. But Artin remained interested in the problem and eventually encouraged Helmut Hasse to work on it. This was an inspired choice, since Hasse was an important figure in number theory at the time: there was a circle of talented mathematicians connected to him, which Roquette refers to as “Hasse’s school.” Hasse replaced Artin’s quadratic extensions with arbitrary algebraic extensions of \(\mathbb{F}_q(x)\) and formulated the general problem in that context.

Much of the book focuses on the work of Hasse and his school, who created most of the tools required to formulate and prove the Riemann hypothesis in this context. In fact, Roquette argues that Hasse had all of the necessary tools to complete the proof; his chapter 10 is dedicated to a “virtual proof” that Hasse’s school could have found any time after 1937. Hasse did prove the Riemman hypothesis for curves of genus one, but never did find the general proof. Instead, it was André Weil who gave a proof in the early 1940s.

The modern way to think about this problem is in terms of algebraic geometry over a finite field. (This was already clear above when I described it as the Riemann hypothesis for curves, rather than for function fields in one variable.) One of the fascinating things about the book is the description of how Hasse’s school developed the structures that I know from algebraic geometry in the language of algebraic function fields. For example, Hasse’s approach involved studying the ring of endomorphisms of the Jacobian of a function field, but there was no Jacobian variety to work with.

Weil’s eventual success was the result of his recasting the entire theory in terms of algebraic geometry over finite fields and being able to deploy the work of the Italian school of algebraic geometry, especially the work of Severi. The first version of his proof mostly just asserted that Severi’s results worked in characteristic \(p\). To fully support that claim required Weil to rewrite the Foundations of Algebraic Geometry (the title of his 1946 book containing the results).

Roquette only mentions, without details, the rest of the story. Weil made conjectures that included a Riemann hypothesis for varieties of any dimension over a finite field, and those conjectures were eventually proved by Pierre Deligne. That story will need another book.

Roquette makes full use of Hasse’s remarkably extensive correspondence to fill in the details of the story, which allows the reader to follow closely as the mathematics is developed. The approach is mostly “internal,” but the correspondence allows us to see a little bit of the personalities and how they were affected by their non-mathematical context. The result is a fascinating and useful book.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.