Axiomatics as a Functional Strategy for Complex Proofs: The Case of Riemann Hypothesis

Abstract

My purpose is to comment some claims of André Weil (1906–1998) in his letter of March 26, 1940 to his sister Simone, in particular, the following quotation: “it is essential, if mathematics is to stay as a whole, to provide a unification, which absorbs in some simple and general theories all the common substrata of the diverse branches of the science, suppressing what is not so useful and necessary, and leaving intact what is truly the specific detail of each big problem. This is the good one can achieve with axiomatics”. For Weil (and Bourbaki), the main problem was to find “strategies” for finding complex proofs of “big problems”. For that, the dialectic balance between general structures and specific details is crucial. I will focus on the fact that, for these creative mathematicians, the concept of structure is a functional concept, which has always a “strategic” creative function. The “big problem” here is Riemann Hypothesis (RH). Artin, Schmidt, Hasse, and Weil introduced an intermediary third world between, on the one hand, Riemann original hypothesis on the non-trivial zeroes of the zeta function in analytic theory of numbers, and, on the other hand, the algebraic theory of compact Riemman surfaces. The intermediary world is that of projective curves over finite fields of characteristic $p\ge 2$. RH can be translated in this context and can be proved using sophisticated tools of algebraic geometry (divisors, Riemann-Roch theorem, intersection theory, Severi-Castelnuovo inequality) coupled with the action of Frobenius maps in characteristic $p\ge 2$. Recently, Alain Connes proposed a new strategy and constructed a new topos theoretic framework à la Grothendieck where Weil’s proof could be transferred by analogy back to the original RH.