An algebraic approach to Connes--Consani arithmetic sites.

For a large number N, the Riemann Hypothesis would give a very precise estimate of the amount of prime numbers smaller than N. Because prime numbers are the foundation of number theory, many mathematical problems depend on it, which is why the Riemann Hypothesis is considered to be one of the most important unresolved problems in mathematics. In the 1940's, André Weil, a famous mathematician and brother of philosopher Simone Weil, has proved a variant of the Riemann Hypothesis regarding estimation problems for a very specific type of polynomials. His proof was related to geometry, more specifically to the study of curves. In a recent series of papers, Alain Connes and Caterina Consani have described an approach to the Riemann Hypothesis by introducing and studying the Arithmetic Site. This is a new geometric object describing the distribution of the prime numbers, constructed with contemporary mathematical techniques. It has properties similar to that of a curve in geometry, so the hope is that eventually Weil's proof can be translated to a proof of the original Riemann Hypothesis. While Connes and Consani focus on a tropical geometry point of view, we will construct alternatives that allow for a more algebraic point of view. In particular, we want to relate their approach to the study of noncommutative algebras, a subject for which the University of Antwerp is well-known.

Keywords:ALGEBRAIC GEOMETRY

Disciplines:Algebraic geometry, Associative rings and algebras, Category theory, homological algebra, Number theory