From monoids to hyperstructures: in search of an absolute arithmetic

2010/06/24 by Alain Connes, Connes, Alain, Caterina Consani +1
Mathematics · #11G40 #14A15 #14G10 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G40 #msc:14A15 #msc:14G10

paper · pdf · doi:10.48550/arxiv.1006.4810

43 pages, 1 figure

arxiv created 2010/06/24 · arxiv updated 2010/06/25

Abstract

We show that the trace formula interpretation of the explicit formulas expresses the counting function N(q) of the hypothetical curve C associated to the Riemann zeta function, as an intersection number involving the scaling action on the adele class space. Then, we discuss the algebraic structure of the adele class space both as a monoid and as a hyperring. We construct an extension R^convex of the hyperfield S of signs, which is the hyperfield analogue of the semifield R₊^max of tropical geometry, admitting a one parameter group of automorphisms fixing S. Finally, we develop function theory over Spec(S) and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over Spec(S).

Related

• Knots, primes and class field theory
• Geometry of the scaling site
• The Scaling Site
• Geometry of the arithmetic site
• The Weil proof and the geometry of the adeles class space