This will be a (necessarily) impressionistic discussion of the oft-hidden arithmetic dimension that underlies much of group theory, number theory, topology, and geometry. It will not be about foundations, but in a sense quite the opposite: by way of three specific examples (A, B, and C below), a praise for the dim trichotomy of “Space-Time-Spec Z” as a fertile source of mathematical ideas. The three mental dimensions are essentially
different: spatial cognition requires a common ground — a field or an absolute point — on which to base static constructs of geometry; the dynamics of group actions we visualize by our sense of time, and provides the methodologies of infinitesimal and variational analysis; while Arithmetics, in a loose sense, is the global glue that holds together the different geometric sites.
To what extent may the three dimensions be mixed? The real question is whether, and how, may the arithmetic axis Spec Z interact with the dynamical dimension: does this (a priori) organically rigid object play any meaningful role for the complementary issue of flexibility? What, for instance, breeds rational points? To what extent, in contrast, are finiteness issues of arithmetic forced upon by a hidden deformation theory?
One could speculate ad infinitum on such vague questions and on their potential relevance to several outstanding conjectures of number theory, including a potential clarification on the folklore idea of “the field with one element.” Instead of this, I will discuss some classical arithmetic existence theorems as well as a few basic analogies, outline the two standard viewpoints on Spec Z as Zariski and etale sheaves, and finally,
present in some detail three (out of many) deeper, yet quite specific, examples of the dynamical theme in arithmetic:
(A) Mori’s magnificent employment of the “Frobenius loop” dimension of Z for providing an additional degree of freedom for deforming a curve inside an algebraic variety, producing by bend-and-break rational curves in sufficiently flexible complex varieties. This has led to a proof of the rational connectivity of Fano varieties, as well as to the characterization of projective space as the most flexible of all algebraic varieties.
(B) The arithmetic Hilbert-Samuel theorem of Arakelov theory is a vast generalization of the prototypical Minkowski existence theorem on lattice points in convex bodies. I will present a caricature of its proof, which contains a curious deformation theme, and mention some striking arithmetic consequences, such as equidistribution of orbits of Galois and the dynamical theory of small algebraic points.
(C) An outlook on class-field theory underlain by a Galois dynamical system of ergodic nature.