Geometry, dynamics, and arithmetic of $S$-adic shifts

Geometry, Dynamics, and Arithmetic of S-adic Shifts

This paper studies geometric and spectral properties of S-adic shifts and their relation to continued fraction algorithms. Pure discrete spectrum for S-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to t...

Arithmetic Geometry Learning Seminar: Adic Spaces

1. January 12th – Motivation (Bhargav Bhatt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. January 26th – Huber rings (Rankeya Datta). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

p-adic Analysis in Arithmetic Geometry

Non–commutative geometry, dynamics, and ∞–adic Arakelov geometry

In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over t...

SOME p-ADIC DIFFERENTIAL EQUATIONS IN ARITHMETIC GEOMETRY

The old: Say X is a family of smooth proper varieties over some base B, all defined over Z and having good reduction modulo p (i.e., everything is still smooth when you reduce the equations modulo p; for given varieties, this is true for all but finitely many p). Then the algebraic de Rham cohomology (i.e., the hypercohomology of the sheaf of algebraic/meromorphic differential forms) of the fib...