Foundations of Arithmetic Differential Geometry

  title={Foundations of Arithmetic Differential Geometry},
  author={Alexandru Buium},
  • A. Buium
  • Published 9 June 2017
  • Mathematics

Arithmetic differential geometry in the arithmetic PDE setting, I: connections

This is the first in a series on papers developing an arithmetic PDE analogue of Riemannian geometry. The role of partial derivatives is played by Fermat quotient operations with respect to several

Semi-galois categories II: An arithmetic analogue of Christol's theorem

Arithmetic Analogues of Hamiltonian Systems

  • A. Buium
  • Mathematics
    Integrable Systems and Algebraic Geometry
  • 2020
The paper reviews various arithmetic analogues of Hamiltonian systems and presents some new facts suggesting ways to relate/unify these examples.

Lie invariant Frobenius lifts on linear algebraic groups

We show that if $G$ is a linear algebraic group over a number field and if $G$ is not a torus then for all but finitely many primes $p$ the $p$-adic completion of $G$ does not possess a Frobenius

Purely arithmetic PDE's over a p-adic field I: delta-characters and delta-modular forms

A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be

Curvature in noncommutative geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past 10 years. This new episode in noncommutative geometry started when a Gauss-Bonnet

Arithmetic Levi–Cività connection

  • A. Buium
  • Mathematics
    Selecta Mathematica
  • 2019
This paper is part of a series of papers where an arithmetic analogue of classical differential geometry is being developed. In this arithmetic differential geometry functions are replaced by integer

Invariant Frobenius lifts and deformation of the Hasse invariant

We show that the $p$-adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts whose "normalized" action on $1$-forms preserves mod $p$ the space of invariant



Collected Papers

THIS volume is the first to be produced of the projected nine volumes of the collected papers of the late Prof. H. A. Lorentz. It contains a number of papersnineteen in all, mainly printed

Differential characters of abelian varieties overp-adic fields

Arithmetic Laplacians

Les variétés sur le corps à un élément

  • Mosc. Math. J
  • 2004

Schemes over F1 and zeta functions

  • Compositio Mathematica, 146 (6),
  • 2010

Arithmetic Differential Equations

Main concepts and results: Preliminaries from algebraic geometry Outline of $\delta$-geometry General theory: Global theory Local theory Birational theory Applications: Spherical correspondences Flat

Arithmetic differential equations on $GL_n$, III: Galois groups

Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned

Le Groupe Fondamental de la Droite Projective Moins Trois Points

Le present article doit beaucoup a A. Grothendieck. Il a invente la philosophie des motifs, qui est notre fil directeur. Il y a quelques cinq ans, il m’a aussi dit, avec force, que le complete

Knots and Primes