# nLab arithmetic differential geometry

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Arithmetic differential geometry is an approach to arithmetic which looks to find analogs of constructions in differential geometry, for example, arithmetic jet spaces. It has been developed primarily by Alexandru Buium.

According to this approach, the classical derivatives of differential geometry are replaced by p-derivations for prime $p$, such as the Fermat quotient

$\delta_p: \mathbb{Z} \to \mathbb{Z}, a \mapsto \delta_p a = \frac{a - a^p}{p}.$

## Comparison with Borger’s absolute geometry

Buium explains that when working with a single prime his approach is consistent with Borger's absolute geometry, which is described as “an algebraization of our analytic theory” (Buium 17, p. 24). However, in the case of multiple primes Borger requires Frobenius lifts to commute, and this diverges from the non-vanishing ‘curvature’ Buium derives from non-commuting lifts. For him, the (“manifold” corresponding to) the integers, $\mathbb{Z}$, is “intrinsically curved”.

## References

Last revised on June 23, 2017 at 07:20:22. See the history of this page for a list of all contributions to it.