nLab arithmetic differential geometry

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”.

Related pages

• arithmetic jet space

• Joyal delta-ring

• Borger's absolute geometry

• D-geometry

• arithmetic geometry

References

• Alexandru Buium, Foundations of arithmetic differential geometry, 2017, AMS, Mathematical Surveys and Monographs Vol. 222, (AMS, Preface and Introduction).

• Alexandru Buium, Arithmetic differential geometry, May 19, 2017, talk slides