nLab
Fermat quotient

Contents

Idea

For pp a prime number, then the Fermat quotient of any integer aa is the quotient

paa pap \partial_p a \coloneqq \frac{a^p - a}{p}

of the difference between the ppth power of aa and aa itself by pp. By Fermat's little theorem this is indeed an integer, i.e. conversely one has for all aa\in \mathbb{Z} that

a p=a+p( pa) a^p = a + p (\partial_p a)

for a uniquely defined integer pa\partial_p a.

(The pp-power operation () p(-)^p here is the one that restricts to the Frobenius homomorphism after p-localization and the one which is a shadow of the power operations in E-infinity arithmetic geometry (Lurie, remark 2.2.7).)

As the notation is meant to suggest, the Fermat quotient as a map \mathbb{Z} \to \mathbb{Z} from the (underlying set of the) ring of integers to itself is analogous to a derivation. It is not quite an ordinary derivation, but satisfies conditions of what has been called a pp-derivation.

In view of this, in the context of arithmetic differential equations the Fermat quotient is interpreted as an analog in arithmetic geometry of actual derivations in algebraic geometry/complex analytic geometry. For more on this see also at Borger's absolute geometry the section Motivation.

References

Revised on June 23, 2017 07:38:17 by Todd Trimble (24.146.226.222)