xxx INTRODUCTION

0.3.6. Drinfeld modules. The theory of Drinfeld modules [52], [62], [130]

exhibits analogies with our theory that deserve being understood. Let us quickly

recall the definition of a Drinfeld module and make a few comments. One starts

with a smooth projective geometrically connected curve X over a finite field Fp™

equipped with a point o o G l . Let A be the ring of regular functions on X\{oo}.

Let i : A — A be a homomorphism into a field A. Let F : A — A be the pm—power

homomorphism. Let A[F] be the non-commutative ring generated by A and a

variable, called F, with relations F • a = F(a) • F = of" • F, a E A (Morally A[F]

is a convolution algebra attached to the action of F on A.) Let e : A[F] —* A be the

ring homomorphism sending e(^2

&iFl)

— a$ and let j : A — • A[F] be the inclusion.

Then a Drinfeld module is an . b rm —algebra homomorphism ip : A — • A[F] such

that eoi\) — i, i\) ^ j o i. The ring A[F] acts on A (with the letter F acting on A as

the endomorphism F and ^4 acting on A by multiplication). Hence A will act on A

via ip. The theory of Drinfeld modules can be viewed therefore as a very non-trivial

example of "algebra / geometry with operators"; it entirely lives in characteristic

p. The ring A is viewed, in this theory, as an analogue of the ring Z of integers.

Its quotient field k plays the role of Q. The completion K of k at oc plays the role

of the real field, R. The completion C ^ of an algebraic closure of K plays the role

of the complex field C. The "first" examples of the theory are Drinfeld modules

\j) : A — » Coo [F] attached to "lattices". Drinfeld modules of ranks 1 and 2 play the

role of one dimensional algebraic groups (multiplicative group and elliptic curves

respectively), viewed as Z—modules (or sometimes modules over rings of integers in

imaginary quadratic fields). They can also be viewed as analogues of formal groups

viewed as Z—modules (or sometimes modules over rings of integers inp—adic fields).

There are analogies between the exponentials in the theory of Drinfeld modules and

our arithmetic Manin maps; both classes of maps have, as kernels, objects that can

be viewed as "lattices". It is a suggestion of Manin (cf. private communication

to the author) that maybe there exists a lifting to characteristic zero of Drinfeld

module theory which is similar in spirit to the arithmetic differential algebra of the

present book.

0.3.7. Dwork's theory. It is interesting to compare our theory of arithmetic

differential equations with what is usually understood by the arithmetic theory of

differential equations (as it appears in the work of Dwork, for instance; cf. [55]).

It turns out that the two theories are "perpendicular" in the very precise sense

that they deal with "differentiation" in two "perpendicular" directions. Indeed

Dwork's theory is about genuine differential equations (i.e. equations involving

d/dt) whereas our theory is about analogues of differential equations (involving,

instead, the Fermat quotient operator S). In a precise sense d/dt and 8 point in

two different directions. Now although the two theories are perpendicular this

doesn't mean they don't interact. Indeed part of Dwork's theory can be rephrased

in terms of crystalline cohomology; cf. the report by Katz [82]. The genuine

differential equations can be read off the Gauss-Manin connection whereas our

arithmetic differential equations are closely related to the action of Frobenius. On

the other hand Gauss-Manin and Frobenius are, as well known, both present in the

crystalline picture. We will actually heavily rely in our proofs on this crystalline

interpretation when we get to discuss the hyperbolic case.