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
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.
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