0.3. COMPARISON WITH OTHER THEORIES

XXXI

0.3.8. Mochizuchi's p—adic Teichmiiller theory. An interesting problem

is to find interactions between our theory here and Mochizuchi's p—adic Teichmiiller

theory [111]. Unlike the Tate and Mumford p—adic uniformizations of curves

with (totally) degenerate reduction mod p both our theory and Mochizuchi's can

be viewed, in a certain sense, as "uniformization" theories in the case of good

reduction. As an example of what this might mean in the case of our theory we

propose to see the "inverse" of our arithmetic Manin map of an elliptic curve E (cf.

our discussion above) as a multivalued map from the affine line A

1

into E. This

can be interpreted, in some bold sense, as a sort of uniformization of E.

0.3.9. Ihara's congruence relations. There an interesting possibility that

a profitable link can be established between our approach here and Ihara's beautiful

work on congruence relations (i.e. liftings to characteristic zero of the correspon-

dence IIU IT on a curve of characteristic p, where II is the graph of the Frobenius);

cf. [73], [74]. A hint as to such a link can be seen, for instance, in Lemma 8.16

of this book. On the other hand we would like to point out what we think is an

important difference between our viewpoint here and the viewpoint proposed by

Ihara in a related paper [75]. Our approach, in its simplest form, proposes to see

the operator

5 — 5V : Z — » Z, a

H-»

5a = ,

P

where p is a fixed prime, as an analogue of a derivation with respect to p. In [75]

Ihara proposed to see the map

(0.31) d : Z -» Yl

Z/PZ,

a- (

a

~

Q

mod p)

P

^ '

as an analogue of differentiation for integers and he proposed a series of very inter-

esting conjectures concerning the "zeroes" of the differential of an integer; these con-

jectures are still completely open. The main difference between Ihara's viewpoint

and ours is that we do not consider the reduction mod p of the Fermat quotients but

the Fermat quotients themselves. This allows the possibility of considering iterates

5r

of our 5 which leads to the possibility of considering arithmetic analogues of

higher order differential equations; and indeed most such equations relevant to our

theory will have order 2. On the other hand it doesn't make sense to consider

iterates of Ihara's operator d. One way out of this dilemma would be to find a map

D : Z — • Z which, composed with the canonical projection Z — • JT Z/pZ, yields

Ihara's map d. (Then one could consider iterates

Dr

of D.) Voloch [132] proved,

however, that no such map D can exist at least if one assumes the conjecture that

there are only finitely many Mersenne primes.

0.3.10. Field with one element. The theory proposed in this book seems

to agree with certain aspects of the "myth of the field with one element" emerging

from the work of Kurokawa, Soule, Deninger, Manin, and others; cf. [86], [128],

[98], [46]. For instance the constants a G Z£r, 5a = 0, of the map 5 : Z£r - Z£r

consist of the roots of unity in

Z^r

together with 0; on the other hand the roots of

unity and 0 are, according to the philosophy of the field with one element, precisely

the Fi— points of the affine line where Fi is the "algebraic closure of the field with

one element, Fi". Not all the predictions of the philosophy of the field with one