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Let be a quasiprojective variety defined over a finite field , thus for instance could be an affine variety

where is -dimensional affine space and are a finite collection of polynomials with coefficients in . Then one can define the set of -rational points, and more generally the set of -rational points for any , since can be viewed as a field extension of . Thus for instance in the affine case (1) we have

The Weil conjectures are concerned with understanding the number

of -rational points over a variety . The first of these conjectures was proven by Dwork, and can be phrased as follows.

Theorem 1 (Rationality of the zeta function)Let be a quasiprojective variety defined over a finite field , and let be given by (2). Then there exist a finite number of algebraic integers (known ascharacteristic valuesof ), such thatfor all .

After cancelling, we may of course assume that for any and , and then it is easy to see (as we will see below) that the become uniquely determined up to permutations of the and . These values are known as the *characteristic values* of . Since is a rational integer (i.e. an element of ) rather than merely an algebraic integer (i.e. an element of the ring of integers of the algebraic closure of ), we conclude from the above-mentioned uniqueness that the set of characteristic values are invariant with respect to the Galois group . To emphasise this Galois invariance, we will not fix a specific embedding of the algebraic numbers into the complex field , but work with all such embeddings simultaneously. (Thus, for instance, contains three cube roots of , but which of these is assigned to the complex numbers , , will depend on the choice of embedding .)

An equivalent way of phrasing Dwork’s theorem is that the (-form of the) zeta function

associated to (which is well defined as a formal power series in , at least) is equal to a rational function of (with the and being the poles and zeroes of respectively). Here, we use the formal exponential

Equivalently, the (-form of the) zeta-function is a meromorphic function on the complex numbers which is also periodic with period , and which has only finitely many poles and zeroes up to this periodicity.

Dwork’s argument relies primarily on -adic analysis – an analogue of complex analysis, but over an algebraically complete (and metrically complete) extension of the -adic field , rather than over the Archimedean complex numbers . The argument is quite effective, and in particular gives explicit upper bounds for the number of characteristic values in terms of the complexity of the variety ; for instance, in the affine case (1) with of degree , Bombieri used Dwork’s methods (in combination with Deligne’s theorem below) to obtain the bound , and a subsequent paper of Hooley established the slightly weaker bound purely from Dwork’s methods (a similar bound had also been pointed out in unpublished work of Dwork). In particular, one has bounds that are uniform in the field , which is an important fact for many analytic number theory applications.

These -adic arguments stand in contrast with Deligne’s resolution of the last (and deepest) of the Weil conjectures:

Theorem 2 (Riemann hypothesis)Let be a quasiprojective variety defined over a finite field , and let be a characteristic value of . Then there exists a natural number such that for every embedding , where denotes the usual absolute value on the complex numbers . (Informally: and all of its Galois conjugates have complex magnitude .)

To put it another way that closely resembles the classical Riemann hypothesis, all the zeroes and poles of the -form lie on the critical lines for . (See this previous blog post for further comparison of various instantiations of the Riemann hypothesis.) Whereas Dwork uses -adic analysis, Deligne uses the essentially orthogonal technique of ell-adic cohomology to establish his theorem. However, ell-adic methods can be used (via the Grothendieck-Lefschetz trace formula) to establish rationality, and conversely, in this paper of Kedlaya p-adic methods are used to establish the Riemann hypothesis. As pointed out by Kedlaya, the ell-adic methods are tied to the intrinsic geometry of (such as the structure of sheaves and covers over ), while the -adic methods are more tied to the *extrinsic* geometry of (how sits inside its ambient affine or projective space).

In this post, I would like to record my notes on Dwork’s proof of Theorem 1, drawing heavily on the expositions of Serre, Hooley, Koblitz, and others.

The basic strategy is to control the rational integers both in an “Archimedean” sense (embedding the rational integers inside the complex numbers with the usual norm ) as well as in the “-adic” sense, with the characteristic of (embedding the integers now in the “complexification” of the -adic numbers , which is equipped with a norm that we will recall later). (This is in contrast to the methods of ell-adic cohomology, in which one primarily works over an -adic field with .) The Archimedean control is trivial:

Proposition 3 (Archimedean control of )With as above, and any embedding , we havefor all and some independent of .

*Proof:* Since is a rational integer, is just . By decomposing into affine pieces, we may assume that is of the affine form (1), then we trivially have , and the claim follows.

Another way of thinking about this Archimedean control is that it guarantees that the zeta function can be defined holomorphically on the open disk in of radius centred at the origin.

The -adic control is significantly more difficult, and is the main component of Dwork’s argument:

Proposition 4 (-adic control of )With as above, and using an embedding (defined later) with the characteristic of , we can find for any real a finite number of elements such thatfor all .

Another way of thinking about this -adic control is that it guarantees that the zeta function can be defined *meromorphically* on the entire -adic complex field .

Proposition 4 is ostensibly much weaker than Theorem 1 because of (a) the error term of -adic magnitude at most ; (b) the fact that the number of potential characteristic values here may go to infinity as ; and (c) the potential characteristic values only exist inside the complexified -adics , rather than in the algebraic integers . However, it turns out that by combining -adic control on in Proposition 4 with the trivial control on in Proposition 3, one can obtain Theorem 1 by an elementary argument that does not use any further properties of (other than the obvious fact that the are rational integers), with the in Proposition 4 chosen to exceed the in Proposition 3. We give this argument (essentially due to Borel) below the fold.

The proof of Proposition 4 can be split into two pieces. The first piece, which can be viewed as the number-theoretic component of the proof, uses external descriptions of such as (1) to obtain the following decomposition of :

Proposition 5 (Decomposition of )With and as above, we can decompose as a finite linear combination (over the integers) of sequences , such that for each such sequence , the zeta functionsare entire in , by which we mean that

as .

This proposition will ultimately be a consequence of the properties of the Teichmuller lifting .

The second piece, which can be viewed as the “-adic complex analytic” component of the proof, relates the -adic entire nature of a zeta function with control on the associated sequence , and can be interpreted (after some manipulation) as a -adic version of the Weierstrass preparation theorem:

Proposition 6 (-adic Weierstrass preparation theorem)Let be a sequence in , such that the zeta functionis entire in . Then for any real , there exist a finite number of elements such that

for all and some .

Clearly, the combination of Proposition 5 and Proposition 6 (and the non-Archimedean nature of the norm) imply Proposition 4.

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