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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 as characteristic values of ), such that
for 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.
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).
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:
for 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:
for 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 :
are entire in , by which we mean that
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:
is entire in . Then for any real , there exist a finite number of elements such that
for all and some .
Selberg’s early work was focused on the study of the Riemann zeta function . In 1942, Selberg showed that a positive fraction of the zeroes of this function lie on the critical line . Apart from improvements in the fraction (the best value currently being a little over 40%, a result of Conrey), this is still one of the strongest partial results we have towards the Riemann hypothesis. (I discuss Selberg’s result, and the method of mollifiers he introduced there, in a little more detail after the jump.)
In working on the zeta function, Selberg developed two powerful tools which are still used routinely in analytic number theory today. The first is the method of mollifiers to smooth out the magnitude oscillations of the zeta function, making the (more interesting) phase oscillation more visible. The second was the method of the Selberg sieve, which is a particularly elegant choice of sieve which allows one to count patterns in almost primes (and hence to upper bound patterns in primes) quite accurately. Variants of the Selberg sieve were a crucial ingredient in, for instance, the recent work of Goldston-Yıldırım-Pintz on prime gaps, as well as the work of Ben Green and myself on arithmetic progressions in primes. (I discuss the Selberg sieve, as well as the Selberg symmetry formula below, in my post on the parity problem. Incidentally, Selberg was the first to formalise this problem as a significant obstruction in sieve theory.)
For all of these achievements, Selberg was awarded the Fields Medal in 1950. Around that time, Selberg and Erdős also produced the first elementary proof of the prime number theorem. A key ingredient here was the Selberg symmetry formula, which is an elementary analogue of the prime number theorem for almost primes.
But perhaps Selberg’s greatest contribution to mathematics was his discovery of the Selberg trace formula, which is a non-abelian generalisation of the Poisson summation formula, and which led to many further deep connections between representation theory and number theory, and in particular being one of the main inspirations for the Langlands program, which in turn has had an impact on many different parts of mathematics (for instance, it plays a role in Wiles’ proof of Fermat’s last theorem). For an introduction to the trace formula, its history, and its impact, I recommend the survey article of Arthur.
Other major contributions of Selberg include the Rankin-Selberg theory connecting Artin L-functions from representation theory to the integrals of automorphic forms (very much in the spirit of the Langlands program), and the Chowla-Selberg formula relating the Gamma function at rational values to the periods of elliptic curves with complex multiplication. He also made an influential conjecture on the spectral gap of the Laplacian on quotients of by congruence groups, which is still open today (Selberg had the first non-trivial partial result). As an example of this conjecture’s impact, Selberg’s eigenvalue conjecture has inspired some recent work of Sarnak-Xue, Gamburd, and Bourgain-Gamburd on new constructions of expander graphs, and has revealed some further connections between number theory and arithmetic combinatorics (such as sum-product theorems); see this announcement of Bourgain-Gamburd-Sarnak for the most recent developments (this work, incidentally, also employs the Selberg sieve). As observed by Satake, Selberg’s eigenvalue conjecture and the more classical Ramanujan-Petersson conjecture can be unified into a single conjecture, now known as the Ramanujan-Selberg conjecture; the eigenvalue conjecture is then essentially an archimedean (or “non-dyadic“) special case of the general Ramanujan-Selberg conjecture. (The original (dyadic) Ramanujan-Petersson conjecture was finally proved by Deligne-Serre, after many important contributions by other authors, but the non-dyadic version remains open.)