Independence of zeros of L-functions over function fields

My paper “The large sieve, monodromy, and zeta functions of algebraic curves, II: independence of the zeros” has just been published online by International Math. Research Notices (the title really should be shorter… but I wanted to emphasize the link with the earlier paper in the series, though that one didn’t know it would have a little brother, so its title does not mention that it is number 1). Note in passing that a paper is masculine for me simply because this is so in French (“un article, un papier”).

The motivation of this work is to try to understand a bit more one famous conjecture about the zeros of the Riemann zeta function, and about those of other L-functions more generally. In the “simplest” case of ζ(s), it is expected that the Riemann Hypothesis holds (of course), so that the zeros (counted with multiplicity) can be written

where we order for convenience the ordinates in increasing order

(choosing an arbitrary ordering of multiple zeros, in case it happens that a zero is not simple). Then, the conjecture is that

are Q-linearly independent. This implies in particular that all zeros are simple, but it is of course much stronger.

If you think about it, this may look like a strange and somewhat arbitrary conjecture. The point is of course that it does turn out naturally in various problems. I know of at least two instances:

(1) Ingham showed that it is incompatible with the conjecture

apparently stated by Mertens; this inequality was of some interest because it implies (very easily) the Riemann Hypothesis. Of course, Ingham’s result made it very doubtful that it could hold, and later Odlyzko and te Riele succeeded in proving that it does not. (Here, μ is the Möbius function). For further closely related recent results, see for instance the work of Ng.

(2) The conjecture has been extended to state that all non-negative ordinates of zeros of primitive Dirichlet L-functions are Q-linearly independent; this was used (I don’t know if it had been introduced before) by Rubinstein and Sarnak in their very nice paper studying the Chebychev bias and generalizations of it. The Chebychev bias refers to the apparent fact that, usually, there are more primes p<X which are congruent to 3 modulo 4 than congruent to 1 modulo 4; it is a strange property, which is not literally true for all X, but should be so only in some sense of logarithmic density, as explained by Rubinstein and Sarnak. It has its unlikely source in the fact that the primes are best counted with a weight log p, and together with all their powers p^k; the counting function incorporating those two changes exhibits no bias, but moving from it to the counting function for primes leads to a discrepancy having to do with squares of primes, and for any modulus q, to a recurrent excess of primes which are non-squares modulo q compared with those which are squares modulo q.

The way I tried to understand this somewhat better is well established: look at what happens for zeta and L-functions over finite fields. There we have two advantages, which have already been used quite often: first, the Riemann Hypothesis is known (by the work of Deligne in general), and second, the L-functions are polynomials (in p^-s) with integral coefficients, instead of analytic functions with infinitely many zeros. In addition, these polynomials have a spectral interpretation: this is the basis of the Katz-Sarnak study of zeta functions and symmetry, motivated by the conjectures relating Random Matrix Theory and the zeta function.

The basic example of zeta functions over finite fields are those of algebraic curves; to be concrete, say we have a polynomial h of degree 2g+1, and an odd prime p; we can then look at the curve C with affine equation

over the field with p elements. Its zeta function (or rather the one of the projective version of this curve) is defined by the formal power series

involving the number of points on the curve over all finite fields of characteristic p. Then it is known that this zeta function is the Taylor expansion of a rational function given by

where the polynomial P_C is of degree 2g (g is the genus of the curve), has integral coefficients, and can be factored as follows:

where

The first identity corresponds to the functional equation, and the last fact is the Riemann Hypothesis in this context (to see why this is so, write T=p^-s and look for the real parts of complex zeros s); it was first proved for curves by André Weil, though special cases go all the way back to Gauss!

So what is the analogue of the Q-linear independence conjecture here? It is not hard to convince oneself that the right analogue is that, if we write

for all j, then the question is whether

are (or not) Q-linearly independent. The restriction to only half the values of α is because of the “functional equation” recalled above that relates α_j with α_2g-j, while the appearance of 1, has to do with the “transfer” from multiplicative to additive forms of the variable.

Now, the point of my paper is that one can indeed show that this independence holds “for most curves”, in a certain sense. Such a restriction is necessary: it is perfectly possible to construct curves where the independence does not hold (I give various examples in the paper). Moreover, I show that one can look at independence involving roots of more than a single curve: so not only are the zeros independent, but they tend to be independent of those of other curves. This implies, in particular, that there is typically no bias between the number of points of one curve and another: if C and D are two “generically chosen” algebraic curves of the same genus over the same finite field with p elements, then

(in fact, properly normalized, the difference is asymptotically normally distributed).

I will just say a few words about the techniques involved: as the title indicates, the basic tool is the large sieve (in a suitable form, introduced in the older brother paper), and the main arithmetico-geometric input comes from monodromy computations. In the cases I treat in detail, those come from an unpublished result of J-K. Yu, recently reproved by C. Hall. The main idea is to first show that one can deduce the desired independence statements, for a given curve (or tuple of curves) provided the splitting field of the zeta function(s) is as large as possible. This part of the argument involves fairly elementary (but cute) theory of representations of finite groups (which go back to results of Girstmair and others). Then one is reduced to showing that the maximality of splitting fields occurs most of the time, and this is provided by the sieve (qualitative forms of such a statement were first proved around 1994 by N. Chavdarov).

Actually, N. Katz pointed out (after the first version was written) that one could use, bypassing any consideration of the splitting field, the properties of the so-called Frobenius tori of Serre. This essentially means that the large sieve, per se, can be avoided, and replaced by an application of the uniform and effective Chebotarev density theorem. Still, I kept the original approach (in addition to discussing briefly this other method) for two reasons:

(1) It also leads quickly to the fact that the α_j themselves are Q-linearly independent (Frobenius tori can not detect this condition); I don’t know any application of this fact, but it seems interesting in principle.

(2) Frobenius tori depend rather strongly on p-adic properties of the zeros; this means that the method does not work if, instead, we try to answer the following question: given a “random” element of SL(n,Z), do the roots of its characteristic polynomial satisfy any multiplicative relation? The “splitting field” technique, in suitable senses of the word “random” (e.g., random walks with respect to a finite generating set) can be made to prove such a statement, using the large sieve technique developed in my recent book.

(3) OK, here’s a third reason that may not apply so much given the possible use of the Frobenius tori: N. Katz had asked a number of times (including, in a truly epiphanic moment for me, at the end of his lecture during the Newton Institute workshop on Random Matrices and L-functions in July 2004) about the possible analogue and meaning, for the usual L-functions over number fields, of the generic maximality of splitting fields of L-functions of algebraic curves, as then known from the qualitative results of Chavdarov; I thought that maybe this link with independence could be a partial answer — but whether it is or not, it is not so clear now.