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for some ambient dimension , and some finite number of polynomials . In order to reduce the number of subscripts later on, let us say that has complexity at most if , , and the degrees of the are all less than or equal to . Note that we do not require at this stage that be irreducible (i.e. not the union of two strictly smaller varieties), or defined over , though we will often specialise to these cases later in this post. (Also, everything said here can also be applied with almost no changes to projective varieties, but we will stick with affine varieties for sake of concreteness.)
One can consider two crude measures of how “big” the variety is. The first measure, which is algebraic geometric in nature, is the dimension of the variety , which is an integer between and (or, depending on convention, , , or undefined, if is empty) that can be defined in a large number of ways (e.g. it is the largest for which the generic linear projection from to is dominant, or the smallest for which the intersection with a generic codimension subspace is non-empty). The second measure, which is number-theoretic in nature, is the number of -points of , i.e. points in all of whose coefficients lie in the finite field, or equivalently the number of solutions to the system of equations for with variables in .
These two measures are linked together in a number of ways. For instance, we have the basic Schwarz-Zippel type bound (which, in this qualitative form, goes back at least to Lemma 1 of the work of Lang and Weil in 1954).
Proof: (Sketch) For the purposes of exposition, we will not carefully track the dependencies of implied constants on the complexity , instead simply assuming that all of these quantities remain controlled throughout the argument. (If one wished, one could obtain ineffective bounds on these quantities by an ultralimit argument, as discussed in this previous post, or equivalently by moving everything over to a nonstandard analysis framework; one could also obtain such uniformity using the machinery of schemes.)
We argue by induction on the ambient dimension of the variety . The case is trivial, so suppose and that the claim has already been proven for . By breaking up into irreducible components we may assume that is irreducible (this requires some control on the number and complexity of these components, but this is available, as discussed in this previous post). For each , the fibre is either one-dimensional (and thus all of ) or zero-dimensional. In the latter case, one has points in the fibre from the fundamental theorem of algebra (indeed one has a bound of in this case), and lives in the projection of to , which is a variety of dimension at most and controlled complexity, so the contribution of this case is acceptable from the induction hypothesis. In the former case, the fibre contributes -points, but lies in a variety in of dimension at most (since otherwise would contain a subvariety of dimension at least , which is absurd) and controlled complexity, and so the contribution of this case is also acceptable from the induction hypothesis.
One can improve the bound on the implied constant to be linear in the degree of (see e.g. Claim 7.2 of this paper of Dvir, Kollar, and Lovett, or Lemma A.3 of this paper of Ellenberg, Oberlin, and myself), but we will not be concerned with these improvements here.
Without further hypotheses on , the above upper bound is sharp (except for improvements in the implied constants). For instance, the variety
where are distict, is the union of distinct hyperplanes of dimension , with and complexity ; similar examples can easily be concocted for other choices of . In the other direction, there is also no non-trivial lower bound for without further hypotheses on . For a trivial example, if is an element of that does not lie in , then the hyperplane
clearly has no -points whatsoever, despite being a -dimensional variety in of complexity . For a slightly less non-trivial example, if is an element of that is not a quadratic residue, then the variety
which is the union of two hyperplanes, still has no -points, even though this time the variety is defined over instead of (by which we mean that the defining polynomial(s) have all of their coefficients in ). There is however the important Lang-Weil bound that allows for a much better estimate as long as is both defined over and irreducible:
Theorem 2 (Lang-Weil bound) Let be a variety of complexity at most . Assume that is defined over , and that is irreducible as a variety over (i.e. is geometrically irreducible or absolutely irreducible). Then
Again, more explicit bounds on the implied constant here are known, but will not be the focus of this post. As the previous examples show, the hypotheses of definability over and geometric irreducibility are both necessary.
The Lang-Weil bound is already non-trivial in the model case of plane curves:
Thus, for instance, if , then the elliptic curve has -points, a result first established by Hasse. The Hasse-Weil bound is already quite non-trivial, being the analogue of the Riemann hypothesis for plane curves. For hyper-elliptic curves, an elementary proof (due to Stepanov) is discussed in this previous post. For general plane curves, the first proof was by Weil (leading to his famous Weil conjectures); there is also a nice version of Stepanov’s argument due to Bombieri covering this case which is a little less elementary (relying crucially on the Riemann-Roch theorem for the upper bound, and a lifting trick to then get the lower bound), which I briefly summarise later in this post. The full Lang-Weil bound is deduced from the Hasse-Weil bound by an induction argument using generic hyperplane slicing, as I will also summarise later in this post.
The hypotheses of definability over and geometric irreducibility in the Lang-Weil can be removed after inserting a geometric factor:
where is the number of top-dimensional components of (i.e. geometrically irreducible components of of dimension ) that are definable over , or equivalently are invariant with respect to the Frobenius endomorphism that defines .
Proof: By breaking up a general variety into components (and using Lemma 1 to dispose of any lower-dimensional components), it suffices to establish this claim when is itself geometrically irreducible. If is definable over , the claim follows from Theorem 2. If is not definable over , then it is not fixed by the Frobenius endomorphism (since otherwise one could produce a set of defining polynomials that were fixed by Frobenius and thus defined over by using some canonical basis (such as a reduced Grobner basis) for the associated ideal), and so has strictly smaller dimension than . But captures all the -points of , so in this case the claim follows from Lemma 1.
Note that if is reducible but is itself defined over , then the Frobenius endomorphism preserves itself, but may permute the components of around. In this case, is the number of fixed points of this permutation action of Frobenius on the components. In particular, is always a natural number between and ; thus we see that regardless of the geometry of , the normalised count is asymptotically restricted to a bounded range of natural numbers (in the regime where the complexity stays bounded and goes to infinity).
Example 1 Consider the variety
for some non-zero parameter . Geometrically (by which we basically mean “when viewed over the algebraically closed field “), this is the union of two lines, with slopes corresponding to the two square roots of . If is a quadratic residue, then both of these lines are defined over , and are fixed by Frobenius, and in this case. If is not a quadratic residue, then the lines are not defined over , and the Frobenius automorphism permutes the two lines while preserving as a whole, giving in this case.
Corollary 4 effectively computes (at least to leading order) the number-theoretic size of a variety in terms of geometric information about , namely its dimension and the number of top-dimensional components fixed by Frobenius. It turns out that with a little bit more effort, one can extend this connection to cover not just a single variety , but a family of varieties indexed by points in some base space . More precisely, suppose we now have two affine varieties of bounded complexity, together with a regular map of bounded complexity (the definition of complexity of a regular map is a bit technical, see e.g. this paper, but one can think for instance of a polynomial or rational map of bounded degree as a good example). It will be convenient to assume that the base space is irreducible. If the map is a dominant map (i.e. the image is Zariski dense in ), then standard algebraic geometry results tell us that the fibres are an unramified family of -dimensional varieties outside of an exceptional subset of of dimension strictly smaller than (and with having dimension strictly smaller than ); see e.g. Section I.6.3 of Shafarevich.
Now suppose that , , and are defined over . Then, by Lang-Weil, has -points, and by Schwarz-Zippel, for all but of these -points (the ones that lie in the subvariety ), the fibre is an algebraic variety defined over of dimension . By using ultraproduct arguments (see e.g. Lemma 3.7 of this paper of mine with Emmanuel Breuillard and Ben Green), this variety can be shown to have bounded complexity, and thus by Corollary 4, has -points. One can then ask how the quantity is distributed. A simple but illustrative example occurs when and is the polynomial . Then equals when is a non-zero quadratic residue and when is a non-zero quadratic non-residue (and when is zero, but this is a negligible fraction of all ). In particular, in the asymptotic limit , is equal to half of the time and half of the time.
Now we describe the asymptotic distribution of the . We need some additional notation. Let be an -point in , and let be the connected components of the fibre . As is defined over , this set of components is permuted by the Frobenius endomorphism . But there is also an action by monodromy of the fundamental group (this requires a certain amount of étale machinery to properly set up, as we are working over a positive characteristic field rather than over the complex numbers, but I am going to ignore this rather important detail here, as I still don’t fully understand it). This fundamental group may be infinite, but (by the étale construction) is always profinite, and in particular has a Haar probability measure, in which every finite index subgroup (and their cosets) are measurable. Thus we may meaningfully talk about elements drawn uniformly at random from this group, so long as we work only with the profinite -algebra on that is generated by the cosets of the finite index subgroups of this group (which will be the only relevant sets we need to measure when considering the action of this group on finite sets, such as the components of a generic fibre).
Theorem 5 (Lang-Weil with parameters) Let be varieties of complexity at most with irreducible, and let be a dominant map of complexity at most . Let be an -point of . Then, for any natural number , one has for values of , where is the random variable that counts the number of components of a generic fibre that are invariant under , where is an element chosen uniformly at random from the étale fundamental group . In particular, in the asymptotic limit , and with chosen uniformly at random from , (or, equivalently, ) and have the same asymptotic distribution.
This theorem generalises Corollary 4 (which is the case when is just a point, so that is just and is trivial). Informally, the effect of a non-trivial parameter space on the Lang-Weil bound is to push around the Frobenius map by monodromy for the purposes of counting invariant components, and a randomly chosen set of parameters corresponds to a randomly chosen loop on which to perform monodromy.
Example 2 Let and for some fixed ; to avoid some technical issues let us suppose that is coprime to . Then can be taken to be , and for a base point we can take . The fibre – the roots of unity – can be identified with the cyclic group by using a primitive root of unity. The étale fundamental group is (I think) isomorphic to the profinite closure of the integers (excluding the part of that closure coming from the characteristic of ). Not coincidentally, the integers are the fundamental group of the complex analogue of . (Brian Conrad points out to me though that for more complicated varieties, such as covers of by a power of the characteristic, the etale fundamental group is more complicated than just a profinite closure of the ordinary fundamental group, due to the presence of Artin-Schreier covers that are only ramified at infinity.) The action of this fundamental group on the fibres can given by translation. Meanwhile, the Frobenius map on is given by multiplication by . A random element then becomes a random affine map on , where chosen uniformly at random from . The number of fixed points of this map is equal to the greatest common divisor of and when is divisible by , and equal to otherwise. This matches up with the elementary number fact that a randomly chosen non-zero element of will be an power with probability , and when this occurs, the number of roots in will be .
Example 3 (Thanks to Jordan Ellenberg for this example.) Consider a random elliptic curve , where are chosen uniformly at random, and let . Let be the -torsion points of (i.e. those elements with using the elliptic curve addition law); as a group, this is isomorphic to (assuming that has sufficiently large characteristic, for simplicity), and consider the number of points of , which is a random variable taking values in the natural numbers between and . In this case, the base variety is the modular curve , and the covering variety is the modular curve . The generic fibre here can be identified with , the monodromy action projects down to the action of , and the action of Frobenius on this fibre can be shown to be given by a matrix with determinant (with the exact choice of matrix depending on the choice of fibre and of the identification), so the distribution of the number of -points of is asymptotic to the distribution of the number of fixed points of a random linear map of determinant on .
Theorem 5 seems to be well known “folklore” among arithmetic geometers, though I do not know of an explicit reference for it. I enjoyed deriving it for myself (though my derivation is somewhat incomplete due to my lack of understanding of étale cohomology) from the ordinary Lang-Weil theorem and the moment method. I’m recording this derivation later in this post, mostly for my own benefit (as I am still in the process of learning this material), though perhaps some other readers may also be interested in it.
Caveat: not all details are fully fleshed out in this writeup, particularly those involving the finer points of algebraic geometry and étale cohomology, as my understanding of these topics is not as complete as I would like it to be.
Many thanks to Brian Conrad and Jordan Ellenberg for helpful discussions on these topics.