You are currently browsing the tag archive for the ‘Lang-Weil bound’ tag.
Much as group theory is the study of groups, or graph theory is the study of graphs, model theory is the study of models (also known as structures) of some language (which, in this post, will always be a single-sorted, first-order language). A structure is a set , equipped with one or more operations, constants, and relations. This is of course an extremely general type of mathematical object, but (quite remarkably) one can still say a substantial number of interesting things about very broad classes of structures.
We will observe the common abuse of notation of using the set as a metonym for the entire structure, much as we usually refer to a group simply as , a vector space simply as , and so forth. Following another common bending of the rules, we also allow some operations on structures (such as the multiplicative inverse operation on a group or field) to only be partially defined, and we allow use of the usual simplifying conventions for mathematical formulas (e.g. writing instead of or , in cases where associativity is known). We will also deviate slightly from the usual practice in logic by emphasising individual structures, rather than the theory of general classes of structures; for instance, we will talk about the theory of a single field such as or , rather than the theory of all fields of a certain type (e.g. real closed fields or algebraically closed fields).
Once one has a structure , one can introduce the notion of a definable subset of , or more generally of a Cartesian power of , defined as a set of the form
for some formula in the language with free variables and any number of constants from (that is, is a well-formed formula built up from a finite number of constants in , the relations and operations on , logical connectives such as , , , and the quantifiers ). Thus, for instance, in the theory of the arithmetic of the natural numbers , the set of primes is a definable set, since we have
In the theory of the field of reals , the unit circle is an example of a definable set,
but so is the the complement of the circle,
and the interval :
Due to the unlimited use of constants, any finite subset of a power of any structure is, by our conventions, definable in that structure. (One can of course also consider definability without parameters (also known as -definability), in which arbitrary constants are not permitted, but we will not do so here.)
We can isolate some special subclasses of definable sets:
- An atomic definable set is a set of the form (1) in which is an atomic formula (i.e. it does not contain any logical connectives or quantifiers).
- A quantifier-free definable set is a set of the form (1) in which is quantifier-free (i.e. it can contain logical connectives, but does not contain the quantifiers ).
Example 1 In the theory of a field such as , an atomic definable set is the same thing as an affine algebraic set (also known as an affine algebraic variety, with the understanding that varieties are not necessarily assumed to be irreducible), and a quantifier-free definable set is known as a constructible set; thus we see that algebraic geometry can be viewed in some sense as a special case of model theory. (Conversely, it can in fact be quite profitable to think of model theory as an abstraction of algebraic geometry; for instance, the concepts of Morley rank and Morley degree in model theory (discussed in this previous blog post) directly generalises the concepts of dimension and degree in algebraic geometry.) Over , the interval is a definable set, but not a quantifier-free definable set (and certainly not an atomic definable set); and similarly for the primes over .
A quantifier-free definable set in is nothing more than a finite boolean combination of atomic definable sets; in other words, the class of quantifier-free definable sets over is the smallest class that contains the atomic definable sets and is closed under boolean operations such as complementation and union (which generate all the other boolean operations). Similarly, the class of definable sets over is the smallest class that contains the quantifier-free definable sets, and is also closed under the operation of projection from to for every natural number , where is the map .
Some structures have the property of enjoying quantifier elimination, which means that every definable set is in fact a quantifier-free definable set, or equivalently that the projection of a quantifier-free definable set is again quantifier-free. For instance, an algebraically closed field (with the field operations) has quantifier elimination (i.e. the projection of a constructible set is again constructible); this fact can be proven by the classical tool of resultants, and among other things can be used to give a proof of Hilbert’s nullstellensatz. (Note though that projection does not necessary preserve the property of being atomic; for instance, the projection of the atomic set is the non-atomic, but still quantifier-free definable, set .) In the converse direction, it is not difficult to use the nullstellensatz to deduce quantifier elimination. For theory of the real field , which is not algebraically closed, one does not have quantifier elimination, as one can see from the example of the unit circle (which is a quantifier-free definable set) projecting down to the interval (which is definable, but not quantifer-free definable). However, if one adds the additional operation of order to the reals, giving it the language of an ordered field rather than just a field, then quantifier elimination is recovered (the class of quantifier-free definable sets now enlarges to match the class of definable sets, which in this case is also the class of semi-algebraic sets); this is the famous Tarski-Seidenberg theorem.
On the other hand, many important structures do not have quantifier elimination; typically, the projection of a quantifier-free definable set is not, in general, quantifier-free definable. This failure of the projection property also shows up in many contexts outside of model theory; for instance, Lebesgue famously made the error of thinking that the projection of a Borel measurable set remained Borel measurable (it is merely an analytic set instead). Turing’s halting theorem can be viewed as an assertion that the projection of a decidable set (also known as a computable or recursive set) is not necessarily decidable (it is merely semi-decidable (or recursively enumerable) instead). The notorious P=NP problem can also be essentially viewed in this spirit; roughly speaking (and glossing over the placement of some quantifiers), it asks whether the projection of a polynomial-time decidable set is again polynomial-time decidable. And so forth. (See this blog post of Dick Lipton for further discussion of the subtleties of projections.)
Now we consider the status of quantifier elimination for the theory of a finite field . If interpreted naively, quantifier elimination is trivial for a finite field , since every subset of is finite and thus quantifier-free definable. However, we can recover an interesting question in one of two (essentially equivalent) ways. One is to work in the asymptotic regime in which the field is large, but the length of the formulae used to construct one’s definable sets stays bounded uniformly in the size of (where we view any constant in as contributing a unit amount to the length of a formula, no matter how large is). A simple counting argument then shows that only a small number of subsets of become definable in the asymptotic limit , since the number of definable sets clearly grows at most polynomially in for any fixed bound on the formula length, while the number of all subsets of grows exponentially in .
Another way to proceed is to work not with a single finite field , or even with a sequence of finite fields, but with the ultraproduct of a sequence of finite fields, and to study the properties of definable sets over this ultraproduct. (We will be using the notation of ultraproducts and nonstandard analysis from this previous blog post.) This approach is equivalent to the more finitary approach mentioned in the previous paragraph, at least if one does not care to track of the exact bounds on the length of the formulae involved. Indeed, thanks to Los’s theorem, a definable subset of is nothing more than the ultraproduct of definable subsets of for all sufficiently close to , with the length of the formulae used to define uniformly bounded in . In the language of nonstandard analysis, one can view as a nonstandard finite field.
The ultraproduct of finite fields is an important example of a pseudo-finite field – a field that obeys all the sentences in the languages of fields that finite fields do, but is not necessarily itself a finite field. The model theory of pseudo-finite fields was first studied systematically by Ax (in the same paper where the Ax-Grothendieck theorem, discussed previously on this blog, was established), with important further contributions by Kiefe, by Fried-Sacerdote, by two papers of Chatzidakis-van den Dries-Macintyre, and many other authors.
As mentioned before, quantifier elimination trivially holds for finite fields. But for infinite pseudo-finite fields, such as the ultraproduct of finite fields with going to infinity, quantifier elimination fails. For instance, in a finite field , the set of quadratic residues is a definable set, with a bounded formula length, and so in the ultraproduct , the set of nonstandard quadratic residues is also a definable set. However, in one dimension, we see from the factor theorem that the only atomic definable sets are either finite or the whole field , and so the only constructible sets (i.e. the only quantifier-free definable sets) are either finite or cofinite in . Since the quadratic residues have asymptotic density in a large finite field, they cannot form a quantifier-free definable set, despite being definable.
Nevertheless, there is a very nice almost quantifier elimination result for these fields, in characteristic zero at least, which we phrase here as follows:
where is an atomic definable subset of (i.e. the -points of an algebraic variety defined over in ) and is a polynomial.
Informally, this theorem says that while we cannot quite eliminate all quantifiers from a definable set over a nonstandard finite field, we can eliminate all but one existential quantifier. Note that negation has also been eliminated in this theorem; for instance, the definable set uses a negation, but can also be described using a single existential quantifier as .) I believe that there are more complicated analogues of this result in positive characteristic, but I have not studied this case in detail (Kiefe’s result does not assume characteristic zero, but her conclusion is slightly different from the one given here). In the one-dimensional case , the only varieties are the affine line and finite sets, and we can simplify the above statement, namely that any definable subset of takes the form for some polynomial (i.e. definable sets in are nothing more than the projections of the -points of a plane curve).
There is an equivalent formulation of this theorem for standard finite fields, namely that if is a finite field and is definable using a formula of length at most , then can be expressed in the form (2) with the degree of bounded by some quantity depending on and , assuming that the characteristic of is sufficiently large depending on .
The theorem gives quite a satisfactory description of definable sets in either standard or nonstandard finite fields (at least if one does not care about effective bounds in some of the constants, and if one is willing to exclude the small characteristic case); for instance, in conjunction with the Lang-Weil bound discussed in this recent blog post, it shows that any non-empty definable subset of a nonstandard finite field has a nonstandard cardinality of for some positive standard rational and integer . Equivalently, any non-empty definable subset of for some standard finite field using a formula of length at most has a standard cardinality of for some positive rational of height and some natural number between and . (For instance, in the example of the quadratic residues given above, is equal to and equal to .) There is a more precise statement to this effect, namely that the Poincaré series of a definable set is rational; see Kiefe’s paper for details.
Below the fold I give a proof of Theorem 1, which relies primarily on the Lang-Weil bound mentioned above.
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.