A Local to global principle for densities

A local to global principle for densities over function fields

Abstract.

Let be a positive integer and be an integrally closed subring of a global function field . The purpose of this paper is to provide a general sieve method to compute densities of subsets of defined by local conditions. The main advantage of the method relies on the fact that one can use results from measure theory to extract density results over . Using this method we are able to compute the density of the set of polynomials with coefficients in which give rise to “good” totally ramified extensions of the global function field . As another application, we give a closed expression for the density of rectangular unimodular matrices with coefficients in in terms of the -polynomial of the function field.

Key words and phrases:
Function fields; density; local to global principles; totally ramified places; rectangular unimodular matrices

1. Introduction

In [bib:poonenAnn, Lemma 20] B. Poonen and M. Stoll formalise a nice sieve method for computing densities using -adic analysis. Essentially, the method consists of writing a given set in terms of local conditions at the completions of ; once this is done, the density of can be computed by determining the measures of certain sets which are associated to the local conditions which define . It is worth mentioning that the result is a powerful evolution of Ekedhal’s Sieve (See [torsten1991infinite]).

In this paper we present the extension of this method to global function fields i.e. univariate function fields over finite fields. Let be a non-trivial integrally closed subring of a function field and be the set of places of where all the functions in are well defined. It is well known (see for example [bib:stichtenoth2009algebraic, Theorem 3.2.6]) that consists exactly of the intersection of all the valuation rings of for . Vice versa, it also holds that an arbitrary intersection of valuation rings of is an integrally closed subring [bib:stichtenoth2009algebraic, Proposition 3.2.5]). We will be interested in computing the density of a subset of .

Before doing so, we first need to specify what we mean by “density” of in the function field context. Over the set of rational integers , the density of a subset is computed by considering the sequence of ratios between the number of points of falling in the hypercube of side and centred at the origin, and . If is this sequence of ratios and is its limit (if exists), then we say that has density . In the case of , we explain how to use Moore-Smith convergence [bib:kelley1955general, Chapter 2] to define a notion of limit over the directed set of positive divisors having support in the complement of (see also [bib:BS]). Once this is understood, Riemann-Roch spaces of positive divisors having support in the complement of will play the role of intervals, and therefore products of such spaces will play the role of hypercubes.

Let be the set of positive divisors having support in the complement of . The striking analogy between and which allows our density definition (see Subsection 1.1) is given by

 Z=⋃B∈N[−B,B[∩ZandH=⋃D∈DSL(D).

In particular the reader should notice that the definition of density we will provide is consistent with the one used in the literature in the case of : if , and is the place at plus infinity with respect to , we have that and therefore

 L(D)=L(nP∞)={f∈Fq[x]:deg(f)≤n},

which induces the natural definition of density in the context of .

The essence of the presented method (Theorem 2.1) is to polarize the difficulty of the problem: in fact, on one hand the -adic formalism allows to easily compute a “candidate” for the density of a certain subset of by using tools from measure theory, on the other hand all the difficulty of the problem is unloaded on proving that the limit of a certain sequence (given by Equation (2.1)) tends to zero. In particular, we show that whenever the local conditions are actually related in a certain way to polynomial equations, the limit can be proven to be always zero (Theorem 2.2).

The entire machinery we build in Section 2 is then used to produce two new results in Sections 3 and 4.

In Section 3 we compute the probability that a “random” polynomial of fixed degree with coefficient in an given integrally closed subring gives rise to a totally ramified extension of for which the equation is “good enough” around the totally ramified place (in terms of Definition 3.1).

Let be positive integers such that and be a domain. The question whether a homomorphism of in can be extended to an automorphism of raised many interesting questions in the past (see for instance Serre’s Conjecture, which is proven in [suslin1974projective, bib:projmod]). In Section 4 we close the problem of computing the density of homomorphisms of in which can be extended to automorphims of . In the case of , these homomorphisms arise from context of convolutional codes (see for example [FORNASINI2004119] or [rosenthal2001connections]) and their density was studied in [bib:guo2013probability] and [micheli2016density]. In Theorem 4.4 we show that the density of unimodular matrices over is a rational number and can be explicitly computed as soon as the complement of the holomorphy set is finite.

1.1. Preliminary definitions and notations

Let be a finite field. In this paper all the function fields are global and have full constant field . We denote by a valuation ring of function field , having maximal ideal . The set of all the places of will be denoted by . If is a proper subset of , we denote by the subset of places of of degree greater than . Moreover, we write to denote the holomorphy ring of i.e. the intersection of all the valuation rings associated to the places of :

 HS=⋂P∈SOP.

Sometimes, we will refer to as the holomorphy set of and to as the holomorphy ring of . Holomorphy rings are integrally closed in and any integrally closed subring of is an holomorphy ring [bib:stichtenoth2009algebraic, Proposition 3.2.5, Theorem 3.2.6]. In the whole paper we consider only holomorphy rings whose holomorphy set has finite complement in the set of all places of . The most immediate example of holomorphy ring is as this consists of the intersection of all the valuation rings of different from the valuation ring at infinity.

Let be the set of divisors of i.e. the free abelian group having as base symbols the elements in the set . For , we denote by the finite subset of for which is non-zero. Moreover, we will write whenever for any in . Let

 Div+(F)={D∈Div(F)|D≥0}

Let be the subset of divisors of having support over the complement of in . As is a directed set, we can define via Moore-Smith Convergence (see [bib:kelley1955general, Chapter 2] and more specifically for this context [bib:HolMS]) a notion of limit over . In this context, we can give an upper density definition for a subset of as follows:

 ¯¯¯¯DS(A):=limsupD∈DS|A∩L(D)d|qℓ(D)d

where is the Riemann-Roch space attached to the divisor and . Analogously, one can give a notion of lower density by considering the inferior limit of the sequence. Whenever these two quantities are equal, we say that a subset of has a well-defined density .

For a valuation ring let us denote by the completion of with respect to the -adic metric. In addition, let us denote by the normalized Haar measure on with respect to the -adic metric. For a subset we denote by the boundary of with respect to the topology induced by the -adic metric. For a multivariate polynomial , we will denote by (resp. ) the degree of in the variable (resp. the degree of the homogenization of ). Whenever has all the coefficients in a given valuation ring , we will denote by (resp. ) the degree of in the variable (resp. the degree of the homogenization of ) in . For a positive integer and a given commutative domain , we will denote by the set of matrices whose determinant is a unit of .

2. The local to global principle for densities over global function fields

In this section we describe the local to global principle which will be used later on. This result is the function field analogue of [bib:poonenAnn, Lemma 20].

Theorem 2.1.

Let be a positive integer, be a subset of places of and the holomorphy ring of . For any , let be a measurable set such that . Suppose that

 (2.1) limt→∞¯¯¯¯DS({a∈HdS|a∈UPfor some P∈St})=0.

Let defined by . Then

• is convergent.

• Let . Then exists and defines a measure on .

• is concentrated at finite subsets of . In addition, if is finite we have:

 ν({T})=(∏P∈TμP(UP))∏P∈S∖T(1−μP(UP)).
Proof.

Throughout the proof will be fixed, so we will denote by . What we need to do is to translate the proof of [bib:loctoglob] to the context of function fields. Essentially, we need to understand how the measure of -adic intervals can be translated into density via the use of Riemann-Roch Theorem [bib:stichtenoth2009algebraic, Theorem 1.5.15]. Once this is done, the same arguments of the proof of [bib:poonenAnn, Lemma 20] will apply to this context. We define a -interval in as the set for some and . A -box will just be a product of -intervals:

 IP={x∈ˆOdP:xj≡ajmodPeP,jfor j∈{1,…d}}.

To simplify notation, we say that a -box is a -cube if it has the form

 CP={x∈ˆOdP:xi≡yimodPeP,i∈{1,…,d}}

for some and . In other words, is the cartesian product of intervals of equal length. We say that is the center of the cube. Let be a finite subset of . Let us now compute the density of the elements in which are mapped in a product of a finite number of -boxes via the natural embedding . Let be such product of -boxes. For any , the -box can be covered with a finite number of disjoint cubes of equal size , as all the congruences can be decomposed in terms of the finest congruence, given by . Therefore, one can write

 I=∏P∈AIP=∏P∈AlP⨆i=1C(i)P

with independently of .

We consider the diagram

 Hdι−−−−→∏P∈AˆOdP⊇I⏐⏐↓πJ⏐⏐↓π′J(H/J)dψ−−−−→∏P∈A(ˆOP/PeP)d=:R

where , the map is the natural inclusion, and is the isomorphism (coming from the Chinese Remainder Theorem) which makes the diagram commutative. On the right-hand side, we can immediately compute the product measure of by looking at its definition, getting . It remains to show that the density of is indeed . For this, let us decompose . Let be a finite set indexing the cubes which cover and let . Any given determines a choice of cubes as follows: for each place we select exactly one cube , having center . We now build a as the product . Clearly, the set of ’s built in this way has cardinality and covers via a disjoint union. If we can now prove that the density of is independent of the choice of , then we will have that

 (2.2) D(ι−1(I))=D(ι−1(Ci))⋅∏P∈AlP.

To achieve this, we now explicitly compute the value . As the diagram above is commutative, we can equivalently compute the density of elements of falling into via the map . Let . Notice that we have

 ι−1(Ci)=π−1Jψ−1π′J(Ci)=zCi+JH.

Observe that for any divisor , the map restricted to is -linear. Let be the genus of . Therefore if we denote by the -th component of , we have

 |L(D)∩(zCi,j+JH)|=|L(D)∩JH|=∣∣ ∣∣L(D−∑P∈AePP)∣∣ ∣∣=qℓ(D−∑P∈AePP),

which for of large degree, equals by Riemann-Roch Theorem. We can finally compute the density of elements mapping in the cube (which is in fact independent of , as we wanted):

 D(ι−1(Ci))=limD∈DS|L(D)d∩π−1Jψ−1π′J(Ci)|qℓ(D)d=q−ℓ(D)dd∏j=1|L(D)∩(zCi,j+JH)|=q−(∑P∈StePdeg(P))d.

Using now Equation (2.2) we get the final claim by comparing with . Since now we have proved the theorem for boxes, all the arguments of the proof of [bib:poonenAnn, Lemma 20] are now straightforward to apply. In fact, suppose for a moment that the set of ’s in for which is different from the empty set is a finite set . Now, let be a finite set of places. Assuming that , one can cover each of the ’s from the interior (resp. ) with a finite set of boxes which well approximate the measure (resp. ). In particular one has

 ν({T})≥(∏P∈TμP(U′P))∏P∈S∖T(1−μP(U′P)),

where the products above are both finite and union of the boxes for each , where the theorem holds. As we can apply the symmetric argument with a set of external approximations we have

 ν({T})≤(∏P∈TμP(U′′P))∏P∈S∖T(1−μP(U′′P)),

from which the claim follows by letting the approximation get sharper and then and tend to .

On the other hand, if is an infinite set, one easily sees that an approximation with finitely many is good enough, as long as condition (2.1) is verified. To see this, let be a finite subset of and let us recall that is the set of places of of degree larger than and then is the subset of consisting of places of degree less than or equal to . Observe that for a positive integer such that contains we can define a partial approximation of

 Wt={a∈HdS∣a∈UP∀P∈T,a∉UP∀P∈Tc∩Sct}.

Notice that contains so . In addition we have that

 D(π−1({T}))≥D(Wt)−¯¯¯¯D(Wt∖π−1({T})).

Now, by letting go to infinity and using condition (2.1) on one gets the claim. ∎

The next Theorem ensures that when the can be expressed in terms of polynomial equations, Condition (2.1) is always verified, similarly to what happens in the case of Ekedhal Sieve for integers [torsten1991infinite].

Theorem 2.2.

Let be a global function field and be a subset of with finite complement. Let be the holomorphy ring of . Let be coprime polynomials. Then

 (2.3) limt→∞¯¯¯¯DS({y∈HdS:f(y)≡g(y)≡0modPfor% someP∈St})=0.
Proof.

If there is nothing to prove so we can suppose . Without loss of generality, we can also suppose . Since will be fixed throughout the proof, we will denote and by and respectively. Let us recall that the places in are in natural correspondence with the prime ideals of , therefore with a small abuse of terminology we will identify this two sets. We first fix large enough, so that for any of degree larger than . Now fix large enough so that . Let us also introduce new notation to simplify the computations. For a divisor , let us define

 at(D):=∣∣{y∈L(D)d:f(y)≡g(y)≡0modPfor someP∈St}∣∣q−dℓ(D),
 cP(D):=∣∣{y∈L(D)d:f(y)≡g(y)≡0modP}∣∣q−dℓ(D).

Our first purpose is to estimate for and large. First, we notice a simple upper bound for :

 at(D)≤∑P:deg(P)>tcP(D).

We now want to estimate the sum above for different regimes of and . In order to do so, let us further split the sum as

 (2.4) ∑P:tdeg(D)cP(D)(\textupII).

Let us estimate (I). First, we want to give a reasonable estimate for in the specified regime. Notice that for each point of satisfying there are at most preimages of in , as the evaluation map is linear and has kernel . Let be the number of -points of the variety defined by and when reduced modulo . Let be the genus of . By observing that and that we get:

 cP(D)≤ NP|L(D−P)|dq−ℓ(D)d ≤NPqd(deg(D)−deg(P)+1)q−(deg(D)+1−gF)d =NPq(gF−deg(P))d.

As can be chosen large enough to avoid the places of bad reduction, we can estimate classically as for some constant . It follows that

 ∑P:t

for some other constant .

Let us estimate (II). Let be the ideal generated by in and let . Since has codimension , is principal. Let be the generator of , which can be chosen with coefficients over by multiplying by an appropriate element in . Let us also assume without loss of generality that . Let now be so large that modulo every prime of degree larger than , we have . Consider now all the elements of ending with a fixed and for which . Let us estimate their contribution to each in the sum (II). Let be the product of all the prime ideals of such that and for which there exists such that (this set is finite as ). If we denote by the number of distinct primes appearing in the factorization of , the contribution of all the -tuples ending with is bounded by . By the definition of , it is clear that . If we denote by the projection of in , we have that

 hH/Ir[x1,…,xd]⊆(f,g)Ir.

Therefore this in turn implies that satisfies . Now, the key observation to get the final estimate for (II) is the following: was chosen large in such a way that the homogeneous degree of is constant modulo for any of degree larger than . Recall now that every prime ideal appearing in the factorization of has degree larger than , therefore

 urdeg(D)

where the constant depends on the leading coefficients of (and independent of ), from which it follows that , for large enough.

The reader should now notice that in (II), an element in ending with (i.e. of the form ), cannot contribute more than for each , as the evaluation map is an injection to . It follows easily that the set of all the -tuples ending with contribute at most to the whole sum (II). Now, using the observations above and recalling that we also have to take into account the size of the set of the -tuples such that for , we finally get

 ∑P:deg(P)>deg(D)cP(D)≤degxi(h)qℓ(D)(d−1)q−ℓ(D)d+∑r∈L(D)d−1∖Tq−ℓ(D)durdegx1(f)
 ≤q−ℓ(D)(deghom(h)degx1(f)−1).

At this point we observe that the estimates of (I) and (II) only hold for large values of and , so it is now important to notice how the order of the limits in (2.3) is actually taken into account: in order for estimate (I) to hold, it is enough to choose so large that the primes of bad reduction are avoided. For estimate (II), take so large that

• its degree is larger than ,

• the homogeneous degree of is constant with respect to all places of of degree larger than ,

• the degree of with respect to is constant for all places of of degree larger than .

We can now safely use the two estimates to complete the proof:

 limt→∞limD→∞at(D)≤limt→∞limD→∞∑P:deg(P)>tcP(D)= limt→∞limD→∞∑P:deg(P)>deg(D)cP(D)+∑P:t

This completes the proof, since the sum above is the tail of a subseries of the zeta function of evaluated at , which is converging. ∎

The reader should notice that the counting technique using in the estimate of (II) is similar to the one used in the case of in the main result of [torsten1991infinite].

3. On the probability of a totally ramified extension of global function fields

In this section we are interested in obtaining the “probability” that a random extension of a given function field is totally ramified in a good way at some place. If is a function field with full constant field and is a finite extension of , we recall that an extension of places is said to be totally ramified if the dimension of as an vector space is equal to .

The notion of “good” total ramification is encoded in the following

Definition 3.1.

Let be a separable totally ramified extension of function fields. We say that is nicely totally ramified with respect to if

• is the defining polynomial of , i.e. .

• there exists a totally ramified extension of places of such that .

The reader with some insight in algebraic geometry can read the condition of nice total ramification as a condition of “non-singularity” around the totally ramified place.

Our purpose is now to compute the density of degree polynomials of for which is irreducible and the extension is nicely totally ramified with respect to .

3.1. A characterization of polynomials defining nice totally ramified extensions

First of all, we convert the nice total ramification property into properties of the coefficients of the defining polynomials. Let us start by stating some well known facts on totally ramified extensions.

Lemma 3.2.

Let be an extension of places of the function field extension . Let be a uniformizer for , and be the minimal polynomial for over . Then

• if is totally ramified, we have that ;

• is an Eisenstein polynomial if and only if is totally ramified.

Proof.

To see this, simply combine [bib:stichtenoth2009algebraic, Proposition 3.1.15] and [bib:stichtenoth2009algebraic, Proposition 3.5.12]. ∎

Proposition 3.3.

Let be a perfect field and be an irreducible separable polynomial of degree over a function field . Let and be a totally ramified extension of places of . Then we have

• if is holomorphic at , then if and only if is an Eisenstein polynomial for some in .

• if is not holomorphic at , then if and only if is Eisenstein.

Proof.

Let us prove (i). Suppose , then there exists such that . Let . The minimal polynomial of is , which implies that if we can prove that , it will follow that is Eisenstein with respect to . It is easily seen that , therefore if is a uniformizer for , we can write

 x=n−1∑i=0aizi

for some . Whence,

 1=vQ(n−1∑i=0aizi)≥mini∈{0,…,n−1}{vQ(ai)+ivQ(z)}.

Since , we get , which forces , as .

Now suppose that is Eisenstein with respect to some . It follows that is a uniformizer for , which implies .

The claim (ii) easily follows by applying (i) to with . ∎

The following corollary shows that the polynomials which are Eisenstein after a Moebius transformation are not more than the ones which are Eisenstein after either a shift or an inversion. This result is not needed in the proof of the main theorem, nevertheless we include it for completeness (in the case of , this question was asked in [bib:heyman2014shifted]).

Corollary 3.4.

Let be a polynomial of degree and a place of . The following are equivalent:

• There exist such that and is Eisenstein;

• One of the following two occurs:

• is Eisenstein with respect to for some .

• is Eisenstein with respect to .

Proof.

Clearly (ii) implies (i). Let us now prove that (ii) follows from (i). Let be the inverse of . Suppose is a zero of and is holomorphic at the totally ramified place lying over . The element is then a zero of the Eisenstein polynomial therefore by 3.2 and the fact that we have that . Suppose is holomorphic at , then it is enough to show that . To this end, we observe that

 y(Q)=h(Q)~y(Q)+s(Q)l(Q)~y(Q)+j(Q)=s(Q)j(Q).

By the above equation it follows that the element

 (3.1) z=y−s/j=hj−slj(l~y+s)~y

has valuation greater than at and as . As is holomorphic at , we have that since

• if , then since

• suppose that , then cannot be zero as .

If we can show that has valuation we are done, as we would have by (3.1)

 OQ=OP[z]=OP[y].

Since , we have

 1≤vQ(z)=1−vQ(l~y+s)≤1

which concludes the proof in the case in which is holomorphic at . If is not holomorphic at , we choose as before and . Now it holds and the previous considerations can be applied again to show .

3.2. On the density of totally ramified extensions of fixed degree

The following lemma gives the exact restriction for the set of possible shifts which are candidates for turning a given poylnomial into an Eisenstein polynomial.

Lemma 3.5.

Let be a valuation ring of a function field and let of degree . Let be a set of representatives of . We have that is Eisenstein with respect to for some if and only if is Eisenstein with respect to for some .

Proof.

One implication is obvious so let us prove the other direction. Suppose that for and that is -Eisenstein for some . Let us first show that, for any , the polynomial is Eisenstein if and only if is Eisenstein. This follows by the fact that the conditions modulo of are easily verified as . The condition modulo reduces to check that , which indeed holds as . The proof is now straightforward, since any can be written as for some and : we have that is -Eisenstein and then is -Eisenstein for . ∎

We are now ready to extract the density of the set the set of polynomials in question using the local to global principle together with some tools from measure theory and linear algebra. With a small abuse of notation, in what follow we will identify with the set of degree polynomials over a holomorphy ring .

Theorem 3.6.

Let be a holomorphy ring of a global function field having full constant field . The set of polynomials of degree such that the ring is a field and is nicely totally ramified with respect to , has density

 R(n,H)=1−∏P∈S(qdeg(P)−1)2(qdeg(P)+1)qdeg(P)(n+2).
Proof.

First, we should observe that we can restrict to the separable case, as the set of inseparable polynomials has density zero. Let us denote the complement of in by . First, we should prove that if ramification occurs, it occurs with probability one at a place of . In fact, the polynomials defining a totally ramified extension at a place of , do not contribute to . This is not surprising, as the definition of the density depends on the choice of . In particular we prove:
Claim 1. The density of degree polynomials in