On the density of coprime -tuples over holomorphy rings
Let be a finite field, be a function field of genus having full constant field , a set of places of and the holomorphy ring of . In this paper we compute the density of coprime -tuples of elements of . As a side result, we obtain that whenever the complement of is finite, the computation of the density can be reduced to the computation of the -polynomial of the function field. In the rational function field case, classical results for the density of coprime -tuples of polynomials are obtained as corollaries.
Keywords: Function fields, Density, Polynomials, Riemann-Roch spaces, Zeta function.
MSC: 11R58, 11M38, 11T06
Let be a finite field with elements and let be an algebraic function field
In what follows we will say that an -tuple of elements of is coprime if its components generate the unit ideal in (in analogy to the case of the ring of integers in [bib:MicheliCesaro]). In this paper we define a notion of density for subsets of , using Moore-Smith convergence for nets [bib:kelley1955general, Chapter 2]. We then wish to study the density of the set of coprime -tuples in , considered as a subset of .
The special case and has been studied for in [bib:sugita2007probability] and more generally in [bib:guo2013probability]. We will explain how to interpret the densities presented in these papers as particular cases of our general framework. In fact, using the Riemann-Roch Theorem and the absolute convergence of the Zeta function of , we are able to show that the density of coprime -tuples of elements of a holomorphy ring exists and is equal to , where is the Zeta function of the holomorphy ring, which will be defined in Section 2.
Finally, we provide an example in the case of the affine ring of coordinates of an elliptic curve to show a concrete application of the main result.
The results in this paper provide a function field version of a classical result for the ring of integers, where the natural density of the set of coprime -tuples of is proven to be equal to , being the classical Riemann zeta function (see for example [bib:nymann1972probability]). Similar results also hold in the rings of integers of arbitrary number fields (see [bib:MicheliCesaro] and [bib:BS]).
Let be an algebraic function field with full constant field , let be the genus of , and let be the set of its places. Let be the holomorphy ring of a nonempty set of places . For a fixed positive integer , we wish to study the set of coprime -tuples of elements of the ring . Let us denote this set by :
where denotes the ideal of generated by the set .
Define furthermore , the set of positive divisors supported away from . For any divisor , the Riemann-Roch space associated to is defined as in [bib:stichtenoth2009algebraic, Def. 1.4.4] by
where is the valuation associated to the place . It follows that
Recall that we have a bijection between the set of places and the maximal ideals of given by (see for example [bib:stichtenoth2009algebraic, Proposition 3.2.9]). In analogy to the natural density of integers, we define the superior density of a subset as
This limit can be defined via Moore-Smith convergence [bib:kelley1955general, Chapter 2]. To be precise, the set of divisors with the usual partial order is a directed set, so the map from to the topological space defined as
is a net. Now, since is Hausdorff, the definition in (1) is well posed. Analogously one can define the inferior density as
Moreover, whenever , we call this value the density of and denote it by . In the case in which is finite, an analogous definition of density can also be found in [bib:poonen, Section 8].
2. The density of
Recall that the Zeta function of the function field is given by
for . Analogously, we define the Zeta function of the holomorphy ring corresponding to the set of places as
We will now state our main result.
The density of the set of coprime tuples of length of the holomorphy ring is .
We first enumerate the set of places of . Let us define
and notice that . Observe that the condition is equivalent to the fact that for each there exists at least one that does not belong to . Consider now the projection and observe that
by the Chinese remainder theorem over the ideals . This gives us a homomorphism
which we can extend to -tuples by
By construction, this homomorphism satisfies
Consider now a divisor . We wish to count the number of elements in . First, we will show that maps surjectively onto if is large enough.
For this, note that the image of under is . The space
consists of all elements in with at least a root at each , so it is equal to (note that the cannot be in the support of ). Hence, its dimension as an -vector space is , which is equal to if is large enough by Riemann-Roch Theorem. On the other hand, the dimension of is then , and so the image has dimension , the same as . Therefore and restricted to are surjective.
We can now count the elements of using (2). As we have just seen, the dimension of the kernel of restricted to is , so each element of is the image under of exactly elements of . Hence we get
if is large enough. It follows that the density of is well-defined and equals
Since , it follows that .
To get an estimate in the other direction, let us write . We have
hence we have the inequalities
Now, passing to the limit in , we get that . Therefore it remains to prove that
In order to prove the last claim, let us denote by the set . Notice that if , then there exists for which . We get the following inclusion:
where by we mean the Cartesian product of copies of the ideal . Fix now a divisor . It follows that
The last equality holds because if . With this containment, we can now estimate the last term of (3):
By the Riemann-Roch Theorem, we have that and , since [bib:stichtenoth2009algebraic, Eq. 1.21 and Theorem 1.4.17]. It follows that the above is less or equal to
Observe that is the tail of a subseries of the Zeta function of evaluated at , which is absolutely convergent (see for example [bib:Mor, Chapter 3]). As goes to infinity, it converges to , from which our claim follows. ∎
The reader should observe that in Theorem 1 both and could possibly be infinite and the result will still hold. Nevertheless, the density depends on the Zeta function of the holomorphy ring, which may be hard to compute. First of all notice that this is not the case when is finite since under this condition is a finite product. The following immediate corollary covers the case in which is finite.
Let be a function field, a set of places of and the holomorphy ring of . Let be the -polynomial of . Then
The corollary follows from Theorem 1, the definition of and the expression of the Zeta function of in terms of the -polynomial. ∎
Observe now that in the case where is finite, the density of coprime -tuples of depends only on the following finite data: the degrees of the places in and the -polynomial of the function field, which again only depends on the -rational points of the curve associated to the function field for (see for example [bib:stichtenoth2009algebraic, Corollary 5.1.17]).
3.1. An example
Let for simplicity. Let and be a polynomial defining an elliptic curve over . Let us define
Let denote the set of (projective) -rational points of (i.e. the places of degree one of the function field of ).
The density of -tuples of coprime elements of is
Observe that the Zeta function of an elliptic curve is
The result follows from Theorem 1 applied to the holomorphy ring where is the place at infinity of with respect to . ∎
The reader should notice that (4) depends only on the number of -rational points of , since the genus of equals one (see Remark 3). The probabilistic interpretation of Corollary 4 is the following: select uniformly at random elements of of degree at most , then the probability that they generate the unit ideal in approaches as .
3.2. The case
In the remaining part of this section we show how the results [bib:sugita2007probability, Theorem 1] and [bib:guo2013probability, Remark 4] about coprime -tuples over fit in our framework.
Denote by the place at infinity of the function field . It is easy to see that the definition of density for given in [bib:guo2013probability, bib:sugita2007probability] agrees with ours for . Hence, we get [bib:guo2013probability, Remark 4] as a corollary to Theorem 1, while [bib:sugita2007probability, Theorem 1] is simply the special case :
Let be an integer. The density of coprime -tuples over is
It is enough to notice that the Zeta function of the function field (i.e. the Zeta function of the projective line) is
and then the Zeta function of the holomorphy ring is
The claim follows by inverting the expression above and evaluating at . ∎
The authors want to thank Andrea Ferraguti for useful discussions and suggestions.
- In this note we will mostly use the language and notation of [bib:stichtenoth2009algebraic].