Table Of ContentA LOCAL TO GLOBAL PRINCIPLE FOR DENSITIES OVER
FUNCTION FIELDS
7
1
0 GIACOMOMICHELI
2
n Abstract. Let dbeapositive integer and Hbean integrallyclosedsubringof
a
aglobalfunctionfieldF. Thepurposeofthispaperistoprovideageneralsieve
J
method to compute densities of subsets of Hd defined by local conditions. The
4
main advantage of the method relies on the fact that one can use results from
] measure theory to extract density results over Hd. Using this method we are
T
abletocomputethedensityofthesetofpolynomialswithcoefficientsinHwhich
N
giveriseto“good”totallyramifiedextensions oftheglobalfunctionfieldF. As
.
h another application, we give a closed expression for the density of rectangular
t unimodular matrices with coefficients in H in terms of the L-polynomial of the
a
m functionfield.
[
1
v
8 1. Introduction
7
1
In [1, Lemma 20] B. Poonen and M. Stoll formalise a nice sieve method for com-
1
0 puting densities using p-adic analysis. Essentially, the method consists of writing a
1. given set U ⊆ Zd in terms of local conditions at the completions of Q; once this is
0
done, the density of U can be computed by determining the measures of certain sets
7
1 Up ⊆ Zp which are associated to the local conditions which define U. It is worth
:
v mentioning that the result is a powerful evolution of Ekedhal’s Sieve (See [2]).
i In this paper we present the extension of this method to global function fields i.e.
X
r univariate function fields over finite fields. Let H be a non-trivial integrally closed
a
subring of a function field F and S be the set of places of F where all the functions
in H are well defined. It is well known (see for example [13, Theorem 3.2.6]) that
H consists exactly of the intersection of all the valuation rings O of F for P ∈ S.
P
Vice versa, it also holds that an arbitrary intersection of valuation rings of F is an
integrally closed subring [13, Proposition3.2.5]). We will be interested in computing
the density of a subset U of Hd.
Key words and phrases. Function fields; density; local to global principles; totally ramified places;
rectangularunimodularmatrices.
1
2 GIACOMOMICHELI
Before doing so, we first need to specify what we mean by “density” of U in the
function field context. Over the set of rational integers Z, the density of a subset
U ⊆ Zd is computed by considering the sequence of ratios between the number of
points of U falling in the hypercube of side 2B and centred at the origin, and (2B)d.
If {aB}B∈N is this sequence of ratios and u is its limit (if exists), then we say that U
has density u. In the case of Hd, we explain how to use Moore-Smithconvergence[3,
Chapter 2] to define a notion of limit overthe directed set of positive divisorshaving
supportinthecomplementofS (seealso[4]). Oncethisisunderstood,Riemann-Roch
spaces of positive divisors having support in the complement of S will play the role
of intervals, and therefore products of such spaces will play the role of hypercubes.
LetD be thesetofpositivedivisorshavingsupportinthe complementofS. The
S
strikinganalogybetweenZandHwhichallowsourdensitydefinition(seeSubsection
1.1) is given by
Z= [−B,B[∩Z and H= L(D).
B[∈N D[∈DS
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 F [x]: if F = F (x),
q q
H = F [x] and P is the place at plus infinity with respect to x, we have that
q ∞
DS ={nP∞}n∈N and therefore
L(D)=L(nP )={f ∈F [x]: deg(f)≤n},
∞ q
which induces the natural definition of density in the context of F [x].
q
The essence of the presented method (Theorem 2.1) is to polarize the difficulty
of the problem: in fact, on one hand the p-adic formalism allows to easily compute
a “candidate” for the density of a certain subset of Hd by using tools from measure
theory,ontheotherhandallthedifficultyoftheproblemisunloadedonprovingthat
the limit of a certainsequence (givenby 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).
TheentiremachinerywebuildinSection2isthenusedtoproducetwonewresults
in Sections 3 and 4.
In Section 3 we compute the probability that a “random” polynomial f of fixed
degree with coefficient in an given integrally closed subring H ⊂ F gives rise to a
totally ramified extension E = F[y]/(f(y)) of F for which the equation f(y) = 0 is
“good enough” around the totally ramified place (in terms of Definition 3.1).
A LOCAL TO GLOBAL PRINCIPLE FOR DENSITIES 3
Let k,m be positive integers such that k < m and R be a domain. The question
whether a homomorphism of Rk in Rm can be extended to an automorphism of Rm
raised many interesting questions in the past (see for instance Serre’s Conjecture,
whichisprovenin[5,6]). InSection4weclosethe problemofcomputing thedensity
of homomorphisms of Hk in Hm which can be extended to automorphims of Hm. In
thecaseofH=F [x],thesehomomorphismsarisefromcontextofconvolutionalcodes
q
(see for example [7] or [8]) and their density was studied in [9] and [10]. In Theorem
4.4weshowthatthe densityofunimodular matricesoverH isa rationalnumberand
can be explicitly computed as soon as the complement of the holomorphy set S is
finite.
1.1. Preliminarydefinitionsand notations. LetF beafinitefield. Inthispaper
q
all the function fields are global and have full constant field F . We denote by O a
q P
valuationringoffunctionfieldF,havingmaximalidealP. Thesetofalltheplacesof
F willbe denotedbyP . IfS is apropersubsetofP ,we denoteby S the subsetof
F F t
placesofS ofdegreegreaterthant. Moreover,wewriteH todenotetheholomorphy
S
ring of S i.e. the intersection of all the valuation rings associated to the places of S:
H = O .
S P
P∈S
\
Sometimes, we will refer to S as the holomorphy set of H and to H as the holo-
S S
morphy ring of S. Holomorphy rings are integrally closed in F and any integrally
closed subring of F is an holomorphy ring [13, 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 F. The most immediate example of holomor-
phy ring is F [x] as this consists of the intersection of all the valuationrings of F (x)
q q
different from the valuation ring at infinity.
Let Div(F) be the set of divisors of F i.e. the free abelian group having as base
symbols the elements in the set P . For D = n P ∈ Div(F), we denote by
F P∈PF P
supp(D) the finite subset of P for which n is non-zero. Moreover, we will write
F P P
D ≥0 whenever n ≥0 for any P in P . Let
P F
Div+(F)={D ∈Div(F)|D ≥0}
Let D be the subset of divisors of Div+(F) having support over the complement of
S
S inP . AsD isadirectedset,wecandefine viaMoore-SmithConvergence(see[3,
F S
Chapter 2] and more specifically for this context [11]) a notion of limit over D . In
S
4 GIACOMOMICHELI
this context, we can give an upper density definition for a subset A of Hd as follows:
S
|A∩L(D)d|
D (A):=limsup
S qℓ(D)d
D∈DS
where L(D) is the Riemann-Roch space attached to the divisor D and ℓ(D) =
dimFq(L(D)). Analogously, one can give a notion of lower density DS by consid-
ering the inferior limit of the sequence. Whenever these two quantities are equal, we
say that a subset A of Hd has a well-defined density D (A)=D (A)=D (A).
S S S S
For a valuation ring O let us denote by O the completion of O with respect
P P P
to the P-adic metric. In addition, let us denote by µ the normalized Haar measure
P
b
on O with respect to the P-adic metric. For a subset U ⊆ O we denote by
P P
∂U the boundary of U with respect to the topology induced by the P-adic metric.
b b
For a multivariate polynomial f ∈ F[x ,...x ], we will denote by deg (f) (resp.
1 n xi
deg (f))the degreeoff inthe variablex (resp. the degreeofthe homogenization
hom i
off). Wheneverf hasallthecoefficientsinagivenvaluationringO ,wewilldenote
P
by degP (f) (resp. degP (f)) the degree of f in the variable x (resp. the degree of
xi hom i
the homogenizationoff) in(O /P)[x ,...,x ]. For a positiveinteger n anda given
P 1 n
commutative domain R, we will denote by GL (R) the set of n×n matrices whose
n
determinant is a unit of R.
2. The local to global principle for densities over global function
fields
Inthis sectionwedescribe thelocalto globalprinciple whichwillbe usedlateron.
This result is the function field analogue of [1, Lemma 20].
Theorem 2.1. Let d be a positive integer, S be a subset of places of F and H the
S
holomorphy ring of S. For any P ∈ S, let U ⊆ Od be a measurable set such that
P P
µ (∂U )=0. Suppose that
P P
b
(2.1) lim D ({a∈Hd |a∈U for some P ∈S })=0.
S S P t
t→∞
Let π :Hd −→2S defined by π(a)={P ∈S :a∈U }∈2S. Then
S P
(i) µ (U ) is convergent.
P P
P∈S
X
(ii) Let Γ⊆2S. Then ν(Γ):=D (π−1(Γ)) exists and ν defines a measure on 2S.
S
A LOCAL TO GLOBAL PRINCIPLE FOR DENSITIES 5
(iii) ν is concentrated at finite subsets of S. In addition, if T ⊆ S is finite we
have:
ν({T})= µ (U ) (1−µ (U )).
P P P P
!
PY∈T P∈YS\T
Proof. Throughout the proof S will be fixed, so we will denote H by H. What we
S
needtodoistotranslatetheproofof[12]tothecontextoffunctionfields. Essentially,
we need to understand how the measure of P-adic intervals can be translated into
density via the use of Riemann-Roch Theorem [13, Theorem 1.5.15]. Once this is
done, the same arguments of the proof of [1, Lemma 20] will apply to this context.
WedefineaP-interval inOP asthe set{x∈OP :x≡a mod PeP}forsomeeP ∈N
and a∈O . A P-box I will just be a product of P-intervals:
P P
b b
b IP ={x∈OPd :xj ≡aj mod PeP,j for j ∈{1,...d}}.
To simplify notation, we saby that a P-box is a P-cube if it has the form
CP ={x∈OPd :xi ≡yi mod PeP, i∈{1,...,d}}
for some eP ∈N and (y1,...b,yd)∈OPd. In other words, CP is the cartesian product
of intervals of equal length. We say that c = (y ,...,y ) is the center of the cube.
P 1 d
b
Let A be a finite subset of S. Let us now compute the density of the elements in
Hd which are mapped in a product of a finite number of P-boxes via the natural
embedding Hd −→ Od. Let I = I be such product of P-boxes. For
P∈A P P∈A P
anyP,theP-boxI canbecoveredwithafinitenumberl ofdisjointcubesofequal
PQ Q P
b
size e , as all the congruences can be decomposed in terms of the finest congruence,
P
given by max{e : j ∈{1,...d}}=:e . Therefore, one can write
P,j P
lP
I = I = C(i)
P P
P∈A P∈Ai=1
Y Y G
with µP(CP(i))=q−ddeg(P)eP independently of i.
We consider the diagram
Hd −−−ι−→ Od ⊇I
P∈A P
πJ Q bπJ′
(H/yJ)d −−−ψ−→ P∈A(OP/yPeP)d =:R
where J = PeP ⊆ H, the mapQι is the natural inclusion, and ψ is the iso-
P∈A b
morphism(coming from the Chinese Remainder Theorem) which makesthe diagram
Q
6 GIACOMOMICHELI
commutative. Ontheright-handside,wecanimmediatelycomputetheproductmea-
suremofI bylookingatits definition,getting m= P∈AlPq−ddeg(P)eP. Itremains
to show that the density of ι−1(I) is indeed m. For this, let us decompose I. Let I
Q P
be a finite set indexing the cubes which cover I and let I = I . Any given
P P∈A P
i = (i ) ∈ I determines a choice of cubes as follows: for each place P ∈ A we
P P∈A Q
select exactly one cube C(iP), having center c(iP). We now build a C as the product
P P i
C(iP). Clearly, the set of C ’s built in this way has cardinality l and
P∈A P i P∈A P
covers I via a disjoint union. If we can now prove that the density of ι−1(C ) is
Q Q i
independent of the choice of i∈I, then we will have that
(2.2) D(ι−1(I))=D(ι−1(C ))· l .
i P
P∈A
Y
To achieve this, we now explicitly compute the value D(ι−1(C )). As the diagram
i
above is commutative, we can equivalently compute the density of elements of Hd
falling into ψ−1π′(C ) via the map π . Let z ∈π−1ψ−1π′((c(i)) ). Notice that
J i J Ci J J P P∈A
we have
ι−1(C )=π−1ψ−1π′(C )=z +JH.
i J J i Ci
ObservethatforanydivisorD ∈D ,themapπ restrictedtoL(D)isF -linear. Let
S J q
g be the genus of F. Therefore if we denote by z the j-th component of z , we
Ci,j Ci
have
|L(D)∩(zCi,j +JH)|=|L(D)∩JH|= L D− ePP =qℓ(D−PP∈AePP),
(cid:12) !(cid:12)
(cid:12) PX∈A (cid:12)
(cid:12) (cid:12)
which for D of large degree, equals qdeg(D(cid:12)(cid:12)−PP∈AePP)+1−g by(cid:12)(cid:12)Riemann-Roch Theo-
rem. We can finally compute the density of elements mapping in the cube C (which
i
is in fact independent of i, as we wanted):
|L(D)d∩π−1ψ−1π′(C )|
D(ι−1(C ))= lim J J i
i D∈DS qℓ(D)d
d
=q−ℓ(D)d |L(D)∩(zCi,j +JH)|=q−(PP∈StePdeg(P))d.
j=1
Y
Using now Equation (2.2) we get the final claim by comparing m with D(i−1(I)).
Sincenowwehaveprovedthetheoremforboxes,allthe argumentsofthe proofof[1,
Lemma 20]arenow straightforwardto apply. Infact, suppose for a momentthat the
setofP’s in S forwhich U is different fromthe empty setis a finite setA. Now,let
P
T be a finite set of places. Assuming that µ (∂U ) = 0, one can cover each of the
P P
U ’s from the interior (resp. Uc) with a finite set of boxes which well approximate
P P
A LOCAL TO GLOBAL PRINCIPLE FOR DENSITIES 7
the measure µ (U ) (resp. µ (Uc)). In particular one has
P P P P
ν({T})≥ µ (U′ ) (1−µ (U′ )),
P P P P
!
PY∈T P∈YS\T
wheretheproductsabovearebothfiniteandU′ unionoftheboxesforeachP,where
P
the theorem holds. As we can apply the symmetric argument with a set of external
approximations U′′ we have
P
ν({T})≤ µ (U′′) (1−µ (U′′)),
P P P P
!
PY∈T P∈YS\T
from which the claim follows by letting the approximation get sharper and then
µ (U′′) and µ (U′ ) tend to µ(U ).
P P P P P
On the other hand, if A is an infinite set, one easily sees that an approximation
with finitely many U is good enough, as long as condition (2.1) is verified. To see
P
this, let T be a finite subset of S and let us recall that S is the set of places of S of
t
degree larger than t and then Sc is the subset of S consisting of places of degree less
t
than or equal to t. Observe that for a positive integer t such that Sc contains T we
t
can define a partial approximationof π−1({T})
W ={a∈Hd |a∈U ∀P ∈T, a∈/ U ∀P ∈Tc∩Sc}.
t S P P t
Notice that W contains π−1({T}) so D(π−1({T})) ≤ D(W ). In addition we have
t t
that
D(π−1({T}))≥D(W )−D(W \π−1({T})).
t t
Now, by letting t go to infinity and using condition (2.1) on D(W \π−1({T})) one
t
gets the claim. (cid:3)
The next Theorem ensures that when the U can be expressed in terms of poly-
P
nomialequations,Condition(2.1)is alwaysverified,similarlyto whathappens inthe
case of Ekedhal Sieve for integers [2].
Theorem 2.2. Let F/F be a global function field and S be a subset of P with
q F
finite complement. Let H be the holomorphy ring of S. Let f,g ∈H [x ,...,x ] be
S S 1 d
coprime polynomials. Then
(2.3) lim D {y ∈Hd : f(y)≡g(y)≡0 mod P for someP ∈S } =0.
S S t
t→∞
(cid:0) (cid:1)
Proof. If d = 1 there is nothing to prove so we can suppose d > 1. Without loss
of generality, we can also suppose deg (f) > 0. Since S will be fixed throughout
x1
8 GIACOMOMICHELI
the proof, we will denote H and D by H and D respectively. Let us recall that
S S
the places in S are in natural correspondence with the prime ideals of H, therefore
with a small abuse of terminology we will identify this two sets. We first fix t large
enough, so that degP (f) = deg (f) for any P of degree larger than t. Now fix D
x1 x1
large enough so that deg(D)>t. Let us also introduce new notation to simplify the
computations. For a divisor D, let us define
a (D):= {y ∈L(D)d : f(y)≡g(y)≡0 mod P for someP ∈S } q−dℓ(D),
t t
c(cid:12)(cid:12)P(D):= {y ∈L(D)d : f(y)≡g(y)≡0 mod P} q−dℓ(D).(cid:12)(cid:12)
Our first purpose is to(cid:12) estimate a (D) for t and D large. First(cid:12), we notice a simple
(cid:12) t (cid:12)
upper bound for a (D):
t
a (D)≤ c (D).
t P
P:deXg(P)>t
We now want to estimate the sum above for different regimes of deg(P) and deg(D).
In order to do so, let us further split the sum as
(2.4) c (D)+ c (D).
P P
P:t<degX(P)≤deg(D) P:deg(XP)>deg(D)
(I) (II)
Let us estimate|(I). First,{wze want to}give|a reasona{zble estima}te for c (D) in the
P
specified regime. Notice that for each point of z ∈Fd satisfying f(z)≡g(z)≡0
qdeg(P)
mod P there are at most |L(D −P)d| preimages of z in L(D)d, as the evaluation
map L(D) → L(D)(P) ⊆ F is linear and has kernel L(D −P). Let N be
qdeg(P) P
the number of F -points ofthe varietydefined by f andg whenreducedmodulo
qdeg(P)
P. Let g be the genus of F. By observing that ℓ(D) ≥ deg(D)+1−g and that
F F
ℓ(D−P)≤deg(D)−deg(P)+1 we get:
c (D)≤N |L(D−P)|dq−ℓ(D)d
P P
≤NPqd(deg(D)−deg(P)+1)q−(deg(D)+1−gF)d
=NPq(gF−deg(P))d.
As t can be chosen large enough to avoid the places of bad reduction, we can
estimate classically N as Cq(d−2)deg(P) for some constant C. It follows that
P
c (D)≤ Cq−2deg(P)
P
P:t<degX(P)≤deg(D) P:t<degX(P)≤deg(D)
e
A LOCAL TO GLOBAL PRINCIPLE FOR DENSITIES 9
for some other constant C.
Let us estimate (II). Let (f,g) be the ideal generated by f,g in F[x ,...,x ] and
1 d
e
let J = (f,g)∩F[x ,...,x ]. Since (f,g) has codimension 2, J is principal. Let
2 d
h ∈ F[x ,...x ] be the generator of J, which can be chosen with coefficients over
2 d
H by multiplying by an appropriate element in H. Let us also assume without loss
of generality that deg (h) > 0. Let now D be so large that modulo every prime
x2
P of degree larger than deg(D), we have degP (h) = deg (h). Consider now all
hom hom
the elements of L(D)d ending with a fixed r =(r ,...,r )∈L(D)d−1 and for which
2 d
h(r) 6= 0. Let us estimate their contribution to each c (D) in the sum (II). Let I
P r
be the product of all the prime ideals P of H such that deg(P) > deg(D) and for
which there exists x ∈ H such that f(x,r ,...,r ) ≡ g(x,r ,...,r ) ≡ 0 mod P
2 d 2 d
(this set is finite as h(r) 6= 0). If we denote by u the number of distinct primes
r
appearing in the factorization of I , the contribution of all the d-tuples ending with
r
r is bounded by u degP (f) = u deg (f). By the definition of h, it is clear that
r x1 r x1
hH[x ,...,x ] ⊆ (f,g) ∩ H[x ,...,x ]. If we denote by (f,g) the projection of
1 d 1 d Ir
(f,g)∩H[x ,...,x ] in (H/I )[x ,...,x ], we have that
1 d r 1 d
hH/I [x ,...,x ]⊆(f,g) .
r 1 d Ir
Therefore this in turn implies that (r ,...r ) satisfies h(r ,...,r ) ≡ 0 mod I .
2 d 2 d r
Now, the key observation to get the final estimate for (II) is the following: deg(D)
waschosenlargein such a waythat the homogeneousdegree ofh is constantmodulo
P for any P of degree larger than deg(D). Recall now that every prime ideal P
appearing in the factorization of I has degree larger than D, therefore
r
u deg(D)<deg(I )≤deg(D)deg (h)+C
r r hom
where the constant C depends on the leading coefficients of h (and independent of
D), from which it follows that u <deg (h), for D large enough.
r hom
The readershouldnow notice that in (II), an elementin L(D)d ending with r (i.e.
of the form (r ,r)), cannot contribute more than 1 for eachc (D), as the evaluation
1 P
map L(D) →L(D)(P) is an injection to F . It follows easily that the set of all
qdeg(P)
thed-tuplesendingwithrcontributeatmostu deg (f)tothewholesum(II).Now,
r x1
using the observationsabove andrecalling that we also haveto take into accountthe
sizeofthe setT ofthe(d−1)-tuplesr =(r ,...,r )suchthath(r)=0forr ∈Hd−1,
2 d
we finally get
c (D)≤deg (h)qℓ(D)(d−1)q−ℓ(D)d+ q−ℓ(D)du deg (f)
P xi r x1
P:deg(PX)>deg(D) r∈L(DX)d−1\T
10 GIACOMOMICHELI
≤q−ℓ(D)(deg (h)deg (f)−1).
hom x1
Atthispointweobservethattheestimatesof(I)and(II)onlyholdforlargevalues
of t and D, 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
t so large that the primes of bad reduction are avoided. For estimate (II), take D so
large that
• its degree is larger than t,
• the homogeneous degree of h is constant with respect to all places of S of
degree larger than deg(D),
• the degree of f with respect to x is constant for all places of S of degree
1
larger than deg(D).
We can now safely use the two estimates to complete the proof:
lim lim a (D)≤ lim lim c (D)=
t P
t→∞D→∞ t→∞D→∞
P:deg(P)>t
X
lim lim c (D)+ c (D)≤
P P
t→∞D→∞
P:deg(PX)>deg(D) P:t<degX(P)≤deg(D)
lim lim q−ℓ(D)(deg (h)deg (f)−1)+ Cq−2deg(P) =
t→∞D→∞ hom x1
P:t<degX(P)≤deg(D)
e
lim Cq−2deg(P).
t→∞
P:t<Xdeg(P)
e
This completes the proof, since the sum above is the tail of a subseries of the zeta
function of F evaluated at 2, which is converging. (cid:3)
The reader should notice that the counting technique using in the estimate of (II)
is similar to the one used in the case of Z in the main result of [2].
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
F/F isafunctionfieldwithfullconstantfieldF andE isafinite extensionofF,we
q q
recall that an extension of places Q|P is said to be totally ramified if the dimension
of O /Q as an O /P vector space is equal to 1.
Q P
The notion of “good” total ramification is encoded in the following
