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A 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 holomorphy 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 considering 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