Chebotarev density theorem in short intervals for extensions of F q (T)
Lior Bary-Soroker, Ofir Gorodetsky, Taelin Karidi, Will Sawin
aa r X i v : . [ m a t h . N T ] N ov CHEBOTAREV DENSITY THEOREM IN SHORT INTERVALSFOR EXTENSIONS OF F q ( T ) LIOR BARY-SOROKER, OFIR GORODETSKY, TAELIN KARIDI, AND WILL SAWIN
Abstract.
An old open problem in number theory is whether Chebotarev den-sity theorem holds in short intervals. More precisely, given a Galois extension E of Q with Galois group G , a conjugacy class C in G and an 1 ≥ ε >
0, one wantsto compute the asymptotic of the number of primes x ≤ p ≤ x + x ε with Frobe-nius conjugacy class in E equal to C . The level of difficulty grows as ε becomessmaller. Assuming the Generalized Riemann Hypothesis, one can merely reachthe regime 1 ≥ ε > /
2. We establish a function field analogue of Chebotarevtheorem in short intervals for any ε >
0. Our result is valid in the limit whenthe size of the finite field tends to ∞ and when the extension is tamely ramifiedat infinity. The methods are based on a higher dimensional explicit Chebotarevtheorem, and applied in a much more general setting of arithmetic functions,which we name G -factorization arithmetic functions. Introduction
The goal of this paper is to provide support to an open problem in the distri-bution of primes with a given Frobenius conjugacy class. We do this by resolvinga function field version of the problem. We start by introducing the problem innumber fields, and then we present our results.1.1.
The Chebotarev Density Theorem in short intervals.
One of the maintheorems in algebraic number theory is the Chebotarev Density Theorem aboutthe distribution of Frobenius conjugacy classes in Galois extensions of global fields.To keep the presentation as simple as possible, we fix the base field to be Q . Let E be a finite Galois extension of Q with Galois group G = Gal( E/ Q ) and withring of integers O E . For a prime number p , we define (cid:18) E/ Q p (cid:19) ⊆ G, Mathematics Subject Classification.
Primary 11N05; Secondary 11T06, 12F10.LBS was partially supported by a grant of the Israel Science Foundation. Part of the workwas done while LBS was a member of Simons CRM Scholar-in-Residence Program.The research of OG was supported by the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n o to be the set of all σ ∈ G for which there exists a prime P of E lying above p suchthat σ ( x ) ≡ x p mod P , for all x ∈ O E . If p is unramified in E , then (cid:16) E/ Q p (cid:17) is called the Frobenius at p and it is a conjugacy class in G .The Chebotarev Density Theorem says that as p varies, the Frobenius equidis-tributes in the set of conjugacy classes (with the obvious weights). More precisely,let π ( x ) = { p ≤ x : p prime number } be the prime counting function. By the Prime Number Theorem, we know thatLi( x ) = R x dt log t ∼ x log x well approximates π ( x ); that is to say, for any A > π ( x ) = Li( x ) + O A ( x/ (log x ) A ) , x → ∞ . For a conjugacy class C ⊆ G , let π C ( x ; E ) = (cid:26) p ≤ x : p prime number and (cid:18) E/ Q p (cid:19) = C (cid:27) be the function that counts primes with Frobenius equals to C . The ChebotarevDensity Theorem [35, Theorem 2.2, Chapter I] says that(1) π C ( x ; E ) ∼ | C || G | Li( x ) , x → ∞ . This theorem is a vast generalization of the Prime Number Theorem for arithmeticprogressions which follows from (1) applied to cyclotomic fields.It is both natural and important for applications to consider the ChebotarevDensity Theorem in short intervals. Balog and Ono [2] studied the non-vanishingof Fourier coefficients of modular forms in short intervals. For this applicationthey prove that(2) π C ( x + y ; E ) − π C ( x ; E ) ∼ | C || G | y log x , x → ∞ , for x − /c ( E )+ ε ≤ y ≤ x , and where c ( E ) > E (and in fact only on [ E : Q ]). Thorner [38, Corollary 1.1] improves the range of y for which (2) holds true.Naively, we expect that (2) holds for any y = y ( x ) ≤ x that grows ‘sufficientlyfast’. From (1), it follows that the average gap between primes with (cid:16) E/ Q p (cid:17) = C is | G || C | log x . Thus it makes sense to only consider y -s satisfying lim x →∞ y log x = ∞ .The Maier phenomenon [24] about primes tells us that (2) fails unless y ≫ (log x ) A for all A >
1. A folklore conjecture says that for any fixed ε > y = x ε theasymptotic formula (2) holds true: HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 3
Conjecture 1.1.
Let E/ Q be a Galois extension with Galois group G , ≥ ε > ,and C ⊆ G a conjugacy class. Then π C ( x + x ε ; E ) − π C ( x ; E ) ∼ | C || G | x ε log x , x → ∞ . When E = Q , Conjecture 1.1 reduces to primes in short intervals, and we referthe reader to the excellent survey [36] for further reading on this case.One approach for Conjecture 1.1 is to study the error term in Chebotarev Den-sity Theorem. Let ∆ E ; C ( x ) = π C ( x, E ) − | C || G | Li( x )and let d E be the absolute value of the discriminant of E . Under the RiemannHypothesis for the Dedekind zeta function ζ E of E , Lagarias and Odlyzko [20] gavethe bound(3) ∆ E ; C ( x ) = O ( | C | x / (log x + log d E | G | )) , where the implied constant is effective and absolute. We borrow the above for-mulation from [34, Theorem. 4]. See [15, Cor. 1] for a calculation of the impliedconstants and [25, Cor. 3.7] for an improved dependence on | C | .From (3), in particular conditionally on the Riemann Hypothesis for ζ E , oneimmediately gets Conjecture 1.1 for any ε > /
2. As discussed above, there areunconditional results. However, the case ε ≤ / The Chebotarev Density Theorem in function fields.
The functionfield Chebotarev Density Theorem has a long history, starting with Reichardt [30]who first established it. Lang [21] gave a square-root cancellation, based on theRiemann Hypothesis for curves over finite fields, and explicit estimates were givenby Cohen and Odoni in the appendix to [10] and by Halter-Koch [16, Satz 2].Fried and Jarden [13, Proposition 6.4.8] and Murty and Scherk [19] gave explicitbounds on the error term.However, unlike the number field case, there are two obstructions in the Cheb-otarev Density Theorem. One obstruction comes from the arithmetic part of theFrobenius and the other appears when considering short intervals. For a moreconcise presentation of the obstruction we introduce some notation.Let q be a power of a prime number p , let F q be the finite field of q elements,and let F q ( T ) be the field of rational functions over F q . We define P n,q as theset of primes of F q ( T ) of degree n . If n >
1, we identify P n,q with the set ofmonic irreducible polynomials in the ring of polynomials F q [ T ], and we identify P ,q with the degree-1 monic polynomials and 1 /T ‘the infinite prime’. The prime LIOR BARY-SOROKER, OFIR GORODETSKY, TAELIN KARIDI, AND WILL SAWIN polynomial theorem says that π q ( n ) = P n,q = q n n (1 + O ( q − n/ )) , and so we use q n n as an estimate for π q ( n ).Given a Galois extension E/ F q ( T ) with Galois group G = Gal( E/ F q ( T )), foreach P ∈ P n,q we define the Frobenius at P ,(4) (cid:18) E/ F q ( T ) P (cid:19) ⊆ G, as in the number field setting: it is the set of σ ∈ G for which there exists a prime P of E lying above P such that σ ( x ) ≡ x | P | mod P , for all x ∈ E which are integral at P and where | P | = q n . As before, if P isunramified in E , then (cid:16) E/ F q ( T ) P (cid:17) is a conjugacy class in G . Given a conjugacyclass C ⊆ G , we set π C ; q ( n ; E ) = (cid:26) P ∈ P n,q : (cid:18) E/ F q ( T ) P (cid:19) = C (cid:27) , the function that counts primes with Frobenius C .To describe the obstruction for a conjugacy class to be a Frobenius of a primeof degree n , we introduce the restriction map. Let F q ν be the field of scalars of E , that is, the algebraic closure of F q in E . Let φ : F q ν → F q ν , φ ( x ) = x q bethe generator of the cyclic group G = Gal( F q ν / F q ). We have the restriction ofautomorphisms map G ։ G , which is surjective. Since G is abelian, if C ⊆ G is a conjugacy class, then all σ ∈ C map to the same power φ C of φ . Then theChebotarev Density Theorem for function fields says that if φ C = φ n , then(5) (cid:12)(cid:12)(cid:12)(cid:12) π C ; q ( n ; E ) − ν | C || G | q n n (cid:12)(cid:12)(cid:12)(cid:12) ≪ ν | C || G | max { genus( E ) , | G | ν } q n/ n and otherwise π C ; q ( n ; E ) = 0. The implied constant is absolute.Next we turn to short intervals. Following Keating and Rudnick [18, § f of degree n with parameter0 ≤ m < n to be I ( f, m ) = { f + g : deg g ≤ m } . The size of the interval is I ( f, m ) = q m +1 . To compare with the number field interval { x ≤ n ≤ x + x ε } , we see that x corresponds to | f | = q n and x ε corresponds to q m +1 , so ε = m +1 n . Having theanalogy with number fields in mind, one would naively expect that (5) implies a HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 5
Chebotarev Density Theorem for the short interval I ( f, m ) whenever m + 1 > n/ ε > / π C ; q ( I ( f, m ); E ) = (cid:26) P ∈ P n,q ∩ I ( f, m ) : (cid:18) E/ F q ( T ) P (cid:19) = C (cid:27) , then unlike in the number field case, we cannot express π C ; q ( I ( F, m ); E ) as thedifference of values of π C ; q ( n ; E ) in order to utilize the error term (5).In fact, there is an obstruction to Chebotarev in short intervals coming from thefact that E is not necessarily linearly disjoint from the cyclotomic field L n − m − associated to a power of the infinite prime (see [32, Chapter 12]). Thus one needsto modify the asymptotic formula according to the intersection of E and L n − m − .Applying (5) to the compositum of EL n − m − would yield a Chebotarev in shortintervals for ε > /
2. We note that the extensions L n − m − are wildly ramified atthe infinite prime.Our main result is a Chebotarev Density Theorem for short intervals with any ε > Theorem 1.2.
For every
B > there exists a constant M B satisfying the followingproperty. Let q be a prime power. Let n > m ≥ if q is odd and n > m ≥ otherwise. Let G be a finite group and let E/ F q ( T ) be a geometric G -extension.Assume that the infinite prime is tamely ramified in the fixed field E ab in E of thecommutator of G . Further assume that genus( E ) , n, | G | ≤ B . Let f ∈ F q [ T ] bemonic of degree n . Then (cid:12)(cid:12)(cid:12)(cid:12) q m +1 π C ; q ( I ( f, m ); E ) − | C || G | n (cid:12)(cid:12)(cid:12)(cid:12) ≤ M B q − / . It follows in particular that for any ε > n →∞ lim q →∞ max f,E (cid:12)(cid:12)(cid:12)(cid:12) q m +1 π C ; q ( I ( f, m ); E ) − | C || G | n (cid:12)(cid:12)(cid:12)(cid:12) = 0 , where E runs over all G -Galois extensions of F q ( T ) of bounded genus that aretamely ramified at infinity and f ∈ F q [ T ] runs over all monic polynomials ofdegree n . Hence we have proved a version of Conjecture 1.1 in the function fieldsetting.It would be desirable to change the order of the limits in (6). As explainedabove, for ε > / ε ≤ , it is open and we know of no approach to attack it. A yet more challengingtask is to fix q and take n → ∞ , and also here the problem is open, and we knowof no approach to attack it.Our method gives more general results, and may be applied for instance toproblems about norms. In Theorem 5.1 we count, in the large- q limit, how many LIOR BARY-SOROKER, OFIR GORODETSKY, TAELIN KARIDI, AND WILL SAWIN polynomials g ∈ I ( f, m ) satisfy ( g ) = Norm E/ F q ( T ) I for some ideal I in O E . Ourmost general result is given in Theorem 4.3, for which the terminology of § F q ( T ). 2. Methods
We outline our approach when E is a geometric extension of F q , which, underthe notation used in (5), means that ν = 1. We introduce a general notion of G -factorization arithmetic functions (Definition 3.1), which are arithmetic functionson F q [ T ], whose value on a polynomial f ( T ) depends only on the Frobenius atthe prime factors of f ( T ). These functions are closely related to Serre’s Frobenianfunctions [33] and to the extensions by Odoni [27, 28] and Coleman [11].Given a short interval, we relate such an arithmetic function ψ to a class function ψ ′ on a subgroup of the wreath product G ≀ S n using a higher dimensional functionfield Chebotarev Density Theorem. The main property of this association is thatthe expected value of ψ on the short interval is asymptotically equal to the averageof ψ ′ on the subgroup, as q → ∞ (Theorem 4.3). The main technical part of thework is to compute the subgroup: it equals to the wreath product G ≀ S n itself.Applying the above to the indicator function of primes with Frobenius C (Ex-ample 3.2) reduces Theorem 1.2 to either a combinatorial computation in G ≀ S n or the classical Chebotarev Density Theorem.Finally, for the subgroup computation, we take an algebraic approach, usingelementary group theory and Artin-Schreier and Kummer theories. Our methodsare in the spirit of the works [9, 4, 3] which assume genus( E ) = 0 and G cyclic.3. G -factorization arithmetic functions For a finite group G we consider the spaceˆΩ G = { σI : σ ∈ G, I ≤ G } of all cosets of subgroups. The group G acts on ˆΩ G by conjugation and we writeΩ G = ˆΩ G /G for the set of conjugacy classes of cosets of subgroups. If I = 1 is the trivialsubgroup, we identify σI ∈ ˆΩ G with σ . So the image of σI in Ω G is the conjugacyclass C = { τ − στ : τ ∈ G } of σ .We want to encode the combinatorial data of degrees, multiplicities, and theFrobenius at the prime factors of a polynomial. A G -factorization type is afunction λ : N × N × Ω G → Z ≥ HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 7 with finite support. We define Λ = Λ G to be the set of all G -factorization types.For λ ∈ Λ we let deg( λ ) = X d,e,ω λ ( d, e, ω ) de, where the sum runs over d, e ∈ N and ω ∈ Ω G . For a monic polynomial f ∈ F q [ T ]with prime factorization f = P e · · · P e r r and for a G -Galois extension E/ F q ( T ) wedefine λ f ; E/ F q ( T ) ( d, e, ω ) = (cid:26) i : deg P i = d, e i = e, (cid:18) E/ F q ( T ) P i (cid:19) = ω (cid:27) . When there is no risk of confusion we simplify the notation and write λ f for λ f ; E/ F q ( T ) . Obviously, we have that deg( f ) = deg( λ f ). Definition 3.1. A G -factorization arithmetic function is a function on G -factorization types. We denote byΛ ∗ = { ψ : Λ → C } the space of G -factorization arithmetic functions.Given a G -Galois extension E/ F q ( T ), each ψ ∈ Λ ∗ induces an arithmetic func-tion ψ E/ F q ( T ) on F q [ T ] by setting ψ E/ F q ( T ) ( f ) = ψ ( λ f ; E/ F q ( T ) ) , for monic f ∈ F q [ T ]. By abuse of notation, ψ E/ F q ( T ) is also called G -factorizationarithmetic function.Definition 3.1 vastly extends some families of arithmetic functions – see [31, 5]for similar definitions in the cases E = F q ( T ) ( G = { e } ) and E = F q ( √− T )( G = Z / Z ).The following example of a G -factorization arithmetic function is crucial for ourmain result. Example 3.2.
Fix a conjugacy class C ⊆ G . Consider the G -factorization arith-metic function1 C ( λ ) = ( , if λ ( d, e, ω ) > ⇒ ω = C and d = deg λ, , otherwise.For any G -Galois extension E/ F q ( T ) and monic f ∈ F q [ T ] we have1 C,E/ F q ( T ) ( f ) = ( , if f is irreducible and (cid:16) E/ F q ( T ) f (cid:17) = C, , otherwise . LIOR BARY-SOROKER, OFIR GORODETSKY, TAELIN KARIDI, AND WILL SAWIN G -factorization arithmetic functions on wreath products Recall the construction of the permutational wreath product : Let S n bethe symmetric group on X = { , , . . . , n } (with left action ( σ, x ) σ.x ), let G be a finite group, and let G X := { ξ : X → G } be the group of functions from X to G with pointwise multiplication. Then S n acts (from the right) on G X by ξ σ ( x ) = ξ ( σ.x ) , σ ∈ S n , x ∈ X. The corresponding semidirect product G ≀ S n := G X ⋊ S n is the wreath product of G and S n . For the reader’s convenience we recall thatthe multiplication is given by( ξ , σ )( ξ , σ ) = ( ξ ξ σ − , σ σ ) , ξ , ξ ∈ G X , σ , σ ∈ S n . The imprimitive action of G ≀ S n on the set G × X , given explicitly by(7) ( ξ, σ ) . ( g, x ) = ( ξ ( σ.x ) g, σ.x ) , ξ ∈ G X , σ ∈ S n , g ∈ G, x ∈ X, makes G ≀ S n into a transitive permutation group.For ( ξ, σ ) ∈ G ≀ S n we attach a G -factorization type: Let σ = σ · · · σ r bethe factorization of σ to disjoint cycles. We include the trivial cycles so that P ri =1 ord( σ i ) = n . For each i = 1 , . . . , r , if we write σ i = ( j · · · j d ), then we set C ( ξ,σ ) ,σ i to be the conjugacy class in G of the element ξ ( j d ) · · · ξ ( j ) . The conjugacy class C ( ξ,σ ) ,σ i is well defined, since ξ ( j a ) · · · ξ ( j ) ξ ( j d ) · · · ξ ( j a +1 ) isconjugate to ξ ( j d ) · · · ξ ( j ). Now we set(8) λ ( ξ,σ ) ( d, e, ω ) = ( , if e > , { i : ord( σ i ) = d, C ( ξ,σ ) ,σ i = w } , if e = 1 . Any ψ ∈ Λ ∗ induces a function ψ G ≀ S n : G ≀ S n → C by ψ G ≀ S n (( ξ, σ )) = ψ ( λ ( ξ,σ ) )and we refer to such functions on G ≀ S n as G -factorization arithmetic functions aswell. Below we show that the set of G -factorization arithmetic functions on G ≀ S n actually coincides with the set of class functions. Example 4.1.
Recall the G -factorization arithmetic function 1 C from Exam-ple 3.2. Then, for ( ξ, σ ) ∈ G ≀ S n we have1 C ( ξ, σ ) = ( , if σ is an n -cycle and C ( ξ,σ ) ,σ = C, , otherwise. HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 9
Next, we prove that conjugation in G ≀ S n preserve the G -factorization type. Let τ ∈ S n and identify it with (1 , τ ) ∈ G ≀ S n . Then τ στ − = ρ · · · ρ r with ρ i = τ σ i τ − .If σ i = ( j · · · j d ), then ρ i = ( τ ( j ) · · · τ ( j d )). Now, as τ ( ξ, σ ) τ − = ( ξ τ − , τ στ − )we have that(9) ξ τ − ( τ ( j d )) · · · ξ τ − ( τ ( j )) = ξ ( j d ) · · · ξ ( j )and so C ( ξ,σ ) ,σ i = C τ ( ξ,σ ) τ − ,ρ i . We thus conclude that λ ( ξ,ρ ) = λ τ ( ξ,ρ ) τ − . Similarly, if η ∈ G X and we identify it with ( η, ∈ G ≀ S n , then(10) η ( ξ, σ ) η − = ( ηξη − σ − , σ )and we have( ηξη − σ − )( j d ) · · · ( ηξη − σ − ( j ))= η ( j d ) ξ ( j d ) η ( j d − ) − · η ( j d − ) · · · η ( j ) − · η ( j ) ξ ( j ) η ( j d ) − = η ( j d ) ξ ( j d ) · · · ξ ( j ) η ( j d ) − . Here we used that σ − i = ( j d · · · j ). In particular, C ( ξ,σ ) ,σ i = C η ( ξ,σ ) η − ,σ i and thus λ ( ξ,ρ ) = λ η ( ξ,ρ ) η − . We thus deduce that if ( ξ, σ ) and ( η, ρ ) are conjugate, then λ ( ξ,ρ ) = λ ( η,ρ ) .The converse is also true. Indeed, λ ( ξ,σ ) = λ ( ζ,ρ ) implies that we have r conjugacyclasses C , . . . , C r (possibly with repetitions) and factorization to disjoint cycles ρ = ρ · · · ρ r and σ = σ · · · σ r such that ord( σ i ) = ord( ρ i ) and C ( ξ,σ ) ,σ i = C i = C ( ζ,ρ ) ,ρ i . Without loss of generality we may assume that σ = ρ and σ i = ρ i for all i (indeed,conjugate by τ ∈ S n such that τ σ i τ − = ρ i for all i and use (9)). Thus, if σ i = ( j · · · j d ), then gξ ( j d ) · · · ξ ( j ) g − = ζ ( j d ) · · · ζ ( j )for some g ∈ G . By (10) it suffices to find η ∈ G X such that ηξη − σ − = ζ . Defining η ( j d ) = g and η ( j a ) = ζ − ( j a +1 ) η ( j a +1 ) ξ ( j a +1 ) , ≤ a ≤ d − σ i , for each σ i gives the desired solution. We thus proved Lemma 4.2.
The elements ( ξ, σ ) , ( ζ , ρ ) ∈ G ≀ S n are conjugate if and only if λ ( ξ,σ ) = λ ( ζ,ρ ) .In particular, every class function on G ≀ S n may be realized as a G -factorizationarithmetic function. We prove the following general theorem which connects the averages of a G -factorization arithmetic function on a short interval to the average on the wreathproduct. A piece of notation is needed: for a non-empty finite set X and a function ψ on X we denote the mean value by h ψ ( f ) i f ∈ X := 1 X X f ∈ X ψ ( f ) . Theorem 4.3.
For every
B > there exists a constant M B satisfying the followingproperty. Let q be a prime power. Let n > m ≥ if q is odd and n > m ≥ otherwise. Let G be a finite group and let E/ F q ( T ) be a geometric G -extension.Assume that the infinite prime is tamely ramified in the fixed field E ab in E of thecommutator of G . Let f ∈ F q [ T ] be monic of degree n , and ψ ∈ Λ ∗ . Assume that genus( E ) , n, | G | ≤ B . Then (cid:12)(cid:12)(cid:12)(cid:10) ψ E/ F q ( T ) ( f ) (cid:11) deg( f − f ) ≤ m − h ψ G ≀ S n ( τ ) i τ ∈ G ≀ S n (cid:12)(cid:12)(cid:12) ≤ M B q − / max deg( λ )= n | ψ ( λ ) | . We postpone the proof of Theorem 4.3 to § Applications of Theorem 4.3
From Theorem 4.3 it follows immediately that in the large- q limit, the averageon a short interval is the same as on the ‘long interval’ — the set of all degree- n monics M n,q = I ( T n , n − . Moreover, Theorem 4.3 reduces the computations of averages of arithmetic func-tions to combinatorics of group theory. This also works vice versa.5.1.
Proof of Theorem 1.2.
We give two proofs to exemplify the ways to applyTheorem 4.3.First proof: The assumptions allow us to apply Theorem 4.3 with the G -factorization arithmetic function 1 C , and to get that the average on a short intervalis the same as over a long interval. The latter is given by (5), as needed.Second proof: The assumptions allow us to apply Theorem 4.3 with the G -factorization arithmetic function 1 C , and to get that the average on a short in-terval is the same as on the wreath product. We compute the latter: UsingExample 4.1, we find that 1 C ( ξ, σ ) = 0 implies that σ = ( j · · · j n ) is an n -cycleand ξ ( j n ) · · · ξ ( j ) ∈ C . So we may choose ξ ( j ) , . . . , ξ ( j n − ) arbitrarily and thenwe have | C | choices for ξ ( j n ). So h C ( ξ, σ ) i ( ξ,σ ) ∈ G ≀ S n = ( n − | G | n − | C | n ! | G | n = 1 n | C || G | , as needed. (cid:3) HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 11
Norms in short intervals.
Here we discuss two G -factorization arithmeticfunctions related to norms, and our results on their mean value in short intervals.For a function field E/ F q ( T ), we define the following arithmetic functions. For f ∈ M n,q , we define b E/ F q ( T ) ( f ) = ( , if ∃ I ⊆ O E : ( f ) = Norm E/ F q ( T ) ( I ) , , otherwise, r E/ F q ( T ) ( f ) = { I ideal in O E : Norm E/ F q ( T ) ( I ) = ( f ) } . The number field versions of r, b were studied extensively: Let E/ Q be a finiteextension. Odoni [26, Thm. 1] computed the asymptotic of the mean value of b E/ Q .When E/ Q is Galois, the work of Ramachandra [29] gives the mean value of b E/ Q in [ x, x + x ε ] for some 0 < ε < r E/ Q , and studied the error term. Werefer to Bourgain and Watt [7, Thm. 2] for the state-of-the-art result on the errorterm when E = Q ( i ), and to Lao [22] for more general E . These results inparticular gives the expected asymptotics for the mean value of r E/ Q in [ x, x + x ε ]for some 0 < ε < b E/ F q ( T ) in long intervals when E/ F q ( T ) is Galois. This is to be donein the most general limit q n → ∞ . Appendix A also treats r E/ F q ( T ) for which therationality of the corresponding Dedekind zeta function gives a closed formula forthe mean value.The result to be presented is a computation of the mean values in short intervals. Theorem 5.1.
For every
B > there exists a constant M B satisfying the followingproperty. Let q be a prime power. Let n > m ≥ if q is odd and n > m ≥ otherwise. Let G be a finite group and let E/ F q ( T ) be a geometric G -extensionwhich has is tamely ramified at the infinite prime. Let f ∈ F q [ T ] monic of degree n . Assume that genus( E ) , n, | G | ≤ B . Then (cid:10) b E/ F q ( T ) ( f ) (cid:11) deg( f − f ) ≤ m = 1 + O B ( q − / ) , (cid:10) r E/ F q ( T ) ( f ) (cid:11) deg( f − f ) ≤ m = (cid:18) n + | G | − n (cid:19) + O B ( q − / ) . To see how Theorem 5.1 is deduced from Theorem 4.3 we need to express r, b as G -factorization arithmetic functions (Example 5.3) and to compute the meanvalue on the wreath product, or alternatively apply the results from Appendix A.Let E/ F q ( T ) be a Galois extension. Given a prime polynomial P ∈ F q [ T ], wedenote by g ( P ; E ), f ( P ; E ) and e ( P ; E ) the number of distinct primes in E lyingabove P , the inertia degree of P in E and the ramification index of P , respectively. Lemma 5.2.
Let E/ F q ( T ) be a geometric G -extension.(1) The functions b E/ F q ( T ) and r E/ F q ( T ) are multiplicative. (2) Let f ∈ M n,q with prime factorization f = Q ki =1 P a i i . Then (11) b E/ F q ( T ) ( f ) = ( , if f ( P i ; E ) | a i for all i, , otherwise , and if we put b i = a i /f ( P i ; E ) and g i = g ( P i ; E ) , then we have (12) r E/ F q ( T ) ( f ) = b E/ F q ( T ) ( f ) · k Y i =1 (cid:18) b i + g i − g i − (cid:19) . Proof.
Let P be a prime polynomial and let P be a prime ideal of O E lying above P . Then Norm E/ F q ( T ) P = ( P ) f ( P ; E ) . By multiplicativity of the norm map and by unique factorization in O E , it followsthat the image of Norm E/ F q ( T ) on the non-zero ideals in O E is the semigroupgenerated by { ( P ) f ( P ; E ) } P ∈P q , which establishes (11). It now immediately followsthat b E/ F q ( T ) is multiplicative.If f and f are relatively prime polynomials, then from unique factorizationof ideals in O E , every ideal I of O E with Norm E/ F q ( T ) I = ( f f ) has a uniquefactorization I = I I , with Norm E/ F q ( T ) I j = ( f j ). Indeed, if I = Q P a i i take I j bethe product of P a i i with P i | f j . Thus, r E/ F q ( T ) ( f f ) = X Norm E/ F q ( T ) I =( f f ) X Norm E/ F q ( T ) I =( f )Norm E/ F q ( T ) I =( f ) r E/ F q ( T ) ( f ) r E/ F q ( T ) ( f ) . This implies that r E/ F q ( T ) is multiplicative. In particular, it suffices to prove (12)for f = P a a prime power.If f ( P ; E ) ∤ a , then b E/ F q ( T ) ( P a ) = 0, hence also r E/ F q ( T ) ( P a ) = 0. Assume nowthat f ( P ; E ) | a and let b = a/f ( P ; E ). Let P , . . . , P g be the primes of O E lyingabove P . Since Norm E/ F q ( T ) P j = P f ( P ; E ) , the solutions to Norm E/ F q ( T ) I = P a ,are of the form I = Q gj =1 P c j j with c j ≥ P gj =1 c j = b . As there are (cid:0) b + g − g − (cid:1) many such sequences of c j , the proof is done. (cid:3) Lemma 5.2 allows us to realize b E/ F q ( T ) , r E/ F q ( T ) as G -factorization arithmeticfunctions. Example 5.3.
Let ω ∈ Ω G , and let Σ ∈ ω . So Σ is a coset of a subgroup of G ,say Σ = σI . Let e ω = | I | , f ω = [ h σ, I i : I ], and g w = | G | /e w f w . Now we define the G -factorization arithmetic functions b ( λ ) = ( , if λ ( d, a, ω ) > ⇒ f ω | a, , otherwise. r ( λ ) = b ( λ ) · Y ( d,a,w ) (cid:18) a/f w + g w − g w − (cid:19) λ ( d,a,w ) (13) HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 13
Let E/ F q ( T ) be a geometric G -Galois extension. Then(14) b E/ F q ( T ) ( f ) = b ( λ f ; E/ F q ( T ) ) and r E/ F q ( T ) ( f ) = r ( λ f ; E/ F q ( T ) ) . Indeed, by (11) and (12), it suffices to note that w = (cid:16) E/ F q ( T ) P (cid:17) , then e w = e ( P ; E ), f w = f ( P ; E ), and g w = g ( P ; E ). Proof of Theorem 5.1.
By Theorem 4.3, it suffices to compute the average of the G -factorization arithmetic functions b and r given in (14) on the group G ≀ S n . Forbrevity we compute them together by computing the average of r s for any s ∈ C (and noting that r = r and b = r ). Put N = | G | and let s ∈ C . We show that (cid:10) r sG ≀ S n ( ξ, σ ) (cid:11) ( ξ,σ ) ∈ G ≀ S n = (cid:18) n + N s − − n (cid:19) . Let σ = σ · · · σ r be the factorization of σ ∈ S n to disjoint cycles and let ξ ∈ G n .Recall that if we write σ i = ( j · · · j ℓ ), then C ( ξ,σ ) ,σ i is defined to be the conjugacyclass of the element ξ ( j ℓ ) · · · ξ ( j ) . Let d, a ≥ ω ∈ Ω G with λ ( ξ,σ ) ( d, a, ω ) >
0. Then a = 1 and ω = C ( ξ,σ ) ,σ i for some i . In particular, e ω = 1, and thus f ω = 1 if and only if C ( ξ,σ ) ,σ i = 1, where e ω and f ω are as defined in Example 5.3.By (13), we have that r sG ≀ S n ( ξ, σ ) = 0 if and only if f ω | a for all ( d, a, ω )with λ ( ξ,σ ) ( d, a, ω ) >
0. So if r sG ≀ S n ( ξ, σ ) = 0, then a = 1, hence f ω = 1, and so C ( ξ,σ ) ,σ i = 1, for all i . As g ω = Ne ω f ω = N , and so (cid:0) a/e + g − g − (cid:1) = N , we deduce from(13) that r sG ≀ S n ( ξ, σ ) = (Q ( d,a,ω ): λ ( ξ,σ ) ( d,a,ω ) > N sλ ( ξ,σ ) ( d,a,ω ) , if C ( ξ,σ ) ,σ i = 1 for all i ,0 , otherwise.Put X n = { ( ξ, σ ) ∈ G ≀ S n : C ( ξ,σ ) ,σ i = 1 , ∀ i } , so that(15) (cid:10) r sG ≀ S n ( ξ, σ ) (cid:11) ( ξ,σ ) ∈ G ≀ S n = P ( ξ,σ ) ∈ X n ( N s ) r ( σ ) G ≀ S n , where r ( σ ) is the number of cycles in σ . For a fixed σ ∈ S n with a factorization σ = σ . . . σ r to disjoint cycles, we have(16) X ξ ∈ G n :( ξ,σ ) ∈ X n ( N s ) r ( σ ) = ( N s ) r N n − r = N n · ( N s − ) r , since if σ i = ( j . . . j d ), then ξ ( j ) , . . . , ξ ( j d − ) can be chosen arbitrarily and ξ ( j d )must be equal to Q d − k =1 ξ ( j k ) − , so we lose one power of N for each orbit. Plugging (16) in (15), we find that(17) (cid:10) r sG ≀ S n ( ξ, σ ) (cid:11) ( ξ,σ ) ∈ G ≀ S n = P σ ∈ S n ( N s − ) r ( σ ) S n . We apply the exponential formula for permutations [37, Cor. 5.1.9] with f ( i ) = N s − the constant function and h defined by h (0) = 1 and h ( i ) = X σ ∈ S i ( N s − ) r ( σ ) . Then the formula gives that E ( x ) := ∞ X i =0 h ( i ) x i i ! = exp( X i ≥ N s − x i i ) . As P i ≥ x i /i = − ln(1 − x ), we can simplify the right hand side using the binomialseries to get that E ( x ) = (1 − x ) − N s − = X i ≥ ( − i (cid:18) − N s − i (cid:19) x i . In particular, by (17) we have (cid:10) r sG ≀ S n ( ξ, σ ) (cid:11) ( ξ,σ ) ∈ G ≀ S n = h ( n ) n ! = ( − n (cid:18) − N s − n (cid:19) = (cid:18) n + N s − − n (cid:19) , as needed. (cid:3) Galois Theory G -factorization arithmetic functions and the Frobenius automor-phism. Let ψ be a G -factorization arithmetic function and E/ F q ( T ) a G -Galoisgeometric extension. The goal of this section is, for a given a = ( a , . . . , a n − ) ∈ F nq ,to naturally construct an element φ a ∈ G ≀ S n such that(18) ψ E/ F q ( T ) ( T n + a n − T n − + · · · + a ) = ψ G ≀ S n ( φ a ) . We start with a general construction which we later specialize to our setting. Let F be a field and π : C → A F a branched covering of smooth geometrically connected F -curves with function field extension E/F ( T ). Assume that E/F ( T ) is Galoiswith Galois group G . HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 15
This gives rise to the following cover of varieties with corresponding functionfields(19) C nπ n (cid:15) (cid:15) E · · · E n A nFs (cid:15) (cid:15) F ( Y , . . . , Y n ) A nF = A n /S n F ( A , . . . , A n ) . Here S n acts on A n by permuting the coordinates: if ( Y , . . . , Y n ) are the coordi-nates of A n and ( A , . . . , A n − ) of A n = A n /S n , then the map s is given by A = ( − n Y · · · Y n , . . . , A n − = − ( Y + . . . + Y n ) . Also, E i is the function field of the i -th copy of C in C n , in particular, for every i there exists an isomorphism(20) ϕ i : E i → E with ϕ i ( Y i ) = T that fixes F . Put ϕ i,j : E i → E j to be ϕ − j ◦ ϕ i . Let D ( T ) F q [ T ] bethe discriminant ideal of π and D ( A , . . . , A n − ) = disc T ( T n + A n − T n − + . . . + A )and put(21) D ( A , . . . , A n − ) = D ( A , . . . , A n − ) Y i D ( Y i ) ∈ F [ A , . . . , A n − ]which is a non-zero polynomial in the A i -s. Then, for a point a ∈ A n ( F ) we have(22) D ( a ) = 0 = ⇒ a is unramified in C n . If we write f ( T ) = T n + a n − T n − + · · · + a , then the condition D ( a ) = 0 isequivalent to f being a separable polynomial that does not vanish on the branchpoints of π which are exactly the roots of D . The Riemann-Hurwitz formula (seee.g. [13, Thm. 3.6.1]) gives that deg D ≪ genus( C ) + | G | . On the other hand,deg D ≪ n . So, if B ≥ max { genus( C ) , | G | , n } then deg D is bounded in terms of B .The extension E · · · E n /F ( A , . . . , A n ) is a Galois extension with Galois groupisomorphic to G ≀ S n . More explicitly, the action of an element ( ξ, σ ) ∈ G ≀ S n on E · · · E n is given by( ξ, σ ) .e i = ξ ( σ ( i ))( ϕ i,σ ( i ) ( e i )) , e i ∈ E i . (23)This is compatible with the imprimitive action (7).If F = F q is a finite field, then any point a ∈ A n ( F ) with D ( a ) = 0 induces aFrobenius conjugacy class φ a ⊆ G ≀ S n , which is the higher dimensional version of(4) and is defined similarly. Now we can prove (18): Proposition 6.1.
Let F = F q , let f ( X ) = T n + a n − T n − + · · · + a ∈ F q [ T ] suchthat the point a = ( a , . . . , a n − ) is unramified in C n and let φ a ⊆ G ≀ S n be theFrobenius conjugacy class. Then (18) holds for every ψ ∈ Λ ∗ .Proof. To ease notation we identify each of the E i with E via the map φ i . Let f = P · · · P r be the prime factorization of f with P i monic irreducible of degree d i . For each i = 1 , . . . , r , let α i, ∈ A ( F q ) be a root of P i and β i, ∈ C ( F q ) with α i, = π ( β i, ) and let α i,j = α q j − i, be the other roots, and respectively β i,j = β q j − i,j , j = 1 , . . . , d i −
1. We replace the indices of C n and of the middle A n in (19) to be I = { ( i, j ) : i = 1 , . . . , r, j = 1 , . . . , d i } .So ( β i,j ) ( i,j ) ∈ I ∈ C n ( F q ) maps under π n to ( α i,j ) ( i,j ) ∈ I ∈ A n ( F q ) which mapsunder s to a = ( a , . . . , a n − ) ∈ A n ( F q ).To this end, let φ a be the corresponding Frobenius element of ( β i,j ) ( i,j ) ∈ I and let h ∈ E i, be a rational function that is regular at all β i,j . Then, by definition,( φ d i a h )( β i, ) = ( h ( β i, )) q di . Write φ a = ( ξ, σ ); then φ d i a = ( Q d i k =1 ξ σ k − , σ d i ). The coordinate σ ∈ S n is inducedfrom the action of the Frobenius automorphism on the roots of f , so since α i,j = α q j − i, , we have that Y i,j = σ j ( Y i, ), j = 0 , . . . , d i − Y i, = σ d i ( Y i, ). The latteralso implies that E i, maps to itself under σ d i , hence( φ d i a h )( β i, ) = ( d i Y k =1 ξ σ k − ( i, h )( β i, ) = ( d i Y j =1 ξ ( i, j ) h )( β i, ) . To conclude, we obtained( d i Y j =1 ξ ( i, j ) h )( β i, ) = ( h ( β i, )) q di , but the Frobenius at P i in E is the unique element of G satisfying this, so Frob P i = Q d i j =1 ξ ( i, j ). Thus, λ φ a = λ f ; E/ F q ( T ) , which completes the proof. (cid:3) Computation of a Galois group.
We keep the notation as in § F is a field and π : C → A F is a branched covering of geometricallyirreducible F -curves that is generically Galois with Galois group G . For a =( a , . . . , a n − ) ∈ A n and for 0 ≤ m < n , we consider the following subspace of A n (24) W = W a,m = { ( w , . . . , w n − ) ∈ A n : w i = a i , i = m + 1 , . . . , n − } ∼ = A m +1 . So if A n is the space of coefficients of polynomials, then W is the short interval I ( T n + a n − T n − + · · · + a , m ). Let U and V be irreducible components of ( s ◦ π n ) − ( W ) and s − ( W ), respectively. Let M , L , and K be the function fields of U , V , and W , where A , . . . , A m are independent variables and the y i -s satisfy HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 17 C nπ n (cid:15) (cid:15) U o o π n (cid:15) (cid:15) M A nFs (cid:15) (cid:15) V o o s (cid:15) (cid:15) L = K ( y , . . . , y n ) A nF W o o K = F ( A , . . . , A m ) Figure 1.
Variety and field diagrams f ( T ) = T n + a n − T n − + · · · + a m +1 T m +1 + A m T m + · · · + A = n Y i =1 ( T − y i ) . Then,
M/K is a Galois extension and if D ( a n − , . . . , a m +1 , A m , . . . , A ) = 0, thenby (22), it is unramified, hence its Galois group is canonically isomorphic to thesubgroup the generic Galois group G ≀ S n given in (23); namely all elements thatgenerically map U to itself. We identify Gal( M/K ) with this subgroup, in partic-ular H = Gal( M/L ) ≤ G n . We have that Gal(
M/K ) equals G ≀ S n if and only if ( s ◦ π n ) − ( W ) is irreducible,i.e. equals to U .We prove that this is indeed the case if the ramification at infinity is tame. Proposition 6.2.
Under the notation above, assume that m ≥ if q is odd and m ≥ if q is even. Further assume that the infinite prime is tamely ramified inthe fixed field E ab in E of the commutator of G . Then Gal(
M/K ) = G ≀ S n . Remark 6.3.
Proposition 4.6 in [3] coincides with the special case of Proposi-tion 6.2 where G = Z / Z , C = A and π ( x ) = x . Cyclic extensions with genus 0were partly treated by Cohen [9, Thm. 12].6.2.1. Reduction steps.
Since by [4, Proposition 3.6], Gal(
L/K ) = S n and sinceGal( M/K ) ≤ G ≀ S n , in order to prove that Gal( M/K ) = G ≀ S n it suffices to showthat H = G n .The proof of Proposition 6.2 is reduced, by elementary finite group theory, tothe following statements: Lemma 6.4.
Proposition 6.2 holds true if G is abelian. Lemma 6.5.
For each i = j the projection on the i, j -th coordinates H → G n → G is surjective. Proof that Lemmas 6.4 and 6.5 imply Proposition 6.2.
From Lemma 6.4 appliedto the abelianization G ab of G , we get that H surjects onto ( G ab ) n . This inparticular means that H is contained in no proper normal subgroup with abelianquotient. By Lemma 6.5, H surjects onto any projection to two coordinates. Tofinish the proof we need that these two group theoretical properties suffice to implythat H = G n and this is indeed the case, as the lemma below shows. (cid:3) Lemma 6.6.
Let H be a subgroup of G n . Suppose that H maps surjectively onto G under each possible projection onto two copies. If H is a proper subgroup of G n , then it is contained in a proper normal subgroup with abelian quotient.Proof. The case n = 2 is trivial, and by induction, we may assume this is true for n − n ≥
3. If any map from H to the product of n − G is notsurjective, then the image of H is contained in a normal subgroup with abelianquotient, so H is as well. So we may assume that the projections from H to anyproduct of n − G are surjective. Now apply Goursat’s lemma to G n − and G . We get that there is a group G ′ , surjections a : G n − → G ′ and b : G → G ′ , suchthat H consists of tuples ( g , . . . , g n ) in G n with a ( g , ..., g n − ) = b ( g n ). Moreoversince H is a proper subgroup, G ′ is non-trivial.Now because the map H → G n − obtained by dropping the i -th coordinate issurjective, for any g n there exists some g i such that ( e, . . . , e, g i , e, . . . , g n ) ∈ H ,and so a ( e, . . . , e, g i , . . . , e ) = b ( g n ). Putting G = { ( g , e, . . . , e ) : g ∈ G } and G = { ( e, g , e, . . . , e ) : g ∈ G } , we conclude that a maps both G and G onto G ′ .Thus as G and G commute, we get that G ′ must be abelian. Thus the pre-imageof G ′ is the desired normal subgroup with abelian quotient. (cid:3) To finish the proof of Proposition 6.2, it remains to prove Lemmas 6.4 and 6.5.Since the former lemma is technical, we start by proving the latter assuming theformer.6.2.2.
Proof of Lemma 6.5 using Lemma 6.4.
We look at the covering Υ of W defined by adjoining two roots y i , y j of the polynomial; i.e., Υ is the quotient spaceof V under the action of S n − , so Gal( V / Υ) ∼ = S n − , the group of all permutationsfixing i, j . Let Γ be the Galois group of U/ Υ. So, the restriction-of-automorphismsmap induces a surjection Γ → Gal(
V /
Υ) = S n − .The covering Υ maps to A by sending to the two roots. The fibers are connectedbecause we just add two congruence conditions. We have the covering C → A with Galois group G , and its fiber product Υ × A ( C ) is geometrically connected,since the fiber of Υ → A is geometrically connected. This implies that Γ surjectsonto G .We apply Goursat’s lemma to these two maps. They are jointly surjective unlesssome quotient of G matches some quotient of S n − . But all normal subgroups of S n − are contained in A n − , so this can only happen if there is some non-trivialrelation with order two quotients of G . This is not possible by the abelian case,hence the proof is done. (cid:3) HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 19
The proof of Lemma 6.4 is more technical and requires some preparation.6.2.3.
Some more group theory.
This section contains well known facts that wesummarize for the convenience of the reader. We start by stating two well knownfacts on the symmetric group. The first is on normal subgroups:
Lemma 6.7.
Let n ≥ . The group S n does not have any normal subgroups ofodd prime index. The second is about the invariant subspaces of the standard representation of S n on F np acting by permuting the coordinates. Lemma 6.8.
The invariant subspaces of F np under S n ( n ≥ ) are:(1) V = { (0 , . . . , } ,(2) V = sp F p { (1 , . . . , } ,(3) V n − = { ( x , . . . , x n ) ∈ F np : P ni =1 x i = 0 } , and(4) V n = F np . Recall that the Frattini subgroup Φ( G ) of a group G is defined by Φ( G ) = T U ≤ m G U , where the intersection is over the maximal subgroups of G . It has theproperty that for every H ≤ G , if H/H ∩ Φ( G ) = G/ Φ( G ), then H = G . If G isfinite and p | | G | , then the subgroup Φ p ( G ) = [ G, G ] G p generated by commutatorsand p -th power of elements is normal and G/ Φ p ( G ) ∼ = ( Z /p Z ) r . Thus,(25) Φ p ( G ) = U ∩ · · · ∩ U r , with U i the kernel of the projection on the i -th coordinate, so U i is normal in G of index p . Lemma 6.9.
Let G be a finite abelian group and H ≤ G . Assume that H/ ( H ∩ Φ p ( G )) ∼ = G/ Φ p ( G ) for every p | | G | . Then H = G .Proof. Since G is abelian, Φ( G ) = T p || G | Φ p ( G ) and G/ Φ( G ) ∼ = Q p || G | G/ Φ p ( G ).So the assumption gives that H/ ( H ∩ Φ( G )) = G/ Φ( G ) and so H = G . (cid:3) Let L be a field and p a prime. If p ∤ char( L ) let ℘ ( x ) = x p and L ◦ = L × ,otherwise let ℘ ( x ) = x p − x and L ◦ = L . We say that elements in L ◦ are p -independent if they are linearly independent in L ◦ /℘ ( L ◦ ), considered as a F p -vectorspace. Lemma 6.10.
Let G be a finite abelian group and H ≤ G . Let L be a field suchthat for every p | | G | , either L contains a primitive p -th root of unity or L is ofcharacteristic p . Let M/L be an H -Galois extension. For a prime divisor p of | G | ,put U , . . . , U r ( p ) as in (25) . Then, for every ≤ i ≤ r there exists α i,p ∈ L suchthat M H ∩ U i = L ( β i,p ) , ℘ ( β i,p ) = α i,p . Moreover, if α ,p , . . . , α r ( p ) ,p are p -independent for all p | | G | , then H = G . Proof.
First we note that H/ ( H ∩ U i ) ≤ G/U i = Z /p Z so Gal( M H ∩ U ) i /L ) ≤ Z /p Z ,and Kummer theory (if char( L ) = p ) or Artin-Schreier theory (otherwise) give usthe required elements α i,p . Now, if the α i,p -s are p -independent, then by (25),Kummer theory and Artin-Schreier theory, we find that( Z /p Z ) r ( p ) ∼ = Gal( M H ∩ Φ p ( G ) /L ) ∼ = H/ ( H ∩ Φ p ( G )) . So by Lemma 6.9, H = G . (cid:3) Rational functions.
We borrow the following from [3, Lem. 4.5].
Lemma 6.11.
Let ˜ f ( T ) ∈ K [ T ] be a separable polynomial and let f ( T ) = ˜ f ( T ) + A ∈ K ( A )[ T ] where A is transcendental over K ( T ) . Then disc( f ) ∈ K [ A ] is notdivisible by A . Lemma 6.12.
Let F be a field and let A = ( A , . . . , A m ) be an m -tuple of variables( m ≥ ). Let α = ( α , . . . , α m ) be an m -tuple of scalars from F . Let f ( T ) ∈ F [ T ] be a polynomial of degree > m . Then F ( A , T ) = f ( T ) + P mi =1 A i T i + P mi =1 A i α i is separable in T .Proof. It suffices to show that F is irreducible in T , because F ′ is linear in A andin particular non-zero. Since F is primitive in T , it is irreducible in T if and onlyif it is irreducible in R = F ( A , . . . , A m )[ A , T ] by Gauss’s lemma. Since F = ( T + α ) A + G , with G = f ( T ) + P i> A i ( T i + α i ), either F is primitive in A , then again byGauss it is irreducible in A and thus in T , or T + α divides G in R . In thelatter case, H = A + G T + α is irreducible in T (again primitivity and linearity) anddeg A ∂ H ∂T = 1 hence it is non-zero, so H is separable in T . Moreover, H| T = − α = 0,so we get that F = ( T + α ) H is separable, as needed. (cid:3) Lemma 6.13.
Let F be an algebraically closed field of characteristic p . Let A =( A , . . . , A m ) be an ( m + 1) -tuple of variables, m ≥ . Let f ( T ) ∈ F [ T ] be a monicpolynomial of degree n > m . Let f ( T ) = f ( T ) + P mi =0 A i T i be a polynomial withcoefficients in K = F ( A ) . Let L be the splitting field of f ( T ) = Q ni =1 ( T − y i ) .Let D ( T ) = r ( T ) r ( T ) ∈ F ( T ) × be a reduced rational function with deg r ≥ deg r ,and r ( T ) = c Q dj =1 ( T − α j ) ( c ∈ F × , α j ∈ F ). We have (26) n X i =1 D ( y i ) = h ( A ) Q dj =1 f ( α j ) where h ∈ F [ A ] is coprime to Q dj =1 f ( α j ) as a polynomial in A .Proof. We first prove (26) in the special case D ( T ) = 1 / ( T − α ) k . Let(27) g ( T ) := f ( T + α ) T n f ( α ) = f ( T + α ) T n + P mi =0 A i (1 + αT ) i T n − i f ( α ) . HEBOTAREV DENSITY THEOREM IN SHORT INTERVALS 21