aa r X i v : . [ m a t h . N T ] A p r SECTIONS OF QUADRICS OVER A F q NASER T. SARDARI AND MASOUD ZARGAR
Abstract.
Given finitely many closed points in distinct fibers of a non-degenerate quadric over A F q ,we ask for conditions under which there is a section passing through the closed points, possibly withhigher order (nilpotence) conditions. This could be thought of as a quadratic version of Lagrangeinterpolation, and it is equivalent to proving strong approximation for non-degenerate quadricsover F q [ t ]. We show that under mild conditions on the quadratic form F over F q [ t ] in d variables, f, g ∈ F q [ t ], λ ∈ F q [ t ] d , if d ≥ f ≥ (4 + ε ) deg g + O (1) we have a solution x ∈ F q [ t ] d to F ( x ) = f such that x ≡ λ mod g , where the the big-Oh notation does not depend on f, g, λ .For d = 4, we show the same is true for deg f ≥ (6 + ε ) deg g + O (1). This gives us a new proof(independent of the Ramanujan conjecture over function fields proved by Drinfeld) that the diameterof any k -regular Morgenstern Ramanujan graphs G is at most (2+ ε ) log k − | G | + O ε (1). In contrastto the d = 4 case, our result is optimal for d ≥
5. Along the way, we prove a stationary phasetheorem over function fields that is of independent interest.
1. Introduction 12. The delta method for small target 73. Bounds on the exponential sums S g,r ( c ) 114. Analytic functions on T d I g,r ( c ) 226. Main contribution to counting function 297. Proof of the main theorem 34References 381. Introduction
Motivation.
We begin by considering a natural geometric problem regarding quadratic formsover F q [ t ]. Suppose F is a quadratic form in d variables over F q [ t ]. Suppose f is a polynomial in F q [ t ].We may then consider the affine variety X f given by setting F ( x ) = f , x ∈ A d F q [ t ] . We may viewthis as a family π : X f → A F q over A F q . Suppose we have a collection of closed points p , . . . , p m in A F q . Choose, for each i , a point λ i := ( λ ( p i ) , . . . , λ d ( p i )) in the fiber X p i := X f × F q [ t ] κ ( p i )over p i . Can we find a section s : A F q → X f of the structure morphism π : X f → A F q thatmaps each p i to λ i with some prescribed higher order (nilpotence) conditions of order m i ? Thisproblem could be thought of as a quadratic version of the classical Lagrange interpolation. Weshow that if F is non-degenerate in d ≥ f ≥ (4 + ε ) P i m i deg p i + O ε,F (1), where the implied constant depends only on ε and thequadratic form F (in fact, we show a stronger result depending on anisotropic cones defined indefinition 1.1). We also show that this condition is optimal. On the other hand, if d = 4, weshow that this is true at least if deg f ≥ (6 + ε ) P i m i deg p i + O ε,F (1). That being said, weconjecture that 4 + ε still suffices in the d = 4 case. In fact, as can be found in another paper bythe two authors [TZ19], we have shown that the optimality of 4 + ε when working with the classof quadratic forms in the construction of Morgenstern Ramanujan graphs follows from a twistedLinnik–Selberg conjecture over function fields. That paper relies heavily on the computations and Date : 27th April 2020. techniques developed in this paper. The setup of the problem is pictorially represented by thefollowing figure. X f A F q π s p p . . . p m • • • . . . λ λ λ m • • • X p X p . . .X pm There is another more algebraic way of formulating the problem; in fact, this other formulation ismore common. By packaging all the closed points p i and all their multiplicities m i > p i ( t ) in F q [ t ] raised to the power of m i ) into one polynomial g ( t ) := Q i p i ( t ) m i , we can use the Chinese remainder theorem to reformulate the problem as anoptimal strong approximation problem for quadratic forms over function fields. More precisely, weask for the following. Suppose we have a quadratic form F in d variable over F q [ t ], and polynomials g, f ∈ F q [ t ]. Additionally, we are given polynomials λ , . . . , λ d ∈ F q [ t ]. We want to know when wehave an integral solution x := ( x , . . . , x d ) ∈ F q [ t ] d to the system(1) ( F ( x ) = f, x ≡ λ mod g, where λ = ( λ , . . . , λ d ) and x ≡ λ mod g means x i ≡ λ i mod g for every 1 ≤ i ≤ d . For a primeideal ̟ of F q [ t ] , we write F q [ t ] ̟ for the completion of F q [ t ] at ̟ . We say all local conditionsfor the system (1) are satisfied, if X f ( K ∞ ) := { x ∈ K d ∞ : F ( x ) = f } 6 = ∅ and F ( x ) = f has alocal solution x ̟ ∈ F q [ t ] d̟ for all prime ideals ̟ of F q [ t ] such that x ̟ ≡ λ mod ̟ ord ̟ ( g ) . In thefollowing K ∞ := F q ((1 /t )), d ( − ) := q ( − ) . K d ∞ is equipped with the norm | x | := max i | x i | for any x = ( x , . . . , x d ) ∈ K d ∞ . Consider the following definition. Definition 1.1 (Anisotropic cone) . We say Ω ⊂ K d ∞ is an anisotropic cone with respect to thequadratic form F ( x ) if there exists fixed positive integers ω and ω ′ such that:(1) If x ∈ Ω then f x ∈ Ω for every f ∈ K ∞ . (2) If x ∈ Ω and y ∈ K d ∞ with | y | ≤ | x | / b ω, then x + y ∈ Ω . (3) b ω ′ | F ( x ) | ≥ | x | . Remark . Whenever considering the equation F ( x ) = f along with an anisotropic cone Ω, weassume that Ω ∩ X f ( K ∞ ) = ∅ .The main result of this paper is the following theorem. Theorem 1.2.
Suppose q is a power of a fixed odd prime number, and let F be a non-degeneratequadratic form over F q [ t ] in d ≥ variables and of discriminant ∆ . Let f, g ∈ F q [ t ] be nonzeropolynomials such that ( f ∆ , g ) = 1 , and let λ ∈ F q [ t ] d be a d -tuple of polynomials at least oneof whose coordinates is relatively prime to g . Finally, suppose that all local conditions for thesystem (1) are satisfied and Ω ∩ X f ( K ∞ ) = ∅ . If d ≥ , then for any anisotropic cone Ω and for deg f ≥ (4 + ε ) deg g + O ε,F, Ω (1) , there is a solution x ∈ Ω ∩ F q [ t ] d to (1) . If d = 4 , this holds atleast for deg f ≥ (6 + ε ) deg g + O ε,F, Ω (1) . As a corollary, we obtain the following strong approximation result.
ECTIONS OF QUADRICS OVER A F q Corollary 1.3 (Strong approximation) . With the notation as above, if d ≥ and all local conditionsto the system (1) are satisfied, for deg f ≥ (4 + ε ) deg g + O ε,F (1) , there is a solution x ∈ F q [ t ] d to (1) . If d = 4 , this holds at least for deg f ≥ (6 + ε ) deg g + O ε,F (1) . In order to obtain this corollary, the main theorem 1.2 implies that it suffices to show that we cancover K d ∞ by finitely many anisotropic cones such that, for any given f , X f ( K ∞ ) intersects at leastone of them. See Lemma 5.1 for a proof of this.Before discussing the optimality our main theorem for d ≥
5, let us make some remarks regardingits proof. Though the proof uses the function field analogue of the circle method to prove theanalogue of the theorem over the integers proved in the first author’s paper [T. 19a], there aredifferences between the two papers. Though our theorem proves the stronger statement that asolution exists in an anisotropic cone, restricting to such an anisotropic cone is essential in ourproof. Choosing a weight function centered at the origin will not suffice for isotropic quadraticforms (which are plentiful in positive characteristic); this would lead to suboptimal results evenfor d ≥
5. In order to obtain optimal results for d ≥
5, it is essential that we choose appropriateweighted sum of solutions within anisotropic cones. In order to deduce strong approximation for F ,we show in Lemma 5.1 that for each f we can construct an anisotropic cone depending only on theclass of f in K ×∞ /K × ∞ such that X f ( K ∞ ) ∩ Ω = ∅ . This is one technicality that arises when workingover positive characteristics as opposed to over Z . Additionally, most of the proofs of the centralresults in the function field case are necessarily different than the ones over the integers. One ofthe main differences between the two papers is that in order to compute the oscillatory integrals, astationary phase theorem over function fields had to be developed which is of independent interest.A feature of the function field case is that by using this stationary phase theorem, we can determinethe oscillatory integrals in terms of (a complicated expression involving) Kloosterman sums at theinfinite place. Another difference with the integer case is that we had to use a different method forthe computation of the main term to our counting function. Along the way, we give proofs of thefunction-field analogues of the results of Heath-Brown (see [HB96a]) needed for the circle methodin this setting. The tools developed in this paper are used in another paper of the authors in orderto study the diameter of Morgenstern Ramanujan graphs [TZ19]. Remark . For F ( x ) = x + . . . + x d , we can take Ω = { x ∈ K d ∞ : ∀ i, deg x > deg x i } . Note thatwhen deg f ≤ g −
3, then the system need not have a solution in F q [ t ] d ∩ Ω. For instance,when λ = (1 , , . . . ,
0) and f ≡ t deg g − g mod g , then a solution implies the existence of( t , . . . , t d ) ∈ F q [ t ] d such that(1 + t g ) + ( t g ) + . . . + ( t d g ) ≡ t deg g − g mod g , that is, t ≡ t deg g − mod g . Since the solution is in Ω, the degree of f is equal to the degree of(1 + t g ) , and so deg f ≥ g −
1) = 4 deg g −
2. This shows that the factor 4 + ε is optimalfor d ≥
5, and is the best possible factor for d = 4. In fact, we conjecture that it is also optimal for d = 4. Conjecture 1.4.
For d = 4 in Theorem 1.2, if deg f ≥ (4+ ε ) deg g + O ε,F, Ω (1) , the same conclusionholds. In other words, the factor ε is optimal for all d ≥ . Let us comment on why there is a difference between the d = 4 case and the d ≥ d = 4 is difficult, even in the case of function fields. InProposition 7.1, we show that the error term of the counting function with respect to the maincontribution satisfies the following bound: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤| r |≤ b Q exc X c =0 | gr | − d S g,r ( c ) I g,r ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X ≤| r |≤ b Q exc X c =0 | gr | − d | S g,r ( c ) || I g,r ( c ) | ≪ ε b Q d +32 + ε | g | d − + ε (1+ | g | − d − + ε ) , NASER T. SARDARI AND MASOUD ZARGAR where P exc denotes summation over exceptional vectors (we do not give the definition of excep-tional vectors here). When d ≥
5, we have 1 + | g | − d − + ε = O (1), while when d = 4 this is of order | g | / which forces upon us a weaker bound on the error and so a suboptimal result in this case. Seethe proof of the main theorem on the final page for the precise reason. We remark, however, thatby using the triangle inequality in the above sum, we seem to be losing some extra cancellation thatwould lead to an improved version of the d = 4 case; we are only using the Weil bound and not usinga possible cancellation in the sums of Kloosterman sums themselves. For example, as can be foundin the paper [TZ19] by the two authors, once we restrict to the Morgenstern quadratic forms (in 4variables), we can reduce the optimality to a twisted version of the Linnik–Selberg conjecture (Con-jecture 1.4 of loc.cit ) which we suspect is true. Over function fields, the classical Linnik–Selbergconjecture is true and is equivalent to the Ramanujan conjecture proved by Drinfeld. See the workof Cogdell and Piatetski-Shapiro [CPS90] for a proof of this. Our twisted Linnik–Selberg conjec-ture is a generalization, and does not seem to easily follow from the usual Ramanujan conjecture.As pointed out in that paper, if we have a more complicated quadratic form in d = 4 variables,then even the reduction to a natural cancellation similar to the Linnik–Selberg conjecture does notseem possible. This partially attests to the difficulty of obtaining the optimal result for d = 4.That being said, since the Ramanujan conjecture over F q ( t ) is, in contrast to that over Q , proved,there is greater hope of proving such a result over function fields. This strong approximation forMorgenstern quadratic forms is intimately connected with the diameter of Morgenstern quadraticforms. We now discuss the connection of strong approximation for quadratic forms to Ramanujangraphs.As mentioned above, another motivation for the consideration of this problem is related to theconstruction of Ramanujan graphs with optimal diameters. We begin by defining Ramanujangraphs. Fix an integer k ≥
3, and let G be a k -regular connected graph with the adjacency matrix A G . It follows that k is an eigenvalue of A G . Let λ G be the maximum of the absolute value of allthe other eigenvalues of A G . By the Alon-Boppana Theorem [LPS88], λ G ≥ √ k − o (1), where o (1) goes to zero as | G | → ∞ . We say that G is a Ramanujan graph if λ G ≤ √ k − . The first explicit construction of Ramanujan graphs is due to Lubotzky–Phillips–Sarnak [LPS88],and independently by Margulis [Mar88]. It is a Cayley graph of PGL ( Z /q Z ) or PSL ( Z /q Z ) with p + 1 explicit generators for every prime p and integer q . The optimal spectral gap on the LPSconstruction is a consequence of the Ramanujan bound on the Fourier coefficients of the weight 2holomorphic modular forms, which justifies their naming. We refer the reader to [Sar90, Chapter 3],where a complete history of the construction of Ramanujan graphs and other extremal propertiesof them are recorded. In particular, Lubotzky–Phillips–Sarnak proved that the diameter of every k -regular Ramanujan graph G is bounded by 2 log k − | G | + O (1) . This is still the best known upperbound on the diameter of a Ramanujan graph. It was conjectured that the diameter is boundedby (1 + ε ) log k − | G | as | G | → ∞ ; see [Sar90, Chapter 3]. However, the first author proved that forsome infinite families of LPS Ramanujan graphs the diameter is bigger than 4 / k − | G | + O (1);see [T. 18]. The first author has conjectured that the diameter of the LPS Ramanujan graphs isasymptotically 4 / k − | G | + o (log k − | G | ); the upper bound follows from an optimal strong ap-proximation conjecture for integral quadratic forms in 4 variables; see [T. 19a, Conjecture 1.3]. Thefollowing theorem of Lubotzky–Phillips–Sarnak links the diameter of the LPS Ramanujan graphsto strong approximation on the sphere. Theorem 1.5 (Due to Lubotzky–Phillips–Sarnak [LPS88]) . Let v := (cid:20) a a a a (cid:21) ∈ G , where G isthe LPS Ramanujan graph associated to p and q . There is a bijection between non-backtrackingpaths ( v , . . . , v h ) of length h from v = id to v h = v in G , and the set of integral solutions to the ECTIONS OF QUADRICS OVER A F q following diophantine equation x + x + x + x = N, (cid:20) x + ix x + ix − x − ix x − ix (cid:21) ≡ λ (cid:20) a a a a (cid:21) mod q for some λ ∈ Z / q, (4) where N = p h . In particular, the distance between id and v in G is the smallest exponent h suchthat (4) has an integral solution. We state a version of the optimal strong approximation conjecture for the sphere, which whencombined with this theorem implies that the diameter of LPS Ramanujan graphs is at most ( + ε ) log k − | G | + O ε (1); see [RS17, T. 17] for further numerical evidence regarding this conjecture. Conjecture 1.6.
Suppose that N , m and λ , . . . , λ are given integers such that N ≡ X i =1 λ i mod m. Assuming that N ≫ m ε , there exists an integral solution ( x , . . . , x ) to the system x + x + x + x = N,x l ≡ λ l mod m for ≤ l ≤ . This conjecture is inspired by the conjecture of Sarnak on the distribution of integral points on thesphere S . Indeed, given R > R ∈ Z , we let C ( R ) denote the maximum volume ofany cap on the ( d − S d − ( R ) of radius R which contains no integral points.Sarnak defined [Sar15] the covering exponent of integral points on the sphere by: K d := lim sup R →∞ log (cid:0) S d − ( R ) ∩ Z d (cid:1) log (vol S d − ( R ) /C ( R )) . In his letter [Sar15] to Aaronson and Pollington, Sarnak showed that 4 / ≤ K ≤
2. To showthat K ≤
2, he appealed to the Ramanujan bound on the Fourier coefficients of weight k modularforms, while the lower bound 4 / ≤ K is a consequence of an elementary number theory argument.Furthermore, Sarnak states some open problems [Sar15, Page 24]. The first one is to show that K < K = 4 / K d = 2 − d − for d ≥ / ≤ K ≤
2; see also [T. 19b] for bounds on the average covering exponent. Browning-Kumaraswamy-Steiner [BKS17] showed that K = 4 /
3, subject to the validity of a twisted version of a conjectureof Linnik about cancellation in sums of Kloosterman sums; see also Remark 6.8 of [T. 19a]. Wehave shown, as will appear in a forthcoming paper, that a twisted version of the Linnik–Selbergconjecture proves the optimal bound for the diameter of Morgenstern Ramanujan graphs. Sincethe untwisted version of the Linnik–Selberg conjecture over function fields has already been provedusing the Ramanujan conjecture over function fields (proved by Drinfeld), we are hopeful that wewill be able to prove the desired twisted version of the conjecture. We will discuss this connectionin a future paper.That being said, our main Theorem 1.2 above can be used to give a new proof, independent of theRamanujan conjecture over function fields, that the diameter of k -regular Morgenstern Ramanujangraphs G are bounded above by (2 + ε ) log k − | G | + O ε (1). Let us first recall the construction ofRamanujan graphs due to Morgenstern. NASER T. SARDARI AND MASOUD ZARGAR
Consider the quaternion algebra A := k + k i + k j + k ij , i = ν, j = x − , ij = − ji , where ν is not a square in F q , and k := F q ( t ). Let us assume that q is odd. The quaternion algebrawe should take for even q can be found in Section 5 of Morgenstern’s paper [Mor94]. Let S := F q [ t ] + F q [ t ] i + F q [ t ] j + F q [ t ] ij be the integral part of A . Given ξ = a + b i + c j + d ij in A , its conjugate is defined as ξ := a − b i − c j − d ij .Furthermore, we have the norm N ( ξ ) := ξξ = a − b ν + ( d ν − c )( t − . As can be found in Lemmas 4.2 and 4.4 of Morgenstern’s [Mor94], it is possible to construct elements ξ , . . . , ξ q +1 of norm t (called elements of basic norm t ) such that every element x of S such that N ( x ) = t n has the unique factorization x = t r uθ . . . θ m , where 2 r + m = n , N ( u ) = 1, θ i are basic norm t , and t does not divide θ . . . θ m . Theorem 5.5 ofMorgenstern’s [Mor94] states that such a x = a + b i + c j + d ij in S of norm t n is a multiple of basicnorms t if and only if a − , b ≡ t − . DefineΛ( t −
1) := x = a + b i + c j + d ij ∈ S : a − , b ≡ t − ,N ( x ) is a power of t,t does not divide x . From the above discussion, it follows that Λ( t −
1) is a free group generated by ξ , . . . , ξ q +12 (ifwe reorder the basic norm t elements so that the rest are conjugates of the first half of them).The construction of the Ramanujan graphs given by Morgenstern is obtained by taking the Cayleygraph of the quotient Γ g := Λ( t − / Λ( g ) with respect to the q + 1 basic norm t elements. Here,given g ∈ F q [ t ] is an irreducible polynomial prime to t ( t − g ) := (cid:26) x = a + b i + c j + d ij ∈ Λ( t −
1) : b, c, d ≡ g ( t ) , ( a, g ) = 1 (cid:27) . See Theorem 4.10 of [Mor94] for details. This Cayley graph is a Cayley graph of either PGL ( F q d )or PSL ( F q d ), where d is the degree of the polynomial g . This is obtained by constructing a map µ : Λ( t − → P GL ( F q d ). See Morgenstern’s paper [Mor94] for a detailed discussion of this point.From the unique factorization of elements in Λ( t −
1) as products of basic norm t elements, we havethe analogue of the above Theorem 1.5 of Lubotzky, Phillips, and Sarnak. Our main Theorem 1.2applied to the (anisotropic) quadratic form F ( a, b, c, d ) = a − b ν + ( d ν − c )( t − k -regular Ramanujan graph G := Γ g ( k = q + 1 here) is atmost (6 + ε ) log q q d + O ε (1). Since PGL ( F q d ) and PSL ( F q d ) are of orders q d − q d and q d − q d ,respectively, this is (2 + ε ) log k − | G | + O ε (1), as required. Similarly, we can deal with the case when q is even. We therefore have the following (known) corollary of our strong approximation result.However, our proof is independent of the Ramanujan conjecture over function fields (that is now awell-known deep theorem of Drinfeld). Corollary 1.7.
The diameter of k -regular Morgenstern Ramanujan graphs G is at most (2 + ε ) log k − | G | + O ε (1) . ECTIONS OF QUADRICS OVER A F q Note that the proof that the diameter satisfies this bound is independent of the Ramanujan con-jecture; however, the fact that the graphs G are indeed Ramanujan graphs still uses the Ra-manujan conjecture. Since by Conjecture 1.4 we expect the optimal bound of 4 + ε to hold atleast for anisotropic quadratic forms in 4 variables as well, we expect the stronger upper bound( + ε ) log k − | G | + O ε (1) to be true.Our method is based on a version of the circle method that is developed in the work of Heath-Brown over the integers [HB96a], and modified by Browning and Vishe for function fields [BV15].We improve the known upper bounds on some oscillatory integrals that come from the infinite place.In fact, we give an exact formula for these integrals in terms of the Kloosterman sums and ouroptimal upper bound are a consequence of Weil’s bound on Kloosterman’s sums.2. The delta method for small target
In this section, we define a weighted sum N ( w, λ ) counting the number of integral solutions ofour problem. We then use the delta method to give an expression for it in terms of exponentialsums and oscillatory integrals. This is done by giving an expansion of the delta function using thedecomposition of T (that we shall define below) found in the paper [BV15] of Browning and Vishe.In this section, we also set up the basic notation that we shall use in this paper.2.1. Notation.
Let K = F q ( t ) and let O = F q [ t ] be its ring of integers. The prime at infinity t − –which we denote by ∞ –gives us the completion K ∞ of K with respect to the norm | a/b | ∞ := q deg a − deg b . We often omit the ∞ from the notation | . | ∞ and simply write | . | . For every d , we define the naturalnorm on K d ∞ by | a | := max i | a i | . This endows K d ∞ and O d ∞ with metric topologies. By consideringthe other places as well, we may construct the ring of adeles as A dK . We do not discuss this con-struction here as it plays a minor role in this paper.Note that we may identify K ∞ with the field F q ((1 /t )) = X i ≤ N a i t i : for a i ∈ F q and some N ∈ Z and put T = { α ∈ K ∞ : | α | < } = X i ≤− a i t i : for a i ∈ F q . Let δ ∈ T . Then T /δ T is the set of cosets α + δ T , of which there are | δ | . In the function field setting, smooth functions f : F → C from a non-archimedian local field F are precisely the locally constant functions. The analogue here of Schwarz functions in real anal-ysis is the notion of Schwarz-Bruhat functions which are the smooth (locally constant) functions f : F → C with compact support. We denote the set of Schwarz-Bruhat functions on F by S ( F ).We can then extend this notion to Schwarz-Bruhat functions on F n by defining such a function tobe one that is a Schwarz-Bruhat function in each coordinate. We could similarly define the spaceof Schwarz-Bruhat functions S ( A nF ) on the adeles A nF . As mentioned above, adeles do not play animportant role in this paper; our focus will be on the infinite place. NASER T. SARDARI AND MASOUD ZARGAR
Characters.
There is a non-trivial additive character e q : F q → C ∗ defined for each a ∈ F q by taking e q ( a ) = exp(2 πi tr( a ) /p ), where tr : F q → F p denotes the trace map. This characterinduces a non-trivial (unitary) additive character ψ : K ∞ → C ∗ by defining ψ ( α ) = e q ( a − ) forany α = P i ≤ N a i t i in K ∞ . In particular it is clear that ψ | O is trivial. More generally, givenany γ ∈ K ∞ , the map α ψ ( αγ ) is an additive character on K ∞ . We then have the followingorthogonality property. Lemma 2.1 (Kubota, Lemma 7 of [Kub74]) . X b ∈O| b | < b N ψ ( γb ) = ( b N , if | (( γ )) | < b N − , , otherwise , for any γ ∈ K ∞ and any integer N ≥ , where (( γ )) is the part of γ with all degrees negative. We also have the following
Lemma 2.2 (Kubota, Lemma 1(f) of [Kub74]) . Let Y ∈ Z and γ ∈ K ∞ . Then Z | α | < b Y ψ ( αγ ) dα = ( b Y , if | γ | < b Y − , , otherwise. In particular, if we set Y = 0, then we obtain the following expression for the delta function on O : δ ( x ) = Z T ψ ( αx ) dα, where δ ( x ) = ( x = 0 , The delta function.
The idea now is to decompose T into a disjoint union of balls (with nominor arcs) which is the analogue of Kloosterman’s version of the circle method in this functionfield setting. This is done via the following lemma of Browning and Vishe [BV15, Lemma 4.2]. Lemma 2.3.
For any
Q > we have a disjoint union T = G r ∈O| r |≤ b Qr monic G a ∈O| a | < | r | ( a,r )=1 n α ∈ T : | rα − a | < b Q − o . The following follows from Lemma 2.3.
Lemma 2.4.
Let Q ≥ and n ∈ O . We have (5) δ ( n ) = 1 b Q X r ∈O| r |≤ b Qr monic X ∗| a | < | r | ψ (cid:16) anr (cid:17) h (cid:16) rt Q , nt Q (cid:17) where we henceforth put X ∗| a | < | r | := X a ∈O| a | < | r | ( a,r )=1 . and h is only defined for x = 0 as: h ( x, y ) = ( | x | − if | y | < | x | otherwise. ECTIONS OF QUADRICS OVER A F q Proof.
We have δ ( n ) = X r ∈O| r |≤ b Qr monic X ∗| a | < | r | ψ (cid:16) anr (cid:17) Z | α | < | r | − b Q − ψ ( αn ) dα. It is easy to check that 1 b Q h (cid:16) rt Q , nt Q (cid:17) = Z | α | < | r | − b Q − ψ ( αn ) dα. The lemma follows by substituting the above formula. (cid:3)
Proof.
Indeed, using Lemma 2.3, we may rewrite the integral expression of the delta function as δ ( x ) = Z T ψ ( αx ) dα = X r ∈O| r |≤ b Qr monic X ∗| a | < | r | Z | rα − a | < b Q − ψ ( αx ) dα = X r ∈O| r |≤ b Qr monic X ∗| a | < | r | ψ (cid:16) axr (cid:17) Z | α | < | r | − b Q − ψ ( αx ) dα, where the last equality follows from a linear change of variables. Note that if we define h ( x, y ) := | x | − Z T ψ ( yx − u ) du, then h (cid:16) rt Q , xt Q (cid:17) = b Q | r | − Z T ψ (cid:16) xurt Q (cid:17) du = b Q Z | α | < | r | − b Q − ψ ( αx ) dα. The last statement follows from Lemma 2.2. (cid:3)
Smooth sum N ( w, λ ) . As previously stated, we want to take a weight function w ∈ S ( K d ∞ )and use it to define a weighted sum over all the solutions whose existence we want to show. Wewill denote such a sum by N ( w, λ ), and then we will use the circle method to give a lower boundfor this quantity. A positive lower bound would prove existence of the desired solutions.Let w be a compactly supported (Schwarz-Bruhat) weight function defined on K d ∞ . Assume that x ∈ O d satisfies the conditions F ( x ) = f and x ≡ λ mod g . We uniquely write x = g t + λ , where t ∈ O d and λ = ( λ , . . . , λ d ) for λ i of degree strictly less than that of g . Define(6) k := f − F ( λ ) g . If F ( x ) = f , then g F ( t ) + 2 g λ T A t = f − F ( λ ) which implies that g | λ T A t − k. Then, F ( t ) + g (2 λ T A t − k ) = 0 . We also define G ( t ) := F ( g t + λ ) − fg = F ( t ) + 1 g (2 λ T A t − k ) . Finally, we define N ( w, λ ) := X t w ( g t + λ ) δ ( G ( t )) , where t ∈ O d . Note that N ( w, λ ) is the weighted number of x ∈ O d satisfying the conditions F ( x ) = f and x ≡ λ mod g . We apply the delta expansion in (5) to δ ( G ( t )) . Note that (2.4) holdsonly for values of O . Moreover, G ( t ) ∈ O if and only if g | λ T A t − k. Using Lemma 2.1, we havefor γ ∈ K ∞ | g | X ℓ ∈O| ℓ | < | g | ψ ( γℓ ) = ( | (( γ )) | < | g | − | g | X ℓ ∈O| ℓ | < | g | ψ (cid:18) (2 λ T A t − k ) ℓg (cid:19) = ( | (( λ T A t − kg )) | < | g | − (cid:12)(cid:12)(cid:12)(cid:12) (( 2 λ T A t − kg )) (cid:12)(cid:12)(cid:12)(cid:12) < | g | − is satisfied precisely when (( 2 λ T A t − kg )) = 0 , that is, when g | λ T A t − k . Consequently, we may rewrite N ( w, λ ) = 1 | g | X ℓ ∈O| ℓ | < | g | X t ψ (cid:18) (2 λ T A t − k ) ℓg (cid:19) w ( g t + λ ) δ ( G ( t )) . Then, applying (5) and splitting the sum over t as a sum of sums over different congruence classesmodulo gr , we obtain N ( w, λ )= 1 | g | b Q X ℓ ∈O| ℓ | < | g | X t X r ∈O| r |≤ b Qr monic X ∗| a | < | r | ψ (cid:18) (2 λ T A t − k ) ℓg + aG ( t ) r (cid:19) w ( g t + λ ) h (cid:18) rt Q , G ( t ) t Q (cid:19) = 1 | g | b Q X ℓ ∈O| ℓ | < | g | X t X r ∈O| r |≤ b Qr monic X ∗| a | < | r | ψ (cid:18) ( a + rℓ )(2 λ T A t − k ) + agF ( t ) gr (cid:19) w ( g t + λ ) h (cid:18) rt Q , G ( t ) t Q (cid:19) = 1 | g | b Q X ℓ ∈O| ℓ | < | g | X r ∈O| r |≤ b Qr monic X ∗| a | < | r | X b ∈O d / ( gr ) X s ∈O d ψ (cid:18) ( a + rℓ )(2 λ T A b − k ) + agF ( b ) gr (cid:19) w ( g ( b + gr s ) + λ ) · h (cid:18) rt Q , G ( b + gr s ) t Q (cid:19) . The Poisson summation formula for f ∈ S ( A dK ) states that X x ∈ K d f ( x ) = X x ∈ K d b f ( x ) , ECTIONS OF QUADRICS OVER A F q where b f ( y ) := Z A dK f ( x ) ψ ( h x , y i ) d x . From this, one deduces (see Lemma 2.1 of [BV15], for example) that for v ∈ S ( K d ∞ ), X t ∈O d v ( t ) = X c ∈O d Z K d ∞ ψ ( h c , t i ) v ( t ) d t . Applying this to the s variable in the above expression of N ( w, λ ), we obtain the expression N ( w, λ )= 1 | g | b Q X ℓ ∈O| ℓ | < | g | X r ∈O| r |≤ b Qr monic X ∗| a | < | r | X b ∈O d / ( gr ) X c ∈O d ψ (cid:18) ( a + rℓ )(2 λ T A b − k ) + agF ( b ) gr (cid:19) · Z K d ∞ ψ ( h c , t i ) w ( g ( b + gr t ) + λ ) h (cid:18) rt Q , G ( b + gr t ) t Q (cid:19) d t = 1 | g | b Q X ℓ ∈O| ℓ | < | g | X r ∈O| r |≤ b Qr monic X ∗| a | < | r | X c ∈O d X b ∈O d / ( gr ) | gr | − d ψ (cid:18) ( a + rℓ )(2 λ T A b − k ) + agF ( b ) − h c , b i gr (cid:19) · Z K d ∞ ψ (cid:18) h c , t i gr (cid:19) w ( g t + λ ) h (cid:18) rt Q , G ( t ) t Q (cid:19) d t We express this in the condensed form(7) N ( w, λ ) = 1 | g | b Q X r ∈O| r |≤ b Qr monic X c ∈O d | gr | − d S g,r ( c ) I g,r ( c ) , where I g,r ( c ) and S g,r ( c ) are defined by(8) I g,r ( c ) := Z K d ∞ h (cid:18) rt Q , G ( t ) t Q (cid:19) w ( g t + λ ) ψ (cid:18) h c , t i gr (cid:19) d t , and(9) S g,r ( c ) := X ℓ ∈O| ℓ | < | g | X | a | < | r |∗ S g,r ( a, ℓ, c )with(10) S g,r ( a, ℓ, c ) := X b ∈O d / ( gr ) ψ (cid:18) ( a + rℓ )(2 λ T A b − k ) + agF ( b ) − h c , b i gr (cid:19) . In the next two sections, we bound from above S g,r and I g,r .3. Bounds on the exponential sums S g,r ( c )In this section, we bound from above an averaged sum of the S g,r ( c ). Indeed, we prove the following. Proposition 3.1.
We have the following upper bound X r ∈O| r | < b X | g | − d | r | − d +12 | S g,r ( c ) | ≪ F,ε | g | ε b X ε , where b X = O ( | f | C ) for some fixed C . Initially, a version of this result was proved by Heath-Brown (Lemma 28 of [HB96b]). This is afunction field analogue of proposition 4.1 of the first author in [T. 19a]. We first prove a lemmaindicating that most S g,r ( a, ℓ, c ) vanish. Lemma 3.2.
Unless c ≡ ar + ℓ ) A λ mod g , we have S g,r ( a, ℓ, c ) = 0 . Consequently, S g,r ( c ) = 0 unless c ≡ αA λ mod g for some α ∈ O .Proof. Write b = r b + b , where b is a vector modulo g and b is a vector modulo r . We maythen rewrite S g,r ( a, ℓ, c ) = X b ψ (cid:18) ( a + rℓ )(2 λ T A b − k ) + agF ( b ) − h c , b i gr (cid:19) X b ψ (cid:18) a + rℓ ) λ T A b − h c , b i g (cid:19) . From Lemma 2.1, the second sum vanishes unless c ≡ a + rℓ ) Aλ mod g , which gives the firststatement in the lemma. Since S g,r ( c ) is a sum of the S g,r ( a, ℓ, c ), we obtain that it is zero unlesspossibly c ≡ αA λ mod g for some α ∈ O . (cid:3) By definition, S g,r ( c ) = X ℓ ∈O| ℓ | < | g | X | a | < | r |∗ X b ∈O d / ( gr ) ψ (cid:18) ( a + rℓ )(2 λ T A b − k ) + agF ( b ) − h c , b i gr (cid:19) . Since the sum over ℓ is zero unless g | λ T A b − k , in which case it contributes a factor of | g | , wehave S g,r ( c ) = | g | X | a | < | r |∗ X b ∈O d / ( gr ) g | λ T A b − k ψ (cid:18) a (2 λ T A b − k ) + agF ( b ) − h c , b i gr (cid:19) . We will give a bound on each of the S g,r ( c ). We do so by first decomposing S g,r ( c ) into the productof two sums and then bounding each of the two sums separately.Write r = r r , where r i ∈ O and gcd( r , ∆ g ) = 1 and such that the prime divisors of r areamong the prime divisors of ∆ g . In particular, gcd( r , gr ) = 1, and so we may write k = gr k + r k and a = r a + r a for some k , k ∈ O and unique a ∈ O / ( r ), a ∈ O / ( r ). Similarly, we may find vectors b ∈O d / ( r ) and b ∈ O d / ( gr ) such that b = gr b + r b . If we set(11) S := X ∗| a | < | r | X b ψ (cid:18) r a λ T A b + a ( gr ) F ( b ) − h c , b i − r a k r (cid:19) , and(12) S := | g | X ∗| a | < | r | X b ∈O d / ( gr ) g | λ T A b − kr ψ (cid:18) r a λ T A b + a gr F ( b ) − h c , b i − r a k gr (cid:19) , then we see from a simple substitution of the above that S g,r ( c ) = S S . ECTIONS OF QUADRICS OVER A F q What we proceed to do is bound S and S .In order to bound S from above, consider the following situation. Let G ( x ) := x T B x , where B is a symmetric matrix B ∈ M d ( O ) with D := det( B ) = 0. Furthermore, let r ∈ O be such thatgcd( r, D ) = 1, and for each e ∈ O / ( r ), c , c ′ ∈ O d / ( r ), define S r ( G, c , c ′ , e ) := X ∗| a | < | r | X b ∈O d / ( r ) ψ (cid:18) a ( G ( b ) + h c ′ , b i + e ) − h c , b i r (cid:19) . We will prove the following lemma.
Lemma 3.3.
With the notation as above, (13) S r ( G, c , c ′ , e ) = (cid:18) Dr (cid:19) τ dr Kl r ( G, c , c ′ , e ) , where τ r := P | x | < | r | ψ (cid:16) x r (cid:17) is the Gauss sum, (cid:0) .. (cid:1) is the Jacobi symbol, and Kl r ( G, c , c ′ , e ) is eithera Kloosterman sum (for even d ) or a Sali´e sum (for odd d ). Furthermore, we have | S | ≤ | r | d +12 τ ( r ) | gcd( r , f ) | / , where τ ( . ) is the divisor function. In order to prove this lemma, we first reduce to the case where r = ̟ k for some irreducible ̟ ∈ O .This is done via the following lemma. Lemma 3.4 (Multiplicativity of S r ( G, c , c ′ , e )) . Suppose r = uv for coprime u, v ∈ O . Then S r ( G, c , c ′ , e ) = S u ( G, ¯ v c , c ′ , e ) S v ( G, ¯ u c , c ′ , e ) . Proof.
Since u and v are coprime, as b ranges over O d / ( u ) and b ranges over O d / ( v ), the vector b = v b + u b ranges over a complete set of vectors modulo uv = r . Similarly, as a ranges over O / ( u ) and a ranges over O / ( v ), a = va + ua ranges over a complete set of polynomials modulo uv = r . Making these substitutions, the sum-mands in S r ( G, c , c ′ , e ) become ψ (cid:18) a ( G ( b ) + h c ′ , b i + e ) − h c , b i r (cid:19) = ψ (cid:18) ( va + ua )( G ( v b + u b ) + h c ′ , v b + u b i + e ) − h c , v b + u b i uv (cid:19) = ψ (cid:18) ( va + ua )( v G ( b ) + u G ( b ) + v h c ′ , b i + u h c ′ , b i + e ) − v h c , b i − u h c , b i uv (cid:19) = ψ (cid:18) a ( v G ( b ) + h v c ′ , b i + e ) − h c , b i u (cid:19) ψ (cid:18) a ( u G ( b ) + h u c ′ , b i + e ) − h c , b i v (cid:19) = ψ (cid:18) a ( G ( v b ) + h c ′ , v b i + e ) − h ¯ v c , v b i u (cid:19) ψ (cid:18) a ( G ( u b ) + h c ′ , u b i + e ) − h ¯ u c , u b i v (cid:19) . Since u and v are coprime, u b and v b range over a complete set of residues modulo v and u ,respectively. As a result, S r ( G, c , c ′ , e ) = S u ( G, ¯ v c , c ′ , e ) S v ( G, ¯ u c , c ′ , e ) , as required. (cid:3) Since the characteristic of our base field is odd, we can diagonalize our quadratic form G modulo r , and write G ( x ) = d X i =1 α i x i . Therefore, S r ( G, c , c ′ , e ) = X ∗| a | < | r | ψ (cid:16) aer (cid:17) d Y j =1 X b ∈O / ( r ) ψ a ( α j b + c ′ j b ) − c j br ! . We complete the square to obtain S r ( G, c , c ′ , e ) = X ∗| a | < | r | ψ (cid:16) aer (cid:17) d Y j =1 X b ∈O / ( r ) ψ aα j (cid:16) b + 2 aα j ( ac ′ j − c j ) (cid:17) − aα j ( ac ′ j − c j ) ) r = X ∗| a | < | r | ψ (cid:16) aer (cid:17) d Y j =1 ψ − aα j ( ac ′ j − c j ) ) r ! X b ∈O / ( r ) ψ aα j (cid:16) b + 2 aα j ( ac ′ j − c j ) (cid:17) r The internal sum is equal to (cid:0) aα j r (cid:1) τ r , and so S r ( G, c , c ′ , e ) = τ dr (cid:18) Dr (cid:19) ψ P j α j c j c ′ j r ! X | a | < | r |∗ (cid:16) ar (cid:17) d ψ a ( e − P j α j c ′ j ) − ¯ a P j α j c j r ! . In light of Lemma 3.4, we proceed to bound S ̟ k ( G, c , c ′ , e ) for k ≥ ̟ ∈ O irreducible. Itsuffices to bound the sums X ∗| a | < | ̟ k | (cid:16) a̟ k (cid:17) d ψ a ( e − P j α j c ′ j ) − ¯ a P j α j c j ̟ k ! . We will be interested only in the case when r = ̟ k | r , G = ( gr ) F , c ′ = 2 r A λ , and e = − r k .In this case, e − X j α j c ′ j ≡ − r k − F ( λ )¯ g ≡ ( gr k − f )¯ g ≡ − f ¯ g mod ̟ k . Similarly, X j α j c j ≡ X j g η j c j mod ̟ k . Making these substitutions and changing a to ag , we obtain X ∗| a | < | ̟ k | (cid:16) a̟ k (cid:17) d ψ − af − ¯ ag P j η j c j ̟ k ! . We will obtain Lemma 3.3 using the function-field analogue of the Weil bound on Kloosterman andSali´e sums, whose proof we sketch in the following.
Lemma 3.5 (Weil bound) . Suppose m, n, c ∈ F q [ t ] , c = 0 , and θ ∈ { , } . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ∗| x | < | c | (cid:16) xc (cid:17) θ ψ (cid:18) mx + nxc (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ τ ( c ) | c | / | gcd( m, n, c ) | / . ECTIONS OF QUADRICS OVER A F q Proof.
By a standard computation as in Lemma 3.4, we may reduce to when c is a prime power ̟ k . Furthermore, we may assume that ̟ ∤ mn ; otherwise we have Ramanujan sums which may beexplicitly computed as in the case of integers and shown to satisfy the above bound (see equations(3.1)-(3.3) of [IK04] for usual Ramanujan sums).First, note that when k = 1, then this is the Weil bound on Kloosterman sums over the finitefield O / ( ̟ ). This is a consequence of Theorem 10 of [Kow18]. For Sali´e sums, it is a consequenceof Theorem 2.19 of loc.cit . We may therefore assume that k ≥ | gcd( m, n, c ) | and summing modulo c/ gcd( m, n, c ), we mayassume without loss of generality that gcd( m, n, c ) = 1. Let us assume furthermore that we have aKloosterman sum, that is, θ = 0.Write x = a + a ̟ ⌊ k/ ⌋ , where a is chosen modulo ̟ ⌊ k/ ⌋ and is relatively prime to ̟ , and a is chosen modulo ̟ ⌈ k/ ⌉ . Furthermore, note that a + a ̟ ⌊ k/ ⌋ ≡ a − a a ̟ ⌊ k/ ⌋ + a a ̟ ⌊ k/ ⌋ mod ̟ k , where the inverses are computed modulo ̟ k . Making these substitutions, we obtain ψ (cid:18) mx + nx̟ k (cid:19) = ψ m ( a + a ̟ ⌊ k/ ⌋ ) + n ( a + a ̟ ⌊ k/ ⌋ ) ̟ k ! = ψ ( a + a ̟ ⌊ k/ ⌋ ) m + n ( a − a a ̟ ⌊ k/ ⌋ + a a ̟ ⌊ k/ ⌋ ) ̟ k ! = ψ ma + na + ̟ ⌊ k/ ⌋ (cid:0) a (cid:0) m − a n (cid:1) + a a ̟ ⌊ k/ ⌋ n (cid:1) ̟ k ! = ψ (cid:18) ma + na ̟ k (cid:19) ψ a (cid:0) m − a n (cid:1) + a a ̟ ⌊ k/ ⌋ n̟ ⌈ k/ ⌉ ! . Summation over a mod ̟ ⌈ k/ ⌉ gives us zero unless m − na ≡ ̟ ⌊ k/ ⌋ . For such a , if k is even, summing over a contributes a factor of | ̟ | k/ . If k is odd, then for such a , summing over a contributes a factor of | ̟ | ⌊ k/ ⌋ X y mod ̟ ψ (cid:18) − a y n̟ (cid:19) . This sum is a Gauss sum, and is of norm | ̟ | / unless ̟ | n , we assumed not to be the case at thebeginning of this proof. Therefore, when k is odd, summing over a contributes a factor of | ̟ | k/ as well. Since ̟ ∤ mn , the congruence above has at most 2 solutions a modulo ̟ ⌊ k/ ⌋ . Puttingthese together, the conclusion follows.For the case of Sali´e sums, that is θ = 1, the proof is similar. See Lemmas 12.2 and 12.3 of [IK04]. (cid:3) Using the reduction above and Lemmas 3.4 and 3.5 above, we obtain for every r | S | ≤ τ ( r ) | r | d +12 | gcd( r , f ) | / . This concludes the proof of Lemma 13.We now bound S from above via the following lemma. The proof uses the Cauchy-Schwarzinequality. Lemma 3.6.
For S as above, | S | ≪ ∆ | g | d | r | d +1 . Proof.
Recall that S := | g | X ∗| a | < | r | X b ∈O d / ( gr ) g | λ T A b − kr ψ (cid:18) r a λ T A b + a gr F ( b ) − h c , b i − r a k gr (cid:19) . Applying the Cauchy-Schwarz inequality to the a variable, we obtain | S | ≤ | g | ϕ ( r ) X ∗| a | < | r | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X b ∈O d / ( gr ) g | λ T A b − kr ψ (cid:18) r a λ T A b + a gr F ( b ) − h c , b i − r a k gr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | g | ϕ ( r ) X ∗| a | < | r | X b , b ′ ∈O d / ( gr ) g | λ T A b − kr ,g | λ T A b ′ − kr ψ (cid:18) r a λ T A ( b − b ′ ) + a gr ( F ( b ) − F ( b ′ )) − h c , b − b ′ i gr (cid:19) Making the substitution u = b − b ′ , we obtain | S | ≤ | g | ϕ ( r ) X ∗| a | < | r | X b , u ∈O d / ( gr ) g | λ T A b − kr ,g | λ T A u ψ (cid:18) r a λ T A u + a gr (2 b T A u + F ( u )) − h c , u i gr (cid:19) . The sum over b is zero unless r | ∆ gcd( u ), which implies that the summation is non-zero only if u ∈ ( r O / (gcd(∆ , r ) gr )) d ≃ ( O / (gcd(∆ , r ) g )) d . Hence, | S | ≤ | g | ϕ ( r ) X ∗| a | < | r | X b ∈O d / ( gr ) g | λ T A b − kr X u ∈ ( O / (gcd(∆ ,r ) g )) d g | λ T A u ≪ ∆ | g | d ϕ ( r ) | r | d ≪ ∆ | g | d | r | d +2 . Taking square roots, we obtain | S | ≪ ∆ | g | d | r | d +1 , as required. (cid:3) We now put together the above results to prove Proposition 3.1.
Proof of Proposition 3.1.
As before, write r = r r , where gcd( r , g ∆) = 1 and the prime divisorsof r are among those of g ∆. By construction, we know that | S g,r ( c ) | = | S || S | . Therefore, from ECTIONS OF QUADRICS OVER A F q Lemmas 3.3 and 3.6, we have X r ∈O| r | < b X | g | − d | r | − d +12 | S g,r ( c ) |≪ ∆ X r ∈O| r | < b X τ ( r ) | r | / | gcd( r , f ) | / ≤ b X ε X r ∈O| r | < b X | r | / | gcd( r , f ) | / = b X ε X r ∈O| r | < b X | gcd( r , f ) | / X r ∈O| r | < b X/ | r | | r | / . The second (internal) sum can be bounded using X r ∈O| r | < b X/ | r | | r | / ≤ X d | ( g ∆) ∞ , | d | < b X/ | r | | d | / b X | r d | ≤ b X/ | r | X d | ( g ∆) ∞ , | d | < b X/ | r | ≪ b X | r | | g ∆ | ε b X ε . Hence, X r ∈O| r | < b X | gcd( r , f ) | / X r ∈O| r | < b X/ | r | | r | / ≪ b X | g ∆ | ε b X ε X r ∈O| r | < b X | gcd( r , f ) | / | r | , from which the conclusion follows since this latter sum is ≪ b X ε . (cid:3) Analytic functions on T d In order to prove our main theorem, it turns out that we need to do analysis not just usingpolynomials over K ∞ , but also using convergent Taylor series. We begin by defining a space ofanalytic functions defined on T d that extends the space of polynomials. Let O ∞ := { x ∈ K ∞ : | α | ≤ } . Define C ω ( T d ) := X ( n ,...,n d ) ∈ N d ≥ a ( n ,...,n d ) x n . . . x n d d : a ( n ,...,n d ) ∈ O ∞ . It is easy to see that the above Taylor expansions are convergent for ( u , . . . , u d ) ∈ T d . When d = 1,aside from polynomials in O ∞ [ x ], examples of analytic functions on T are11 − x := ∞ X k =0 x k , and (1 + x ) / := ∞ X k =0 (cid:18) / k (cid:19) x k . This square root function is defined since the base characteristic is odd. We define the partialderivatives ∂∂x i for 1 ≤ i ≤ d on C ω ( T d ) to be the formal derivation operator which acts on the monomials as: ∂∂x i x n . . . x n d d = n i x n . . . x n i − i . . . x n d d and extend them by linearity to power series.It is easy to check that it sends C ω ( T d ) to itself. Let C ω ( T m , T n ) := { Φ = ( φ , . . . , φ n ) : φ j ∈ C ω ( T m ) and φ j (0) ∈ T } . For Φ ∈ C ω ( T m , T n ) define the Jacobi matrix J Φ := h ∂φ i ∂x j i , where 1 ≤ i ≤ n and 1 ≤ j ≤ m. For m = n define the Jacobi determinant to be det( J Φ) . We also have the following change of variablesformula, which readily follows from Igusa [Igu00, Lemma 7.4.2].
Lemma 4.1.
Let Γ ⊂ K n ∞ be a box defined by the inequalities | x i | < b R i , for some real numbers R , . . . , R n . Let f : Γ → C be a continuous function. Then for any M ∈ GL n ( K ∞ ) we have Z Γ f ( u ) d u = | det M | Z M v ∈ Γ f ( M v ) d v . The analytic automorphism of T d . In this section, we define the group of the analyticautomorphism of T d . We use this group in order to simplify and reduce the computations of ouroscillatory integrals into Gaussian integrals. Recall that by Schwarz’s Lemma the analytic auto-morphisms of the disk in the complex plane which fixes the origin are just rotations. Unlike thedisk in the complex plane the analytic group of automorphisms of the disk T d is enormous. Define A ∞ ( T d ) := (cid:8) Φ ∈ C ω ( T d , T d ) : | det( J Φ(0)) | ∞ = 1 , and Φ(0) = 0 (cid:9) . Proposition 4.2. A ∞ ( T d ) is a group under the composition of functions and it preserve the Haarmeasure on T d . First, we prove a lemma on diagonalizing symmetric matrices over K that we use in the proof ofthe preceding proposition. It is easy to see that GL d ( O ∞ ) ⊂ C ω ( T d , T d ) . Lemma 4.3.
Suppose that A ∈ M d × d ( K ∞ ) and A ⊺ = A. Then there exists γ ∈ GL d ( O ∞ ) suchthat γ ⊺ Aγ = D [ η , . . . , η d ] , where D [ η , . . . , η d ] is the diagonal matrix with some η , . . . , η d ∈ K ∞ on its diagonal.Proof. We proceed by induction on d. The lemma is trivial for d = 1 . Without loss of generality,we assume that A ∈ M d × d ( O ∞ ) and A = 0 mod t − . Let ¯ A denote A mod t − which is a matrixwith F q coefficients. Since q = 2 , there exists a matrix g ∈ GL d ( F q ) which diagonalizes ¯ A, and wehave g ⊺ ¯ Ag = D [¯ η , . . . , ¯ η d ] . Suppose that η = 0 . Let A := g ⊺ Ag = (cid:2) a , . . . , a d (cid:3) = (cid:2) a i,j (cid:3) , where a i is the i th column vector of A , and a i,j is the i th and j th coordinate of A . Let H := − a , a , . . . − a ,d a , I d − × d − . Note that a , ∈ O ∗∞ is invertible. Hence H ∈ GL d ( O ∞ ) . Moreover, it is easy to check that H ⊺ A H = a . . . A , where A ⊺ = A ∈ M ( d − × ( d − ( O ∞ ) . The lemma follows from the induction hypothesis on A . (cid:3) ECTIONS OF QUADRICS OVER A F q Proof of Proposition 4.2.
By the product rule of the Jacobian it is easy to see that A ∞ ( T d ) is closedunder the composition of functions. The identity function is the identity element of A ∞ ( T d ) . It isenough to construct the inverse of Φ ∈ A ∞ ( T d ) . We prove the existence of the inverse by solvinga recursive system of linear equations. First, we explain it when d = 1 . We have Φ = P ∞ i =1 a i x i ,where | a | ∞ = 1 . We wish to find Ψ = P ∞ i =1 b i x i ∈ C ω ( T d ) such that Ψ ◦ Φ( x ) = x. This impliesthat b = a − and the following system of equations hold for each n ≥
20 = b n a n + n − X i =1 b i (some polynomial in a , . . . , a n − i +1 ) . The above system of recursive linear equations have a unique solution where b n ∈ O ∞ . For general d , suppose that Φ := ( φ ( x , . . . x d ) , . . . , φ d ( x , . . . , x d )) ∈ A ∞ ( T d ). By the definition of A ∞ ( T d ) , we have det( J Φ(0)) ∈ GL d ( O ∞ ) . Let ¯Ψ := J Φ(0) − ∈ GL d ( O ∞ ) . We note that J ( ¯Ψ ◦ Φ(0)) = I d × d . Without loss of the generality, we assume that J (Φ(0)) = I d × d . We wish to find Ψ :=( ψ ( x , . . . x d ) , . . . , ψ d ( x , . . . , x d )) ∈ A ∞ ( T d ) such that ψ i (cid:0) φ ( x , . . . , x d ) , . . . , φ d ( x , . . . , x d ) (cid:1) = x i for every 1 ≤ i ≤ d. Suppose that φ i := X ( n ,...,n d ) ∈ N d ≥ a i, ( n ,...,n d ) x n . . . x n d d ,ψ i := X ( n ,...,n d ) ∈ N d ≥ b i, ( n ,...,n d ) x n . . . x n d d , where 1 ≤ i ≤ d. Let | ( n , . . . , n d ) | := P di =1 n i . For ( n , . . . , n d ) ∈ N d ≥ , with | ( n , . . . , n d ) | ≥ , wehave(14) 0 = b i, ( n ,...,n d ) + X ( m ,...,m d ) < ( n ,...,n d ) b i, ( m ,...,m d ) (some polynomial in a j, ( k ,...,k d ) ) , where ( k , . . . , k d ) ≤ | ( n , . . . , n d ) | . Similarly, the above system of recursive linear equations have aunique solution where b i, ( n ,...,n d ) ∈ O ∞ . Finally, by the definition of A ∞ ( T d ) , we have | det( J Φ(0)) | ∞ =1 . This implies | det( J Φ( x )) | ∞ = 1 for every x ∈ T d . This completes the proof of our lemma. (cid:3)
Next, we prove a version of the Morse lemma for functions in C ω ( T d ) . Proposition 4.4 (Morse lemma over K ∞ ) . Assume that φ ( u ) is an analytic function on T d witha single critical point at and the Hessian H φ , where | det( H φ (0)) | ∞ = 1 . Then there exists Ψ ∈A ∞ ( T d ) with J Ψ(0) = I d × d such that φ (Ψ) = φ (0) + Ψ ⊺ H φ (0)Ψ . Proof.
By Lemma 4.3 there exists a matrix g ∈ GL d ( O ∞ ) such that g ⊺ H φ (0) g = D [ λ , . . . , λ d ] . Since H φ (0) ∈ GL d ( O ∞ ) then λ i ∈ O ∞ and | λ i | ∞ = 1 . By changing the variables with g , we assume that H φ (0) is a diagonal matrix. First, we explain it for d = 1 . We have φ ( x ) = φ (0)+ λx + x P ∞ n =0 a n x n , where | λ i | ∞ = 1 . Let ψ ( x ) := x (cid:0) x ∞ X n =0 λ − a n x n (cid:1) / = x (cid:16) ∞ X k =0 (cid:18) / k (cid:19)(cid:0) x ∞ X n =0 λ − a n x n (cid:1) k (cid:17) ∈ A ∞ ( T ) , where we used the taylor expansion (1 + x ) / := P ∞ k =0 (cid:0) / k (cid:1) x k . It is easy to check that φ = φ (0) + λψ . This completes the proof of the lemma for d = 1 . For general d, we proceed by induction on d. We explain our induction hypothesis next. Assume that φ ( x , . . . , x d ) = φ (0) + X i,j ≥ x i x j ( δ i,j λ i + h i,j ( x , x , . . . , x d )) , for some h i,j ( x , . . . , x d ) ∈ C ω ( T d ) and λ i ∈ O ∞ , where h i,j (0) = 0 and | λ i | ∞ = 1 . Then φ = φ (0) + X j ≥ λ j ψ j , where ψ j = x j + h j ( x , . . . , x d ) such that h j ( x , . . . , x d ) has a critical point at 0. The inductionhypothesis holds for d = 1 . We assume that it holds for d − , and we prove it for d. We write φ ( x , . . . , x d ) = φ (0) + x ( λ + h , ( x , . . . , x d )) + X j ≥ x x j h ,j ( x , x , . . . , x d )+ X i,j ≥ x i x j ( δ i,j λ i + h i,j ( x , x , . . . , x d )) , for some h i,j ( x , . . . , x d ) ∈ C ω ( T d ) , where h i,j (0) = 0 . Define ψ := x (cid:0) λ − h , (cid:1) / + (cid:0) λ − X j ≥ x j h ,j ( x , x , . . . , x d ) (cid:1)(cid:0) λ − φ (cid:1) − / . We have φ = φ (0) + λ ψ + X i,j ≥ x i x j ( δ i,j λ i + h ′ i,j ( x , x , . . . , x d )) , (15)for some h ′ i,j ( x , . . . , x d ) ∈ C ω ( T d ) , where h i,j (0) = 0 . By the induction induction hypothesis for d −
1, we have φ = φ (0) + λ ψ + X j ≥ λ j ψ j , where ψ j = x j + h j ( x , . . . , x d ) such that h j ( x , . . . , x d ) has a critical point at 0. This concludesour lemma. (cid:3) Stationary phase theorem over function fields.
In this section, we prove a version of thestationary phase theorem in the function fields setting that we use for computing the oscillatoryintegrals I g,r ( c ).Let f ∈ K ∞ and define(16) G ( f ) := min( | f | − / ∞ ,
1) if ord( f ) is even, | f | − / ∞ ε f if ord( f ) ≥ odd, ε f := G ( f ) | G ( f ) | and G ( f ) := P x ∈ F q e q ( a f x ) is the gauss sum associated to a f the top degreecoefficient of f. Suppose that φ ∈ C ω ( T d ) has a single critical point at 0 with the Hessian H φ , where | det( H φ (0)) | ∞ = 1 . Proposition 4.5.
Suppose the above assumptions on φ and f. We have Z T d ψ ( f φ ( u )) d u = ψ ( f φ ( )) d Y i =1 G ( f λ i ) , where λ i ∈ O ∞ for ≤ i ≤ d are diagonal element of g ⊺ H φ (0) g for some g ∈ GL d ( O ∞ ) obtainedin Lemma 4.3. ECTIONS OF QUADRICS OVER A F q We begin the proof of the above proposition by proving some spacial cases of the proposition forthe quadratic polynomials.4.2.1.
Gaussian integrals over function field.
We define the analogue of the Gaussian integrals overthe function field K and give an explicit formula for them. Lemma 4.6.
For every f ∈ K ∞ , we have Z T ψ ( f u ) du = G ( f ) . Proof.
First, suppose that ord( f ) = 2 k, where k ≥ . We partition T into the cosets of t − k T . Let α + t − k T ⊂ T . We show that Z α + t − k T ψ ( f u ) du = 0for α / ∈ t − k T . We have Z α + t − k T ψ ( f u ) du = Z t − k T ψ ( f ( α + v ) ) dv = ψ ( f α ) Z t − k T ψ ( f (2 αv + v )) dv = ψ ( f α ) Z t − k T ψ ( f αv ) dv = 0 , where we used Lemma 2.2, ord( f v ) ≤ − αf ) ≥ k. Therefore, Z T ψ ( f u ) du = Z t − k T ψ ( f u ) du = Z t − k T du = | f | − / ∞ = G ( f ) . On the other hand, if ord( f ) = 2 k − , where k ≥ . Similarly, for α / ∈ t − k +1 T Z α + t − k T ψ ( f u ) du = Z v ∈ t − k T ψ ( f ( α + v ) ) du = ψ ( f α ) Z t − k T ψ ( f (2 αv + v )) dv = ψ ( f α ) Z t − k T ψ ( f αv ) dv = 0 , where we used Lemma 2.2, ord( f v ) ≤ − αf ) ≥ k. Hence Z T ψ ( f u ) du = Z t − k +1 T ψ ( f u ) du = q − k G ( f ) = G ( f ) . The last equality follows from the following. Indeed, by the definition of the integral, we have Z t − k +1 T ψ ( f u ) du = lim m → + ∞ q − m − k +1 X a − m t − m − k +1 + ... + a − t − k : a i ∈ F q ψ (( a − m t − m − k +1 + . . . + a − t − k ) f )= lim m → + ∞ q − m − k +1 X a − m ,...,a − ∈ F q e q ( a f a − )= q − k X x ∈ F q e q ( a f x ) . It is well-known, that G ( f ) = q / ε f . Consequently, q − k G ( f ) = | f | − / ∞ ε f . We have thereforeproved the result for ord( f ) = 2 k − k ≥ f ) ≤ −
1, then ord( f u ) < − u ∈ T . Consequently, Z T ψ ( f u ) du = Z T du = 1 . This concludes the proof. (cid:3)
Next, we give a formula for the Gaussian integral associate to any symmetric matrix A ∈ M d × d ( K ∞ ) . Define G ( A ) := Z T d ψ ( u ⊺ A u ) . Lemma 4.7.
We have G ( A ) = d Y i =1 G ( λ i ) , where λ i ∈ K ∞ for ≤ i ≤ d are diagonal element of g ⊺ Ag for some g ∈ GL d ( O ∞ ) obtained inLemma 4.3.Proof. By Lemma 4.3, there exists g ∈ GL d ( O ∞ ) such that g ⊺ Ag = D [ λ , . . . , λ d ] . By the changeof the variable formula in Lemma 4.1, we have G ( A ) = Z T d ψ ( u ⊺ A u ) d u = Z T d ψ (cid:0) ( g − u ) ⊺ g ⊺ Ag ( g − u ) (cid:1) d u = Z T d ψ d X i =1 λ i v i ! d v = d Y i =1 G ( λ i ) , where (cid:2) v . . . v d (cid:3) = v = g − u . This completes the proof of the lemma. (cid:3)
Finally, we give a proof of the Proposition 4.5.
Proof of Proposition 4.5 .
By Proposition 4.4, there exists Ψ ∈ A ∞ ( T d ) such that φ (Ψ) = φ (0) +Ψ ⊺ H φ (0)Ψ . By Proposition 4.2, Ψ is a measure preserving automorphism of T d . Hence, Z T d ψ ( f φ ( u )) d u = Z T d ψ ( f ( φ (0) + Ψ ⊺ H φ (0)Ψ)) d Ψ . By Lemma 4.7, Z T d ψ ( f ( φ (0) + Ψ ⊺ H φ (0)Ψ)) d Ψ = ψ ( f φ ( )) d Y i =1 G ( f λ i ) , where λ i ∈ O ∞ for 1 ≤ i ≤ d are diagonal element of g ⊺ H φ (0) g for some g ∈ GL d ( O ∞ ) obtained inLemma 4.3. This concludes the proof of our proposition. (cid:3) Bounds on the oscillatory integrals I g,r ( c )In this section, we give explicit formulas for the oscillatory integrals I g,r ( c ) in terms of the Klooster-man sums (Sali´e sums). By Lemma 4.3, we suppose that F ( γ u ) = P η i η i u i , where γ ∈ GL d ( O ∞ ) . Recall the additive character ψ : K ∞ → C ∗ from § h ( x, y ) = ( | x | − if | y | < | x | Test function.
In this section, we define the test function w that we use for estimating theoscillatory integrals I g,r ( c ) at the end of this section. Recall the definition 1.1 of an anisotropiccone. Lemma 5.1.
Let F ( x ) be a non-degenerate quadratic form in d ≥ variables. We may then cover K d ∞ with four anisotropic cones such that for any given f , X f ( K ∞ ) intersects at least one of them. ECTIONS OF QUADRICS OVER A F q Proof.
We show that each class in K ×∞ /K × ∞ , which consists of representative 1 , ν, t, νt , where ν ∈ F × q is a quadratic non-residue, gives us an anisotropic cone, at least one of which intersects X f ( K ∞ ) for any given f . Indeed, since being an anisotropic cone is preserved by linear change ofcoordinates, we may assume without loss of generality that F is diagonal and the coefficients of F are also among these representatives. Furthermore, we may assume without loss of generalitythat f is one of the representatives 1 , ν, t, νt by uniformly scaling the coordinates (note that, bydefinition, anisotropic cones are invariant under scaling). After these reduction, by taking the setof x ∈ K d ∞ such that | F ( x ) | ≥ q | x | , we obtain an anisotropic cone. Showing that the class of f in K ×∞ /K × ∞ is represented by an element of this anisotropic cone follows from a simple case-by-case analysis. Suppose the class of f is νt . If one of the coefficients of F is νt , then we havea solution in the anisotropic cone. Otherwise, the coefficients are among 1 , ν, t and at least twoof the coefficients are equal since d ≥
4. If − K ∞ can bewritten as the sum of two squares. Since at least one coefficient repeats, this implies that we canrepresent any element. On the other hand, − ν = −
1. If both 1 and − K ∞ . Therefore, let us assume otherwise. We are reduced to showing that there is a solutionin the anisotropic cone to the equations t ( x + x + x + 1) = ± x , t ( x + x + 1) = ± ( x + x ), t ( x + 1) = ± ( x + x + x ), and x + . . . + x = ± t for any choice of signs. x + x + 1 = 0 is solvablemodulo any odd prime, and so the first and second equations have a solution in the anisotropiccone. Take a, b ∈ F × q such that a + b = − − ab = 0). For thethird equation, let ( x , x , x , x ) = (cid:16) , at (cid:0) ∓ t (cid:1) / , bt (cid:0) ∓ t (cid:1) / , t (cid:17) . Note that such squarerootsexist in K ∞ because q is odd (see the beginning of Section 4 for the formula). For the final equation ± t = x + . . . + x let ( x , x , x , x ) = (cid:16) at (cid:0) ± t (cid:1) / , bt (cid:0) ± t (cid:1) / , t, (cid:17) . The other classes canbe dealt with similarly; at the beginning, you can multiply the quadratic form by ν or t and scalethe coordinates to reduce it to the above case that f has class νt . Note that the construction ofthe anisotropic cone associated to f depends only on the class of f in K ×∞ /K × ∞ . (cid:3) Remark . This lemma shows that given any f , we can find an anisotropic cone intersecting X f ( K ∞ ). This fact combined with our main theorem implies strong approximation for F (Corol-lary 1.3).Fix an anisotropic cone Ω with respect to F ( x ) (such that Ω ∩ X f ( K ∞ ) = ∅ ). Lemma 5.2.
Suppose that x ∈ Ω and y / ∈ Ω . Then | x ± y | ≥ max ( | x | , | y | ) / b ω. Proof.
It follows from property (2). (cid:3)
For non-degenerate quadratic form F ( x ) = x ⊺ A x , we say F ∗ ( x ) = x ⊺ A − x is the dual of F ( x ) . Note that F ( x ) = F ∗ ( A x ) . Let Ω ∗ := A Ω . Lemma 5.3. Ω ∗ is an anisotropic cone with respect to F ∗ . Proof.
It follows from the definition of Ω ∗ , F ∗ and anisotropic cones. (cid:3) Let w be the characteristic function of a ball centered at x ∈ V f ∩ Ω : w ( x ) = ( | x − x | < | t − α f | / , α > max deg( η i ) + ω is any large enough fixed integer such that n y ∈ K d ∞ : | y − A x | < | t − α f | / o ⊂ Ω ∗ . Note that if w ( x ) = 0 , then x ∈ Ω . Moreover, w ( g t + λ ) = ( | t − t | < b R, x = g t + λ , and R := ⌈ deg( f ) / − deg ( g ) − α / ⌉ . Bounding I g,r ( c ) . Recall that G ( t ) := F ( g t + λ ) − fg = F ( t ) + 1 g (2 λ T A t − k ) , where k = f − F ( λ ) g . In this section, we assume that Q := ⌈ deg( f ) / − deg ( g ) ⌉ + max i (deg( η i )) + ω ′ . We have(18) I g,r ( c ) = Z K d ∞ h (cid:18) rt Q , G ( t ) t Q (cid:19) w ( g t + λ ) ψ (cid:18) h c , t i gr (cid:19) d t = Z | t − t | < b R | G ( t ) | < b Q | r | b Q | r | ψ (cid:18) h c , t i gr (cid:19) d t . Let κ := max i | c i g | . Lemma 5.4.
Suppose that κ < | r | b R , then I g,r ( c ) = ψ (cid:16) h c , t i gr (cid:17) I g,r (0) . Proof.
Since max i ( | c i | ) < | gr | b R and | t − t | < b R , ψ (cid:16) h c , t i gr (cid:17) = ψ (cid:16) h c , t i gr (cid:17) . Hence, we have I g,r ( c ) = ψ (cid:18) h c , t i gr (cid:19) Z | t − t | < b R | G ( t ) | < b Q | r | b Q | r | d t = ψ (cid:18) h c , t i gr (cid:19) I g,r (0) . This completes the proof of our lemma. (cid:3)
Lemma 5.5.
Let
Q, R and t be as above, and suppose that | t − t | < b R. Then | G ( t ) | < b Q | r | isequivalent to | F ( t ) − k/g | < b Q | r | . Moreover, if | G ( t ) | < b Q | r | , then | G ( t + ζ ) | < b Q | r | for every ζ ∈ K d ∞ , where | ζ | ≤ min( | r | , b R ) . Proof.
Since t ∈ Ω , by property (3) in Lemma 5.1, | t | ≤ | f | / b ω ′ / / | g | . Recall that Q = ⌈ deg( f ) / − deg ( g ) ⌉ +max i (deg( η i ))+ ω ′ . Since | λ || g | < , and | t | < | f | / b ω ′ / / | g | then | g (2 λ T A t ) | < b Q. Hence, for | t − t | < b R , | G ( t ) | < b Q | r | is equivalent to | F ( t ) − k/g | < b Q | r | . Moreover, supposethat | ζ | ≤ min( | r | , b R ) , and | t − t | < b R , then | G ( t + ζ ) − G ( t ) | ≤ max ( | F ( ζ ) | , | ζ ⊺ A ( t + λ /g ) | ) ≤ max( | ζ ⊺ A ζ | , b Q | ζ | ) ≤ b Q | r | , where we used | λ || g | < , | A | = \ max i (deg( η i )) . Hence, if | G ( t ) | < b Q | r | , then(19) | G ( t + ζ ) | ≤ max( | G ( t ) | , | G ( t + ζ ) − G ( t ) | ) < b Q | r | . This concludes the proof of our lemma. (cid:3)
We say c is an ordinary vector if κ ≥ b Q/ b R. (20) Lemma 5.6.
Suppose that c is an ordinary vector and | r | ≤ b Q . Then, (21) I g,r ( c ) = 0 . ECTIONS OF QUADRICS OVER A F q Proof.
By (18) and (19), we have I g,r ( c ) = Z | t | < b R | G ( t ) | < b Q | r | b Q | r | ψ (cid:18) h c , t i gr (cid:19) d t = b Q | r | Z | t | < b R | G ( t ) | < b Q | r | | r | , b R ) d Z | ζ | < min( | r | , b R ) ψ (cid:18) h c , t + ζ i gr (cid:19) d ζ d t . Since | c | ≥ | g | b Q/ b R, and | r | < b Q then R | ζ | < min( | r | , b R ) ψ (cid:16) h c , ζ i gr (cid:17) d ζ = 0 . This concludes the lemma. (cid:3)
We say c = 0 is an exceptional vector if κ < b Q/ b R. For the exceptional vectors c , we represent I g,r ( c )in terms of the Kloosterman sums (Sali´e sums) at ∞ . For α ∈ K ∞ with | α | ∞ = b l , defineKl ∞ ( α, ψ ) := Z | x | ∞ = b l ψ (cid:16) αx + x (cid:17) dx, and Sa ∞ ( α, ψ ) := Z | x | ∞ = b l ε x ψ (cid:16) αx + x (cid:17) dx, where ε x were defined in (16). By Weil’s estimate on the Kloosterman sums and the Sali´e sums,we show that Kl ∞ ( ψ, α ) ≪ | α | / , and Sa ∞ ( ψ, α ) ≪ | α | / . Proposition 5.7.
Suppose that c is an exceptional vector and κ ≥ η | r | b R and d ≥ , where η > b ω isa fixed large enough constant integer. For c ∈ Ω ∗ , we have (22) | I g,r ( c ) | ≪ F, Ω b Q d (cid:16) | c | b Q | gr | (cid:17) − d − . Otherwise, c / ∈ Ω ∗ and I g,r ( c ) = 0 . We give the proof of the above proposition after proving some auxiliary lemmas. For α ∈ K and l ∈ Z , define B ∞ ( ψ, l, α ) := Z | x | ∞ = b l ψ ( αx + x ) dx, e B ∞ ( ψ, l, α ) := Z | x | ∞ = b l ε x ψ ( αx + x ) dx. We write α = t l + k α ′ (1 + ˜ α ) and x = t l x ′ (1 + ˜ x ) for unique ˜ α, ˜ x ∈ T and α ′ , x ′ ∈ F q . Note thatfor k = 0 , we have B ∞ ( ψ, l, α ) = Kl ∞ ( ψ, α ) and e B ∞ ( ψ, l, α ) = Sa ∞ ( ψ, α ). In the following lemma,we give an explicit formula for B ∞ ( ψ, l, α ) in terms of the Kloosterman sums; see [CPS90, Lemma3.4] for a similar calculation. Lemma 5.8.
We have B ∞ ( ψ, l, α ) := ( q − b l if max( l + k, l ) < − , and k = 0 , − b l if max( l + k, l ) = − , and k = 0 , if max( l + k, l ) > − , and k = 0 . Kl ∞ ( ψ, α ) := ( q − b l if l < − , b l Kl( α ′ , F q ) if l = − , b l P x ′ = α ′ ψ (cid:16) t l x ′ (1 + ˜ α ) / (cid:17) G (2 x ′ t l ) if α ′ is a quadratic residue, if α ′ is not a quadratic residue. Similarly, e B ∞ ( ψ, l, α ) := ( q − b l if max( l + k, l ) < − , and k = 0 , − b l if l + k = − , and k > , b lτ ψ ( ε ) if l = − , and k < , if max( l + k, l ) > − , and k = 0 . where τ ψ := P a ∈ F q e q ( a ) χ ( a ) , where χ is the quadratic character in F q . Finally, Sa ∞ ( ψ, l, α ) := ( q − b l if l < − , b l Sa( α ′ , F q ) if l = − , b l P x ′ = α ′ ψ (cid:16) t l x ′ (1 + ˜ α ) / (cid:17) G (2 x ′ t l ) if α ′ is a quadratic residue, if α ′ is not a quadratic residue.Proof. Suppose that k >
0. We have B ∞ ( ψ, l, α ) = Z | x | ∞ = b l ψ ( αx + x ) dx = b l X x ′ ∈ F ∗ q Z T ψ (cid:16) t l + k α ′ (1 + ˜ α ) x ′ (1 + ˜ x ) + t l x ′ (1 + ˜ x ) (cid:17) d ˜ x. Fix ˜ α ∈ T and α ′ , x ′ ∈ F q , and define the analytic function u (˜ x ) as u (˜ x ) := α ′ (1 + ˜ α ) x ′ (1 + ˜ x ) + t − k x ′ (1 + ˜ x ) − (cid:2) α ′ (1 + ˜ α ) x ′ + t − k x ′ (cid:3) , where ˜ x ∈ T . We note that u (0) = 0 , and | ∂u∂ ˜ x (0) | = | − α ′ (1+˜ α )(1+˜ x ) x ′ + t − k x ′ | = 1 . Hence u ∈ A ∞ ( T ) . Bychanging the variable to u (˜ x ), we have B ∞ ( ψ, l, α ) = b l X x ′ ∈ F ∗ q ψ (cid:16) α ′ (1 + ˜ α ) t l + k x ′ + x ′ t l (cid:17) Z T ψ ( t l + k u ) du = ( q − b l if l + k < − , − b l if l + k = − , k <
0. Fix ˜ α ∈ T and α ′ , x ′ ∈ F q , and define the analytic function v (˜ x ) as v (˜ x ) := t k α ′ (1 + ˜ α ) x ′ (1 + ˜ x ) + x ′ (1 + ˜ x ) − (cid:2) t k α ′ (1 + ˜ α ) x ′ + x ′ (cid:3) , where ˜ x ∈ T . We note that | ∂v∂ ˜ x (0) | = | − t k α ′ (1+˜ α )(1+˜ x ) x ′ + x ′ | = 1 . Hence v ∈ A ∞ ( T ) . By changing thevariable to v (˜ x ), we have B ∞ ( ψ, l, α ) = b l X x ′ ∈ F ∗ q ψ (cid:16) α ′ (1 + ˜ α ) t l + k x ′ + x ′ t l (cid:17) Z T ψ ( t l v ) dv = ( q − b l if l < − , − b l if l = − , k = 0 . Fix ˜ α ∈ T and α ′ , x ′ ∈ F q . Suppose that x ′ = α ′ in F q , and define theanalytic function w (˜ x ) as w (˜ x ) := α ′ (1 + ˜ α ) x ′ (1 + ˜ x ) + x ′ (1 + ˜ x ) − (cid:2) α ′ (1 + ˜ α ) x ′ + x ′ (cid:3) , where ˜ x ∈ T . We note that | ∂w∂ ˜ x (0) | = | − α ′ (1+˜ α )(1+˜ x ) x ′ + x ′ | = | − α ′ (1+˜ α ) − (1+˜ x ) x ′ (1+˜ x ) x ′ | = 1 and w ∈ A ∞ ( T ) . Otherwise x ′ = α ′ in F q . Define x := (1 + ˜ α ) / − ∈ T and h (˜ x ) := α ′ (1 + ˜ α ) x ′ (1 + ˜ x ) + x ′ (1 + ˜ x ) − (cid:2) x ′ (1 + ˜ α ) / (cid:3) . ECTIONS OF QUADRICS OVER A F q It is easy to see that h ( x ) = 0, ∂h∂ ˜ x ( x ) = 0 and ∂ h∂ ˜ x ( x ) = x ′ (1+˜ α ) / . Hence x is a critical pointwith | ∂ h∂ ˜ x ( x ) | = 1 . By the stationary phase theorem, we have B ∞ ( ψ, l, α ) = b l X x ′ = α ′ ψ (cid:16) α ′ (1 + ˜ α ) t l x ′ + x ′ t l (cid:17) Z T ψ ( t l w ) dw + b l X x ′ = α ′ ψ (cid:16) t l x ′ (1 + ˜ α ) / (cid:17) G (2 x ′ t l )Suppose that α ′ is a quadratic non-residue in F q . Then, from above it follows that B ∞ ( ψ, l, α ) = ( q − b l if l < − , b l Kl( α ′ , F q ) if l = − , α ′ is a quadratic residue in F q . We have B ∞ ( ψ, l, α ) = ( q − b l if l < − , b l Kl( α ′ , F q ) if l = − , b l P x ′ = α ′ ψ (cid:16) t l x ′ (1 + ˜ α ) / (cid:17) G (2 x ′ t l ) otherwise.This concludes the proof of the first part of the lemma. The argument for e B ∞ ( ψ, l, α ) is similar.Recall that ε x = 1 unless l is odd, which is the quadratic character evaluated at the top coefficientof t x . The second part of the lemma follows from the same lines, and we skip the details. (cid:3) Proof of Proposition 5.7.
By Lemma 5.5, | G ( t ) | < b Q | r | is equivalent to | F ( t ) − k/g | < b Q | r | for | t − t | < b R. By Lemma 2.2, we have Z T ψ (cid:16) αrt Q ( F ( t ) − k/g ) (cid:17) dα = ( , if | F ( t ) − k/g | < b Q | r | , , otherwise.We replace the above integral for detecting | F ( t ) − k/g | < b Q | r | . Hence, by (18) I g,r ( c ) = b Q | r | Z T Z | t − t | < b R ψ (cid:18) h c , t i gr + αrt Q ( F ( t ) − k/g ) (cid:19) d t dα. Recall that F ( γ y ) = P η i η i y i for some γ ∈ GL d ( O ∞ ) . We change variables to y = y ... y d = γ − t , and obtain h c , t i gr + αrt Q ( F ( t ) − k/g ) = − αkrgt Q + 1 r (cid:16) X i c ′ i y i g + αη i y i t Q (cid:1) , where c ′ ... c ′ d = γ ⊺ c . Let y := γ − t . Then γ is a bijection between n t ∈ K d ∞ : | t − t | < b R o and n y ∈ K d ∞ : | y − y | < b R o . Hence, I g,r ( c ) = b Q | r | R T ψ ( − αkrgt Q ) I g,r ( α, c ) dα, where I g,r ( α, c ) := d Y i =1 Z | y i − y i | < b R ψ (cid:18) r (cid:18) c ′ i y i g + αη i y i t Q (cid:19)(cid:19) dy i , where y ... y d = y . We write z i := y i − y i . We have I g,r ( α, c ) := d Y i =1 Z | z i | < b R ψ (cid:18) r (cid:18) c ′ i ( z i + y i ) g + αη i ( z i + y i ) t Q (cid:19)(cid:19) dz i , The phase function has a critical point at − c ′ i t Q gη i α − y i . This critical point is inside the domain ofthe integral, if | κ i | < b R, where κ i := c ′ i t Q gη i α + 2 y i . Note that κ i is a function of α. Given α ∈ T , wepartition the indices into: CR := n ≤ i ≤ d : | κ i | < b R o ,N CR := n ≤ i ≤ d : | κ i | ≥ b R o . For i ∈ N CR, we change the variables to v i = z i + κ − i z i . It is easy to check that this change ofvariables belongs to A ∞ ( t < R ) . For i ∈ CR , we change the variables to w i = z i + κ i / . Hence, I g,r ( α, c ) = Y i ∈ NCR ψ (cid:18) r (cid:18) c ′ i y i g + αη i y i t Q (cid:19)(cid:19) Z | v i | < b R ψ (cid:16) αη i rt Q κ i v i (cid:17) dv i × Y i ∈ CR ψ ( − t Q c ′ i rg η i α ) Z | w i | < b R ψ (cid:16) αη i rt Q w i (cid:17) dw i . (23)By Lemma 2.2 and Lemma 4.6, we have(24) Z | v i | < b R ψ (cid:16) αη i rt Q κ i v i (cid:17) dv i = ( b R, if | αη i rt Q κ i | < / b R , , otherwise, Z | w i | < b R ψ (cid:16) αη i rt Q w i (cid:17) dw i = b R G (cid:18) αη i t R rt Q (cid:19) . Suppose that c ′ / ∈ Ω ∗ . By Lemma 5.2, max ≤ i ≤ d | κ i | ≥ | y | / b ω ≥ b R. On the other hand, recall that κ := max i | c i g | . Since c ′ ... c ′ d = γ ⊺ c and γ ∈ GL d ( O ∞ ) , κ = max i | c i g | = max i | c ′ i g | . By Lemma 5.2,max ≤ i ≤ d (cid:12)(cid:12)(cid:12) κ i αη i t Q (cid:12)(cid:12)(cid:12) = max ≤ i ≤ d (cid:18) c ′ i g + 2 αη i y i t Q (cid:19) ≥ κ/ b ω By our assumption, κ ≥ η | r | b R . Since η > b ω, max ≤ i ≤ d | αη i rt Q κ i | ≥ / b R. By equations (23) and (24), wehave I g,r ( c ) = 0 for c ′ / ∈ Ω ∗ . Next, we suppose that c ′ ∈ Ω ∗ and prove inequality (22). By equations (23) and (24), I g,r ( c ) = 0unless | α | = b l, where b l := κ b Q | A t | . Note that | α | = b l ≫ κ. By equations (23) and (24), we have I g,r ( α, c ) = b R d d Y i =1 δ b R ≤ κ i < | r | b Q b R | αηi | ψ (cid:18) r (cid:18) c ′ i y i g + αη i y i t Q (cid:19)(cid:19) + δ κ i < b R ψ ( − t Q c ′ i rg η i α ) G (cid:18) αη i t R rt Q (cid:19)! . (25) ECTIONS OF QUADRICS OVER A F q The contribution of the first term on the right hand side is zero unless b R ≤ | r | b Q b R | αη i | , which implies | α | ≤ | r | b R b Q b R | η i | ! ≪ | r | b R .
By comparing the preceding inequality with α ≫ κ , we have κ ≪ | r | b R . By choosing η large enough,this contradicts with our assumption κ ≥ η | r | b R . Therefore, for large enough ηI g,r ( c ) = b Q b R d | r | Z | α | = b l ψ ( − αkrgt Q ) Y κ i < | α | ψ ( − t Q c ′ i rg η i α ) G (cid:18) αη i t R rt Q (cid:19) dα. By (16), we have Y i G (cid:18) αη i t R rt Q (cid:19) = ± ε vα Y i min , b l b R | η i || r | b Q ! − / , where v = 0 , η i and α and quadratic residue of their topcoefficients. Hence, I g,r ( c ) = b Q b R d | r | X κ< b l< ± Y i min , b l b R | η i || r | b Q ! − / Z | α | = b l ψ ( − αkrgt Q ) ψ ( − t Q F ∗ ( c )4 rg α ) ε vα dα, where F ∗ ( c ) = P i c ′ i η i . By Lemma 5.8, we have Z | α | = b l ψ ( − αkrgt Q ) ψ ( − t Q F ∗ ( c )4 rg α ) ε vα dα = ( | rgt Q k | B ∞ ( ψ, l + deg( krgt Q ) , kF ∗ ( c )4 r g ) for v = 0 , | rgt Q k | ˜ B ∞ ( ψ, l + deg( krgt Q ) , kF ∗ ( c )4 r g ) for v = 1 , = | rgt Q k | Kl ∞ ( ψ, kF ∗ ( c )4 r g ) if 2 l = deg( t Q F ∗ ( c ) kg ) , and v = 0 | rgt Q k | Sa ∞ ( ψ, kF ∗ ( c )4 r g ) if 2 l = deg( t Q F ∗ ( c ) kg ) , and v = 10 otherwise.Therefore, by using the Weil bound on the Kloosterman sums (Sali´e sums), we have | I g,r ( c ) | ≪ b Q b R d | r | | F ∗ ( c ) | / b R | f | / | r | ! − d/ (cid:12)(cid:12)(cid:12)(cid:12) rgt Q k (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) f F ∗ ( c ) r g (cid:12)(cid:12)(cid:12)(cid:12) / ≪ b Q d (cid:16) | F ∗ ( c ) | / b Q | gr | (cid:17) − d − , where we used | f | / ≫ b Q | g | . Since | c | ≪ | F ∗ ( c ) | / for c ∈ Ω ∗ , this concludes Proposition 5.7. (cid:3) Main contribution to counting function
In this section, we study the main contribution to the counting function N ( w, λ ). We first beginby estimating the contribution in N ( w, λ ) from the terms where c = 0. In order to do so, wefirst prove the following lemma which gives an estimate on the the norm of I g,r ( ) for | r | not toolarge. We then show that the contribution from the other terms is small. Finally, we show thatcontribution from 0 can be written in terms of local densities. Lemma 6.1.
Suppose ε > . With the notation as before and for ≤ | r | ≤ b Q − ε , we have I g,r ( ) = C F b Q d for some non-negative constant C F and sufficiently large (depending only on ε ) b Q . C F > if thesystem of equations has a solution in K ∞ . Proof.
It follows from equation 18 that I g,r ( ) = b Q | r | Z | t − t | < b R | G ( t ) | < b Q | r | d t = b Q | r | Z | g t + λ − x |≤| t − α f | / | F ( g t + λ ) − f | < b Q | r || g | d t . Making the substitution x = g t + λ gives us the equality I g,r ( ) = b Q | r || g | d Z | x − x |≤| t − α f | / : | F ( x ) − f | < b Q | r || g | d x . Let f = α f u , where α f ∈ { , ν, t, νt } is the quadratic residue of f . Furthermore, by Lemma 2.2and Fubini, we may rewrite this as I g,r ( ) = b Q | r || g | d Z | x − x |≤| t − α f | / Z T ψ (cid:18) F ( x ) − frg t Q α (cid:19) dαd x = b Q | r || g | d Z T Z | x | < b D ψ (cid:18) F ( x + x ) − frg t Q α (cid:19) d x dα = b Q b D d | r || g | d Z T Z T d ψ (cid:18) F ( x + ( t − E /u ) x ) − f / ( t E u ) rg t Q − E /u α (cid:19) d x dα = b Q b D d | r || g | d Z T Z T d ψ (cid:18) F ( x + ( t − E /u ) x ) − α f / ( t E ) rg t Q − E /u α (cid:19) d x dα where D := E + deg u, E := ⌈ ( − α + deg α f + 1) ⌉ and the last equality follows from scaling the x coordinate by a factor of b D . Making the substitution β = αrg t Q − E /u , we obtain the equality I g,r ( ) = b Q b D d | g | d − c D Z | β | < d D b Q | r || g | Z T d ψ (cid:0) ( F ( x + ( t − E /u ) x ) − α f /t E ) β (cid:1) d x dβ. Note that the integral is equal to c D b Q | r || g | vol ( x ∈ T d : | F ( x + ( t − E /u ) x ) − α f /t E | ≤ b Q | r || g | c D )! ≥ . Consequently, the first integral is a non-negative real number and can be viewed as a density. Notethat x = 0 is a zero of F ( x + ( t − E /u ) x ) − α f /t E . Also c D b Q | r || g | ≫ b Q ε . The next lemma also showsthat since D > Q , the shift by ( t − E /u ) x is also not important. Consequently, by Lemma 6.2proved next, we can choose b Q large enough (depending on ε and the F ) to ensure that b Q ε is largeenough so that the integral corresponds to taking integrals for | β | in a ball of radius larger thanthe threshold after which it is stable. The conclusion follows. (cid:3) We prove the following lemma that was used in the proof of the previous lemma.
Lemma 6.2.
Let L be an integer, α ∈ T , and let Q be a polynomial over K ∞ such that Q ( x ) − α is nonsingular and Q ( ) = α . Consider Z T d Z | β |≤ b L ψ (( Q ( x ) − α ) β ) dβd x . The limit of this as L → ∞ exists and is a strictly positive number σ ∞ > . Moreover, the integralsstabilize when L is sufficiently large.Proof. As in the computation in the proof of the previous lemma, we have the equality Z T d Z | β |≤ b L ψ (( Q ( x ) − α ) β ) dβd x = b L vol (cid:16)n x ∈ T d : | Q ( x ) − α | ≤ b L − o(cid:17) . ECTIONS OF QUADRICS OVER A F q Note that vol( t − L T ) = b L − . Each x ∈ T such that | Q ( x ) − α | ≤ b L − gives us a coset x + t − L T d ofsolutions in Q − ( t − L T ). Hence, using vol( t − L T d ) = b L − d , we havevol( Q − ( t − L T )) = b L − d |{ x + t − L T d ∈ T d /t − L T d : | Q ( x ) − α | ≤ b L − }| . Therefore, b L vol (cid:16)n x ∈ T d : | Q ( x ) − α | ≤ b L − o(cid:17) = b L − d +1 |{ x + t − L T d ∈ T d /t − L T d : | Q ( x ) − α | ≤ b L − }| = |{ x + t − L T d ∈ T d /t − L T d : | Q ( x ) − α | ≤ b L − }| b L d − . By Hensel’s Lemma, for large enough L , this latter quantity stabilizes. Since there is a solution in T to the equation Q ( x ) = α , namely , the above quantity is strictly positive as well. The conclusionfollows. (cid:3) We now show that when b Q − ε ≤ | r | ≤ b Q , then the contribution of the terms in N ( w, λ ) when c = and corresponding to such r is small. This follows from the following more general statement forall c . Lemma 6.3. X b Q − ε ≤| r |≤ b Q | gr | − d | S g,r ( c ) || I g,r ( c ) | ≪ ε, ∆ | g | ε b Q d +32 + ε Proof.
Suppose b Q − ε ≤ | r | ≤ b Q . It is easy to see from the definition of I g,r ( c ) that for such r , | I g,r ( c ) | ≪ b Q d + ε . Using this, we obtain X b Q − ε ≤| r |≤ b Q | gr | − d | S g,r ( c ) || I g,r ( c ) | = X b Q − ε ≤| r |≤ b Q | r | − d − | g | − d | r | − d +12 | S g,r ( c ) || I g,r ( c ) |≤ b Q d + ε X (1 − ε ) Q For d ≥ and every c , the sum X r | r | − d S g,r ( c ) is absolutely convergent. Proof. Using Lemmas 3.3 and 3.6, we obtain | r | − d | S g,r ( c ) | ≪ ∆ τ ( r ) | r | − d | r | d +12 | r | d/ | gcd( r , f ) | / = τ ( r ) | r | − d/ | gcd( r , f ) | / | r | / ≤ | r | − d/ ε | f | / | r | / . Hence, X | r |≤ b X | r | − d | S g,r ( c ) | ≪ ∆ ,g,f,ε X ≤ N ≤ X b N − d/ ǫ X | r |≤ b N | r | / ≪ ∆ ,g,f,ε X ≤ N ≤ X b N − d/ / ε . The last summation is a partial sum of a geometric series, and so the associated infinite sum isconvergent since d ≥ (cid:3) Lemma 6.5. For any ε > , we have X r :1 ≤| r |≤ b T | r | − d S g,r ( ) = X r | r | − d S g,r ( ) + O ε, ∆ ( | g | d b T / − d + ε ) . Proof. Write X r | r | − d S g,r ( ) = X r :1 ≤| r |≤ b T | r | − d S g,r ( ) + X | r | > b T | r | − d S g,r ( ) . The triangle inequality gives us (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | r | > b T | r | − d S g,r ( ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X b N = b T b N − d X | r | = b N | S g,r ( ) | . From S g,r ( ) = S S and Lemmas 3.3 and 3.6, we have | S g,r ( ) | ≪ ∆ | g | d τ ( r ) | r | d/ | r | / | r || gcd( r , f ) | / , using which we obtain ∞ X N = T b N − d X | r | = b N | S g,r ( ) | ≪ ∆ | g | d ∞ X N = T b N − d/ X | r | = b N τ ( r ) | r | / | r || gcd( r , f ) | / ≤ | g | d ∞ X N = T b N − d X | r |≤ b N τ ( r ) | r | − / | gcd( r , f ) | / ≤ | g | d ∞ X N = T b N / − d + ε X | r |≤ b N | gcd( r , f ) | / | r | = | g | d ∞ X N = T b N / − d +2 ε = O ε,F ( | g | d b T / − d + ε ) , where we have used that d ≥ 4. Using this, we obtain that X ≤| r |≤ b T | r | − d S g,r ( ) = X r | r | − d S g,r ( ) + O ε, ∆ ( | g | d b T / − d + ε ) . From Lemma 6.4, the infinite sum is absolutely convergent. The conclusion follows. (cid:3) ECTIONS OF QUADRICS OVER A F q We now want to show that the infinite sum X r | r | − d S g,r ( )can be entirely written in terms of number theoretic information. Lemma 6.6. Suppose that all local conditions are satisfied. Then X r | gr | − d S g,r ( ) = Y ̟ σ ̟ ≫ F | f | − ε , where ̟ ranges over the monic irreducible polynomials in F q [ t ] , and σ ̟ := lim k →∞ | (cid:8) x mod ̟ k + ν ̟ ( g ) : F ( x ) ≡ f mod ̟ k + ν ̟ ( g ) , x ≡ λ mod ̟ ν ̟ ( g ) (cid:9) || ̟ | ( d − k , and is strictly positive.Proof. Define for each N ≥ N )! := Y | f |≤ b Nf monic f. Write X r | ( N )! | r | − d S g,r ( )= X r | ( N )! | r | − d X a mod gr ( a,r )=1 X b ∈O d / ( gr ) ψ (cid:18) a (2 λ T A b − k + gF ( b )) gr (cid:19) = 1 | ( N )! | d X r | ( N )! X a mod gr ( a,r )=1 (cid:12)(cid:12)(cid:12)(cid:12) ( N )! r (cid:12)(cid:12)(cid:12)(cid:12) d X b ∈O d / ( gr ) ψ (cid:18) a (2 λ T A b − k + gF ( b )) gr (cid:19) = 1 | ( N )! | d X r | ( N )! X a mod gr ( a,r )=1 X b ∈O d / ( g ( N )!) ψ (cid:18) a (2 λ T A b − k + gF ( b )) gr (cid:19) = 1 | ( N )! | d X b ∈O d / ( g ( N )!) X r | ( N )! X a mod gr ( a,r )=1 ψ (cid:18) a (2 λ T A b − k + gF ( b )) gr (cid:19) . Since X r | ( N )! X a mod gr ( a,r )=1 = X a mod g ( N )! , X r | ( N )! | r | − d S g,r ( ) = 1 | ( N )! | d X a mod g ( N )! X b ∈O d / ( g ( N )!) ψ (cid:18) a (2 λ T A b − k + gF ( b )) g ( N )! (cid:19) . Furthermore, this latter quantity is equal to | g | (cid:12)(cid:12) { b ∈ O d / ( g ( N )!) : 2 λ T A b − k + gF ( b ) ≡ g ( N )! } (cid:12)(cid:12) | ( N )! | d − . Let us write ( N )! = ̟ a . . . ̟ a ℓ ℓ . Then2 λ T A b − k + gF ( b ) mod g ( N )! is the same as having F ( g b + λ ) − f ≡ mod ̟ a i +2 ν ̟i ( g ) i for each i = 1 , . . . , ℓ . We conclude that X r | ( N )! | r | − d S g,r ( )= | g | d Y ̟ | ( N )! (cid:12)(cid:12) { b ∈ O d / ( ̟ ν ̟ (( N )!)+ ν ̟ ( g ) ) : F ( ̟ ν ̟ ( g ) b + λ ) ≡ f mod ̟ ν ̟ (( N )!)+2 ν ̟ ( g ) } (cid:12)(cid:12) | ̟ ν ̟ (( N )!)+ ν ̟ ( g ) | d − = | g | d Y ̟ | ( N )! (cid:12)(cid:12) { x ∈ O d / ( ̟ ν ̟ (( N )!)+ ν ̟ ( g ) ) : F ( x ) ≡ f mod ̟ ν ̟ (( N )!)+ ν ̟ ( g ) , x ≡ λ mod ̟ ν ̟ ( g ) } (cid:12)(cid:12) | ̟ ν ̟ (( N )!)+ ν ̟ ( g ) | d − . We know that the infinite sum is absolutely convergent. Letting N → ∞ gives us X r | gr | − d S g,r ( ) = Y ̟ σ ̟ , where σ ̟ are as in the statement of the lemma. Suppose that gcd( ̟, g ∆) = 1 . We have σ ̟ = X k ≥ | ̟ | − kd S g,̟ k ( ) = 1 + X k ≥ | ̟ | − kd S g,̟ k ( ) . By Lemma 3.3 | ̟ | − kd S g,̟ k ( ) ≤ | ̟ | − k d − ( k + 1) | gcd( ̟ k , f ) | / Hence, σ ̟ = ( O (1 / | ̟ | ) = 0 , if ̟ | f, O (1 / | ̟ | ) = 0 , otherwise.This implies that | f | ε ≫ Y gcd( ̟,g ∆)=1 σ ̟ ≫ | f | − ε . Suppose that ̟ | g . Since, gcd(∆ f, g ) = 1, by Hensel’s lemma σ ̟ = 1 . Finally, suppose that ̟ | ∆. Then by Hensel’s lemma1 ≫ ∆ Y ̟ | ∆ σ ̟ ≫ ∆ . This concludes the proof of our lemma. (cid:3) Proof of the main theorem In this section, we prove our main theorem. Though we obtain a theorem for d ≥ 4, it is onlyoptimal when d ≥ 5. We assume that we have a non-degenerate quadratic form over F q [ t ] in d ≥ d ≥ d = 4 variables. Wefirst give a bound on the contributions of the nonzero exceptional vectors to our counting function. ECTIONS OF QUADRICS OVER A F q Proposition 7.1. For any non-degenerate quadratic form F over F q [ t ] in d ≥ variables, and forany ε > , we have X ≤| r |≤ b Q exc X c =0 | gr | − d | S g,r ( c ) || I g,r ( c ) | ≪ ε b Q d +32 + ε | g | d − + ε (1 + | g | − d − + ε ) , where P exc denotes summation over exceptional vectors. We prove this proposition by rewriting X ≤| r |≤ b Q exc X c =0 | gr | − d S g,r ( c ) I g,r ( c ) = E + E , where E := exc X c =0 X ≤| r |≤ b R | c | η | g | | gr | − d S g,r ( c ) I g,r ( c )and E := exc X c =0 X b R | c | η | g | < | r |≤ b Q | gr | − d S g,r ( c ) I g,r ( c ) , and then showing that E and E satisfy the above bound. This division of the sum into two partsis suggested by Proposition 5.7. Lemma 7.2. | E | ≪ ε,F, Ω b Q d +32 + ε | g | d − + ε (1 + | g | − d − + ε ) . Proof. By Proposition 5.7, we know that for | r | ≤ b R | c | η | g | | I g,r ( c ) | ≪ F, Ω b Q d b Q | c || gr | ! − d − . Using this, we obtain | E | ≪ b Q d exc X c =0 X ≤| r |≤ b R | c | η | g | | gr | − d | S g,r ( c ) | b Q | c || gr | ! − d − = b Q d +12 exc X c =0 (cid:18) | c || g | (cid:19) − d − X ≤| r |≤ b R | c | η | g | | g | − d | r | − d +12 | S g,r ( c ) | . By Proposition 3.1, X ≤| r |≤ b R | c | η | g | | g | − d | r | − d +12 | S g,r ( c ) | ≪ ε,F, Ω | g | ε b R | c | η | g | ! ε ≪ ε,F, Ω | g | ε b Q | c || g | ! ε . Here, we are also using the fact that R and Q are of the same order up to a constant dependingon the quadratic form and Ω. Consequently, | E | ≪ b Q d +12 exc X c =0 (cid:18) | c || g | (cid:19) − d − | g | ε b Q | c || g | ! ε = b Q d +32 + ε | g | d − + ε exc X c =0 | c | − d − . Note that the exceptional vectors c are all congruent to αA λ modulo g for some varying polynomial α . By assumption, at least one coordinate of λ is relatively prime to g , say the first one. Sinceevery exceptional c is congruent to αA λ mod g for some α depending on c , the first coordinatevaries through all polynomials modulo g as c and so as α varies. Furthermore, since c is exceptional, | c | ≤ O F, Ω (1) | g | . Consequently, exc X c =0 | c | − d − + ε ≪ ε,F, Ω X = | α | < | g | | α | − d − + ε ≪ ε,F, Ω | g | − d − + ε , from which we obtain | E | ≪ ε,F, Ω b Q d +32 + ε | g | d − + ε (1 + | g | − d − + ε ) . (cid:3) Similarly, we have the same bound on E . Lemma 7.3. | E | ≪ ε,F, Ω b Q d +32 + ε | g | d − + ε (1 + | g | − d − + ε ) . Proof. In this case, | r | > b R | c | η | g | for which we have the trivial bound | I g,r ( c ) | ≪ ε,F, Ω b Q d + ε . Using this, we obtain | E | ≪ ε,F, Ω b Q d + ε exc X c =0 X b R | c | η | g | < | r |≤ b Q | gr | − d | S g,r ( c ) | = b Q d + ε exc X c =0 X b R | c | η | g | < | r |≤ b Q | r | − d − | g | − d | r | − d +12 | S g,r ( c ) | = b Q d + ε exc X c =0 Q X k =1+log q b R | c | η | g | (cid:16) q k (cid:17) − d − X | r | = q k | g | − d | r | − d +12 | S g,r ( c ) | . By Proposition 3.1, for each k , X | r | = q k | g | − d | r | − d +12 | S g,r ( c ) | ≪ ε,F | g | ε ( q k ) ε . Hence | E | ≪ ε,F, Ω b Q d + ε | g | ε exc X c =0 Q X k =1+log q b R | c | η | g | (cid:16) q k (cid:17) − d − + ε ≪ ε,F, Ω b Q d + ε | g | ε exc X c =0 b Q | c || g | ! − d − + ε = b Q d +32 + ε | g | d − + ε exc X c =0 | c | − d − + ε . As before, exc X c =0 | c | − d − + ε ≪ ε,F, Ω X = | α | < | g | | α | − d − + ε ≪ ε,F, Ω | g | − d − + ε , ECTIONS OF QUADRICS OVER A F q from which the conclusion follows. (cid:3) We are now ready to prove our main theorem. Note that from remark 3 this is optimal for d ≥ Proof of the main theorem 1.2. Recall that(26) N ( w, λ ) = 1 | g | b Q X r ∈O| r |≤ b Qr monic X c ∈O d | gr | − d S g,r ( c ) I g,r ( c ) . By Lemma 5.6, Lemma 6.3, and Proposition 7.1, we have N ( w, λ ) = 1 | g | b Q X r ∈O| r |≤ b Q − ε r monic | gr | − d S g,r ( ) I g,r ( ) + O ε,F, Ω b Q d +32 + ε | g | d − + ε (1 + | g | − d − + ε ) | g | b Q ! = 1 | g | b Q X r ∈O| r |≤ b Q − ε r monic | gr | − d S g,r ( ) I g,r ( ) + O ε,F, Ω (cid:16) b Q d − + ε | g | d − + ε (1 + | g | − d − + ε ) (cid:17) . By Lemma 6.1, I g,r ( ) = C F b Q d for some constant C F > b Q sufficiently large depending on ε and F . Hence for such b Q ,1 | g | b Q X r ∈O| r |≤ b Q − ε r monic | gr | − d S g,r ( ) I g,r ( ) = C b Q d − | g | X r ∈O| r |≤ b Q − ε r monic | gr | − d S g,r ( ) . On the other hand, by Lemma 6.5 and Lemma 6.6, X r ∈O| r |≤ b Q − ε r monic | gr | − d S g,r ( ) = Y ̟ σ ̟ + O (cid:16) b Q − d − + ε (cid:17) . As a result, we finally obtain N ( w, λ ) = C b Q d − | g | Y ̟ σ ̟ + O (cid:16) b Q − d − + ε (cid:17)! + O ε,F, Ω (cid:16) b Q d − + ε | g | d − + ε (1 + | g | − d − + ε ) (cid:17) = C b Q d − | g | Y ̟ σ ̟ + O ε,F, Ω (cid:16) b Q d − + ε | g | d − + ε (1 + | g | − d − + ε ) (cid:17) = C b Q d − | g | Y ̟ σ ̟ O ε,F, Ω | f | ε b Q d − + ε | g | d − + ε (1 + | g | − d − + ε ) b Q d − !! = C b Q d − | g | Y ̟ σ ̟ O ε,F, Ω | g | d − ε (1 + | g | − d − + ε ) | f | d − − ε !! = C b Q d − | g | Y ̟ σ ̟ O ε,F, Ω (1 + | g | − d − + ε ) (cid:18) | g | ε | f | (cid:19) d − !! . Therefore, if d ≥ 5, we can take | f | ≫ | g | ε , while if d = 4, we can take | f | ≫ | g | ε . Note that inthe third equality, we have also used Lemma 6.6 ensuring us that the product of the local densitiesis ≫ | f | − ε . (cid:3) Acknowledgment. N.T. 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