aa r X i v : . [ m a t h . N T ] M a r UNIFORMLY COUNTING RATIONAL POINTS ONCONICS
EFTHYMIOS SOFOS
Abstract.
We provide an asymptotic estimate for the number of ra-tional points of bounded height on a non–singular conic over Q . Theestimate is uniform in the coefficients of the underlying quadratic form. Introduction
Let Q ( x ) ∈ Z [ x , x , x ] be a non–singular quadratic form. We denote by Z the integer vectors x that are primitive, i.e. that satisfy gcd( x ) = 1 . Our main concern in this paper regards the number of primitive integerzeros of Q contained on an expanding region of R . It is therefore only thecase that Q is isotropic that we are interested in and we will proceed underthis assumption for the rest of the paper.For any arbitrary norm k . k : R → R ≥ define the counting function N ( Q, B ) := { x ∈ Z : Q ( x ) = 0 , k x k ≤ B } . A very special case of the work [7] establishes the asymptotic formula N ( Q, B ) ∼ c Q B, valid for B → ∞ . This confirms the Manin conjecture and furthermore c Q = c Q ( k . k ) is the constant predicted in [10].Let h Q i denote the maximum modulus of the coefficients of Q. As pointedout in [2], one expects the existence of absolute constants β, γ > N ( Q, B ) = c Q B + O (cid:16) B − γ h Q i β (cid:17) . Our aim is to establish such an estimate and furthermore to state explicitlyadmissible values for β and γ. We begin by recalling existing results related to this subject. Let w : R → R ≥ be a smooth weight function of compact support and let N w ( Q, B ) := X x ∈ Z Q ( x )=0 w (cid:0) B − x (cid:1) . Mathematics Subject Classification.
Primary 11D45; Secondary 14G05.
Key words and phrases. rational points, quadratic forms.
It is proved in [8, Cor.2] that there exists a positive constant c such thatone has N w ( Q, B ) = c Q,w B + O Q,w (cid:16) B exp {− c p log B } (cid:17) , as B → ∞ . The proof is carried out via a modification of the circle method.Let ∆ Q and δ Q be the discriminant and the greatest common divisor ofthe 2 × Q respectively. In [1, Cor. 2], itis proved that N ( Q, B ) ≪ τ ( | ∆ Q | ) Bδ Q / | ∆ Q | / ! , where τ denotes the divisor function. It should be stressed that the impliedconstant is absolute.We provide the definition of the leading constant c Q before stating ourmain result. We define the Hardy–Littlewood local densities following [8].Let(1.1) σ ∞ := σ ∞ ( Q, k . k ) = lim ǫ → ǫ Z | Q ( x ) |≤ ǫ k x k≤ x , and similarly for any prime p, let(1.2) σ p := σ p ( Q ) = lim n →∞ p n N ∗ Q ( p n ) , where for any positive integer n,N ∗ Q ( p n ) := { x (mod p n ) : p ∤ x , Q ( x ) ≡ p n ) } . The Peyre constant is then defined as c Q = 12 σ ∞ Y p σ p where the product is taken over the set of primes and is convergent. Let C ⊆ P be the smooth projective curve defined by Q. The existence of thefactor is due to the fact that the anticanonical line bundle is twice thegenerator of the Picard group Pic ( C ) ∼ = Z , where α ( C ) is the volume of acertain polytope contained in the cone of effective divisors.Next, let(1.3) K := 1 + sup x = k x k ∞ k x k , and notice that K is a constant depending only on the choice of norm k . k . A norm k . k : R → R ≥ is called isometric to the supremum norm k . k ∞ when there exists an invertible matrix g ∈ GL ( R ) such that k x k = k g x k ∞ for all x ∈ R . We have the following result.
NIFORMLY COUNTING RATIONAL POINTS ON CONICS 3
Main Theorem 1.1.
Let Q be a ternary non–singular integer quadraticform with a rational zero and let k . k be any norm isometric to the maximumnorm. Then N ( Q, B ) = c Q B + O (cid:16) ( BK ) (log BK ) h Q i (cid:17) , for B ≥ . The implied constant in the estimate is absolute.
The proof of Theorem 1.1 reveals that for any ǫ >
0, at the expenseof an implied constant that depends on ǫ, one can replace the term h Q i appearing in the error term by h Q i + ǫ as well as h Q i ǫ δ Q (see (6.2)).Further improvements may follow using [9, Theorem 1]. We hope it willbe apparent to the reader that the main value of Theorem 1.1 lies in itsgenerality rather than the exponent of h Q i obtained.The proof is conducted in two stages. Firstly, in § §
5, we prove Theo-rem 1.1 for conics of a special shape, using the fact that since C ( Q ) = ∅ , thereis a morphism P → C. The conditions involving the resulting parametris-ing functions lead to a lattice counting problem. One should comment thatthe choice of the parametrising functions is not unique and that choosingthem appropriately plays a significant rˆole. An amount of work regardingthis issue has taken place, as the papers [4] and [11] reveal. The second stageis performed in §
6. Here we apply a unimodular transformation to a conicof general shape to transform the problem into the one we have alreadytreated.
Notation.
The implied constants in the O ( . ) notation will be absolutethroughout this paper, except where specifically indicated, via the use of asubscript. The norm notation k . k will be reserved for norms of elements of R while k . k ∞ will be used for the matrix supremum norm in R × , definedby k ( a i,j ) ≤ i,j ≤ k ∞ := max ≤ i,j ≤ | a i,j | , as well as the supremum norm of R . We denote the generalised divisor function by τ k ( n ) , which is defined to bethe number of representations of n as the product of k natural numbers.The well–known bound τ k ( n ) ≪ k,ǫ n ǫ , valid for each ǫ > , shall be used.By P *( s,t ) (mod n ) , we shall mean a summation for s, t ∈ [1 , n ] , subject tothe condition gcd( s, t, n ) = 1 . Preliminary estimates
Throughout § §
5, we denote by Q the quadratic forms of which (0 , , Q ( x ) = ax + bxy + dxz + eyz + f z , E.SOFOS where a, . . . , f ∈ Z . We will denote by ∆ Q its discriminant,∆ Q = ae − deb + f b . It is our intention in the aforementioned sections to prove the following spe-cial version of Theorem 1.1. Its proof hinges upon the classical parametri-sation of a conic by the lines going through a given point.
Proposition 2.1.
Let Q be a non–singular integer ternary quadratic formas above. Then for any norm isometric to the maximum norm and for any ǫ > , one has N ( Q, B ) = c Q B + O ǫ (cid:16) ( BK ) log ( BK )min n | ∆ Q | , δ Q o ( | ∆ Q | + h Q i ) h Q i ǫ (cid:17) , for B ≥ . Let Π be the matrix Π := b e − a − d − f b e and define the three binary quadratic forms q , q , q such that(2.1) q ( s, t ) = Π s stt where q = ( q , q , q ) T . One can verify that Det (Π) = ∆ Q and that inparticular the matrix Π is invertible. Hence one gets(2.2) adj (Π) q ( s, t ) = ∆ Q s stt . Notice that for(2.3) g ( s, t ) := as + dst + f t ,L ( s, t ) := bs + et, one has(2.4) q ( s, t ) = sL ( s, t ) ,q ( s, t ) = − g ( s, t ) ,q ( s, t ) = tL ( s, t ) . For each integer n, let(2.5) ρ ∗ ( n ) := { ( s, t ) ∈ [0 , n ) : n | q ( s, t ) , gcd( s, t, n ) = 1 } , and note that ρ ∗ is a multiplicative function. Equations (2.4) imply thatthis expression equals ρ ∗ ( n ) = { ( s, t ) ∈ [0 , n ) : n | ( L ( s, t ) , g ( s, t )) , gcd( s, t, n ) = 1 } . NIFORMLY COUNTING RATIONAL POINTS ON CONICS 5
Lemma 2.2. (i)
The function ρ ∗ is supported on the divisors of ∆ Q gcd( b,e ) . (ii) For all integers n we have ρ ∗ ( n ) ≤ n gcd( b, e ) . Proof. (i) It suffices to show that for each prime p and integer ν ≥ ρ ∗ ( p ν ) = 0 we have that ν + min { v p ( b ) , v p ( e ) } ≤ v p (∆ Q ) . Let ( s, t ) be counted by ρ ∗ ( p ν ). We may assume without loss of generalitythat v p ( b ) ≤ v p ( e ) . Since gcd( b, e ) | ∆ Q our claim in the case ν ≤ v p ( b ) istrivial. If ν > v p ( b ) then we may write b = p v p ( b ) b ′ , e = p v p ( e ) e ′ with p ∤ b ′ e ′ . Plugging these values in the congruence L ( s, t ) ≡ p ν ) yields(2.6) b ′ s ≡ − p v p ( e ) − v p ( b ) e ′ t (cid:0) mod p ν − v p ( b ) (cid:1) and hence p ∤ t since otherwise we would have p | ( s, t ) which would contradictthe definition of ρ ∗ ( p n ). We deduce that t (cid:16) ae p − v p ( b ) − deb ′ p − v p ( b ) + f b ′ (cid:17) ≡ b ′ g ( s, t ) ≡ (cid:0) mod p ν − v p ( b ) (cid:1) and therefore p ν + v p ( b ) | ae − deb + f b = ∆ Q which concludes the proof ofthe first part.(ii) It suffices to prove that for all primes p and integers ν ≥ ρ ∗ ( p ν ) p ν ≤ p min { v p ( b ) ,v p ( e ) } . Let ( s, t ) be counted by ρ ∗ ( p ν ) . We may assume as previously that we have v p ( b ) ≤ v p ( e ). In the case that ν ≤ v p ( b ), then (2.7) is a consequence of thetrivial bound ρ ∗ ( p ν ) ≤ p ν . In the opposite case we proceed as in the proofof part (i). Then equation (2.6) shows that the value of s/t (cid:0) mod p ν − v p ( b ) (cid:1) is uniquely determined and can be lifted to at most p v p ( b ) values (mod p ν ) , which proves (2.7) in all cases. (cid:3) We record a generalisation of M¨obius inversion that will be used later.
Lemma 2.3.
Let A be a finite subset of Z and n a fixed integer. Then { ( s, t ) ∈ A : gcd( s, t ) = 1 } = ∞ X m =1gcd( m,n )=1 µ ( m ) (cid:26) ( s, t ) ∈ A : gcd( s, t, n ) = 1 ,m | s, m | t (cid:27) . E.SOFOS
Proof.
Define A : Z → { , } as the indicator function of A . M¨obiusinversion gives X gcd( s,t,n )=1gcd( s,t )=1 A ( s, t ) = ∞ X m =1 µ ( m ) X gcd( s,t,n )=1 m | s,m | t A ( s, t ) . Our assertion is proved upon noticing that only m coprime to n are takeninto account in the summation. (cid:3) Parametrisation of the conic
In this section, we begin by showing how the problem of counting pointson conics can be rephrased using the parametrisation functions q ( s, t ) . Thiswill lead us to count primitive integer points in regions of R . Let(3.1) N ( Q, B ) := (cid:8) ( s, t ) ∈ Z : t > , k q ( s, t ) k ≤ λB (cid:9) , where λ = gcd( q ( s, t )) ∈ Z . Lemma 3.1.
One has N ( Q, B ) = N ( Q, B ) + O (1) , where the impliedconstant is absolute.Proof. Let C ⊂ P be the curve given by Q = 0 and denote the point (0 , , C by ξ. The tangent line to C through ξ, is given by L ξ := { ez = bx } . Let L be the set of projective lines in P that pass through ξ and L ( Q ) bethe corresponding subset of lines that are defined over Q . Define U ⊂ C asthe open subset formed by deleting ξ from C. Letting U := L \ { L ξ } , wenote that the sets U ( Q ) and U ( Q ) are in bijection.The general element of L ( Q ) is given by L s,t := { sz = tx } for integer pairs ( s, t ) such that gcd( s, t ) = 1 . The condition ( s, t ) = ( b,e )gcd( b,e ) ensures that we have a point in U ( Q ) . One can ignore this, since the contri-bution of such s, t is O (1) . The bijection between lines with t > t < s, t ) with t = 0 is O (1) due to the condition gcd( s, t ) = 1.One can make explicit the bijection between U ( Q ) and U ( Q ) as follows.Recall the definition of L, g in (2.3). A computation reveals that the line L s,t intersects C in the point ( x, y, z ) if and only we have zg ( s, t ) + ytL ( s, t ) = 0 NIFORMLY COUNTING RATIONAL POINTS ON CONICS 7 or z = 0 holds. In the latter case, one gets the point ξ, which is to be ignored.In the former case, we have − g ( s, t ) xt = − g ( s, t ) sz = syL ( s, t ) t, by the equation for L s,t . The primitive integer vectors ( x, y, z ) represent apoint in C ( Q ) if and only if( x, y, z ) = ± ( sL ( s, t ) /λ, − g ( s, t ) /λ, tL ( s, t ) /λ ) , where λ = gcd( sL ( s, t ) , − g ( s, t ) , tL ( s, t )) . Making use of (2.4) concludesthe proof of the lemma. (cid:3)
Let us define for any T ∈ R ≥ and n, σ, τ ∈ N , (3.2) M ∗ σ,τ ( T, n ) := (cid:26) ( s, t ) ∈ Z : ( s, t ) ≡ ( σ, τ ) (mod n ) ,t > , k q ( s, t ) k ≤ T (cid:27) . Lemma 3.2.
One has N ( Q, B ) = X kλ | ∆ Q / gcd( b,e ) µ ( k ) X *( σ,τ ) (mod kλ ) kλ | ( L ( σ,τ ) ,g ( σ,τ )) M ∗ σ,τ ( Bλ, kλ ) . Proof.
Any integer λ that appears in (3.1), satisfies λ | q ( s, t ) for some co-prime integers s, t , so part ( i ) of Lemma 2.2 implies that λ | ∆ Q gcd( b,e ) . Wetherefore get N ( Q, B ) = X λ | ∆ Q / gcd( b,e ) (cid:26) ( s, t ) ∈ Z : λ | q ( s, t ) , gcd( q ( s,t ) λ ) = 1 ,t > , k q ( s, t ) k ≤ Bλ (cid:27) . Using Lemma 2.3 with n = 1 , gives(3.3) N ( Q, B ) = X kλ | ∆ Q / gcd( b,e ) µ ( k ) M ∗ ( Bλ, kλ ) , where for any T ≥ , n ∈ N , we have defined M ∗ ( T, n ) := (cid:26) ( s, t ) ∈ Z : n | q ( s, t ) , t > , k q ( s, t ) k ≤ T (cid:27) . Partitioning into congruence classes (mod n ) yields M ∗ ( T, n ) = X *( σ,τ ) (mod n ) n | ( L ( σ,τ ) ,g ( σ,τ )) M ∗ σ,τ ( T, n ) , which, when used along with (3.3), yields the proof of the lemma. (cid:3) E.SOFOS Counting lattice points
The quantity appearing in (3.2) involves integer points ( s, t ) which areprimitive. We will use M¨obius inversion to deal with this condition. Thiswill lead us to count integer points in a dilated region. In order to do so,one needs certain information regarding this region, which is the purpose ofthe next lemma.Recall the definition (2.1). Denote by V the region(4.1) V := { ( s, t ) ∈ R : t > , k q ( s, t ) k ≤ } . Lemma 4.1. V is bounded and in particular, it is contained in the rectanglegiven by | s | , | t | ≪ h Q i (cid:18) K | ∆ Q | (cid:19) . The length of the boundary of V , denoted by | ∂V | , satisfies | ∂V | ≪ h Q i (cid:18) K | ∆ Q | (cid:19) , where the implied constant is absolute. Furthermore any line parallel to oneof the coordinate axes intersects V in a set of points which, if not empty,consists of at most O (1) intervals, where the implied constant is absolute.Proof. For each ( s, t ) ∈ V, one gets from (2.2) that | s | , | t | ≪ K k adj (Π) k ∞ | ∆ Q | − . Using the estimates k adj (Π) k ∞ ≪ k Π k ≪ h Q i concludes the proof of thefirst assertion. The norm k . k is isometric to the supremum norm and hence V is the intersection of the interior of 3 plane conic sections. Therefore V is a finite union of at most O (1) convex sets, where the implied constantis absolute, thus showing that | ∂V | is bounded by an absolute constantmultiplied with the length of the box that contains V . Our last assertionis a consequence of [6] as the set V is semi–algebraic owing to the the factthat k . k is isometric to the supremum norm. (cid:3) Define for any T ∈ R ≥ and n, σ, τ ∈ N such that gcd( σ, τ, n ) = 1,(4.2) M σ,τ ( T, n ) := (cid:26) ( s, t ) ∈ Z : ( s, t ) ≡ ( σ, τ ) (mod n ) ,t > , k q ( s, t ) k ≤ T (cid:27) . Lemma 4.2.
For any
T, n, σ, τ as above with gcd( σ, τ, n ) = 1 and n | q ( σ, τ ) , one has M ∗ σ,τ ( T, n ) = X ≤ m ≤ (2 T K /n ) gcd( m,n )=1 µ ( m ) M ¯ mσ, ¯ mτ (cid:18) Tm , n (cid:19) , NIFORMLY COUNTING RATIONAL POINTS ON CONICS 9 where ¯ m denotes the inverse of m (mod n ) . Proof.
The condition k q ( s, t ) k ≤ T implies by Lemma 4.1, that the numberof ( s, t ) counted by M ∗ σ,τ ( T, n ) is finite. Therefore Lemma 2.3 may be appliedto yield(4.3) M ∗ σ,τ ( T, n ) = ∞ X m =1gcd( m,n )=1 µ ( m ) M ¯ mσ, ¯ mτ (cid:18) Tm , n (cid:19) . If m > (2 K T /n ) , then each ( s, t ) taken into account by M ¯ mσ, ¯ mτ (cid:0) Tm , n (cid:1) , satisfies k q ( s, t ) k ∞ < n , due to (1.3). The assumptions on σ, τ, n, implythat n | q ( s, t ) which is only possible if q ( s, t ) = . Due to (2.2), one has t = 0 which contradicts the definition of (4.2). This shows that only integers m ≤ (2 K T /n ) make a non–zero contribution to (4.3), which concludesthe proof of the lemma. (cid:3) Recall the definitions (4.1) and (4.2).
Lemma 4.3.
For any
T, n, σ, τ as above, we have M σ,τ ( T, n ) = vol ( V ) Tn + O K T ) n h Q i| ∆ Q | ! . Proof.
The quantity M σ,τ ( T, n ) equals the number of integer points in theregion T n V − (cid:16) σn , τn (cid:17) , where V is defined in (4.1). We thus deduce that M σ,τ ( T, n ) = ♯ (cid:26) Z ∩ V T n (cid:27) + O (cid:18) | ∂V | T n (cid:19) , where | ∂V | denotes the length of the boundary of V. The assumptions ofthe theorem in [5, pg.180] are fulfilled due to Lemma 4.1, thus yielding ♯ (cid:26) Z ∩ V T n (cid:27) = vol ( V ) Tn + O K T ) n h Q i| ∆ Q | ! . This estimate, when combined with the second assertion of Lemma 4.1,finishes the proof. (cid:3) The asymptotic formula
We are now in possession of the required lemmata to show the validityof Proposition 2.1. Before proceeding to the proof we should remark thatwe shall show the asymptotic formula of Proposition 2.1 with a different constant in place of c Q , and at the end of this section we will explain whythe two constants coincide.Let us now define the new constant, which we denote by c ′ Q . Recall thedefinitions (2.5) and (4.1). Let σ ′∞ := vol ( V )and for any prime p, let σ ′ p := (cid:18) − p (cid:19) (cid:16) p (cid:17) X d ≥ ρ ∗ (cid:0) p d (cid:1) p d . Lemma 2.2 shows that the product Q p σ ′ p taken over all primes p convergesand we may thus define c ′ Q := σ ′∞ Y p σ ′ p . Notice that Lemma 4.3 implies that(5.1) σ ′∞ ≪ h Q i K | ∆ Q | . In light of Lemma 3.1, it suffices to prove Proposition 2.1 for N ( Q, B )in place of N ( Q, B ) . Combining Lemma 3.2 and Lemma 4.2, gives(5.2) N ( Q, B ) = X kλ | ∆ Q / gcd( b,e ) µ ( k ) X *( σ,τ ) (mod kλ ) kλ | q ( σ,τ ) X m ≤ (2 BK /k ) gcd( m,kλ )=1 µ ( m ) M ¯ mσ, ¯ mτ (cid:18) Bλm , kλ (cid:19) . Now notice that for L := ( K B ) kλ h Q i| ∆ Q | , the bound (5.1) and Lemma 4.3 imply that M ¯ mσ, ¯ mτ (cid:18) Bλm , kλ (cid:19) = ( σ ′∞ Bm k λ + O (cid:0) L m (cid:1) if m ≤ L O (1) otherwise.The contribution to (5.2) coming from those m with m > L is therefore ≪ ǫ ( BK ) | ∆ Q |h Q i ǫ . We have used the bound τ k ( n ) ≪ k,ǫ n ǫ as well aspart ( ii ) of Lemma 2.2. The contribution of the remaining m is σ ′∞ B X kλ | ∆ Q µ ( k ) ρ ∗ ( kλ ) k λ X m ≤L gcd( m,kλ )=1 µ ( m ) m + O ǫ (cid:16) ( BK ) (log BK ) h Q i ǫ gcd( b, e ) (cid:17) . Extending the summation over m to infinity, the error introduced in themain term is ≪ ǫ ( BK ) h Q i ǫ gcd( b, e ) , where we have made use of NIFORMLY COUNTING RATIONAL POINTS ON CONICS 11 (5.1). The fact that gcd( b, e ) | ∆ Q and gcd( b, e ) | δ Q provides the error termin Proposition 2.1. Using the fact that ρ ∗ is multiplicative and supportedon the divisors of ∆ Q we deduce that X k,λ ∈ N µ ( k ) ρ ∗ ( kλ ) k λ X m ∈ N gcd( m,kλ )=1 µ ( m ) m = Y p − p + (cid:18) − p (cid:19) X d ∈ N ρ ∗ (cid:0) p d (cid:1) p d ! , which shows that the leading constant is equal to c ′ Q , as desired.We proceed to explain why the leading constants c Q and c ′ Q are equal.One can indeed produce an elementary, yet lengthy, argument of this asser-tion, performing a parametrisation argument over Z /p n Z for appropriatelychosen primes p and positive integers n, instead of over Q . However, asthe referee kindly pointed out, it is shown in [10, Sections 3 and 6.2] that c Q = c ′ Q follows from [7]. More precisely, the fact that points are equidis-tributed on the projective line implies that the leading constants agree forany height, including the one coming from the embedding of the projectiveline as a conic. This concludes the proof of Proposition 2.1.6. The proof of Theorem 1.1
In this section we complete the proof of Theorem 1.1 by transforming thegeneral form Q into one to which Proposition 2.1 applies. The next lemmashows that one can find a suitable transformation with the lowest possibleheight. Lemma 6.1.
Let a ∈ Z prim . Then there exists M ∈ SL ( Z ) whose secondcolumn is a and whose entries have maximum modulus O ( k a k ∞ ) . Proof.
By renaming indices if needed, we may assume that0 < | a | ≤ | a | ≤ | a | . Let us notice that an integer solution to the equation a t y = 1 exists, owingto the coprimality of a . The previous inequality implies that we can pick s, t ∈ Z such that max {| y − a t | , | y − a s |} ≤ | a | . Then the integer vector x := y + s ( a , − a ,
0) + t ( a , , − a )satisfies a t x = 1 and k x k ∞ ≪ k a k ∞ . We now let x ′ i := x i gcd( x ,x ) , i = 1 , x ′ , x ′ ) = 1 . We knowtherefore that an integer solution ( x, y ) of x ′ x + x ′ y = x can be found.Considering y − tx ′ in place of y if needed, we can prove as previouslythat we can find ( x, y ) that satisfy the previous equationin addition to max {| x | , | y |} ≪ k x k ∞ . A direct calculation may then reveal that the matrix M := x ′ a − x − x ′ a − y a gcd( x , x ) possesses the required properties. (cid:3) Proof of Theorem 1.1.
It is given that the quadratic form Q possesses arational zero. One can therefore find, using Cassels [3], a non–trivial integerzero ξ := ( x , y , z ) ∈ Z of Q such that k ξ k ∞ ≪ h Q i . We now transformthe form Q using a = ξ in the previous lemma. It provides an integer matrix M of determinant 1 and of size(6.1) k M k ∞ ≪ h Q i such that the quadratic form Q ′ defined by Q ′ ( x ) := Q ( M x ) , possesses the zero (0 , , . We define the norm given by k x k ′ := k M x k and notice that h Q ′ i ≪ h Q i . The fact that M is unimodular implies that the integer vector x is primitiveif and only if M x is. It therefore follows that N ( Q, B ) = N ′ ( Q ′ , B ) , where the notation N ′ indicates a use of the norm k . k ′ . Recall the definition(1.3) of K . Using the inequality k M − k ∞ ≤ k M k ∞ and writing x = M − ( M x ) for all x = , implies that k x k ∞ ≤ k M k ∞ K k x k ′ . Therefore (6.1) shows that for K ′ := 1 + sup x =0 k x k ∞ k x k ′ , we have K ′ ≪ K h Q i . Finally, notice that the discriminants ∆ Q , ∆ Q ′ as well as the greatest com-mon divisors δ Q and δ Q ′ of the 2 × Q and Q ′ remain invariant under the unimodular transformation M. We are now in a position to apply Proposition 2.1 to the form Q ′ withall involved quantities modified as indicated hitherto. We are provided withthe error term(6.2) ≪ ǫ ( BK ) log ( BK ) min n | ∆ Q | , δ Q o h Q i ǫ . NIFORMLY COUNTING RATIONAL POINTS ON CONICS 13
The bound | ∆ Q | ≪ h Q i implies that this is ≪ ǫ ( BK ) log ( BK ) h Q i + ǫ so that using the value ǫ = we obtain the error term appearing in Theo-rem 1.1. Recall the definition (1.1) and (1.2) of the local densities. It remainsto show that they satisfy σ ∞ ( Q ′ , k . k ′ ) = σ ∞ ( Q, k . k )and σ p ( Q ′ ) = σ p ( Q )for any prime p. The fact that the matrix M is invertible (mod p n ) showsthat N ∗ Q ( p n ) = N ∗ Q ′ ( p n ) is valid, which when used in (1.2) proves the latterequality. The former is proved by performing the unimodular linear changeof variables x = M X in (1.1). Hence Z | Q ( x ) |≤ ǫ k x k≤ x = Z | Q ′ ( X ) |≤ ǫ k X k ′ ≤ X , which finishes the proof of Theorem 1.1. Acknowledgements.
The author would like to express his gratitude to T.Browning for suggesting the problem and for his valuable assistance duringthe course of the project. He is furthermore indebted to Dr. ChristopherFrei for useful comments regarding an earlier version of this paper.
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