The representation of integers by positive ternary quadratic polynomials
aa r X i v : . [ m a t h . N T ] M a y The representation of integers by positive ternaryquadratic polynomials
Wai Kiu Chan ∗ , James Ricci ∗∗ Abstract
An integral quadratic polynomial is called regular if it represents every integerthat is represented by the polynomial itself over the reals and over the p -adicintegers for every prime p . It is called complete if it is of the form Q ( x + v ), where Q is an integral quadratic form in the variables x = ( x , . . . , x n )and v is a vector in Q n . Its conductor is defined to be the smallest positiveinteger c such that c v ∈ Z n . We prove that for a fixed positive integer c ,there are only finitely many equivalence classes of positive primitive ternaryregular complete quadratic polynomials with conductor c . This generalizes theanalogous finiteness results for positive definite regular ternary quadratic formsby Watson [18, 19] and for ternary triangular forms by Chan and Oh [8]. Keywords:
Representations of quadratic polynomials
1. Introduction
Let f ( x ) = f ( x , . . . , x n ) be an n -ary quadratic polynomial in variables x = ( x , . . . , x n ) with rational coefficients. It takes the form f ( x ) = Q ( x ) + ℓ ( x ) + m where Q ( x ) is a quadratic form, ℓ ( x ) is a linear form, and m is a constant. Givena rational number a , it follows from the Hasse Principle that the diophantineequation f ( x ) = a (1.1) ∗ Author: Wai Kiu Chan, Wesleyan University, Department of Mathematics and ComputerScience, 265 Church St., Middletown, CT, 06459, USA; Email, [email protected]; Phone,+001 860 685 2196 ∗∗ Corresponding Author: James Ricci, Daemen College, Department of Mathematics andComputer Science, 4380 Main St., Amherst, NY, 14226, USA; Email, [email protected];Phone, +001 716 566 7833
Preprint submitted to Elsevier May 5, 2015 s soluble over the rationals if and only if it is soluble over the p -adic numbersfor each prime p and over the reals. However, this local-to-global approachbreaks down when we consider integral representations. Indeed, there are plentyof examples of quadratic polynomials for which (1.1) is soluble over each Z p and over R , but not soluble over Z . Borrowing a term coined by Dickson forquadratic forms, we call a quadratic polynomial f ( x ) regular if for every rationalnumber a ,(1.1) is soluble over Z ⇐⇒ (1.1) is soluble over each Z p and over R .G.L. Watson [18, 19] showed that there are only finitely many equivalenceclasses of primitive positive definite regular ternary quadratic forms. A list ofrepresentatives of these classes has been compiled in [11] by Jagy, Kaplansky,and Schiemann. Their list contains 913 ternary quadratic forms, and 891 areverified by them to be regular. Later B.-K. Oh [15] proves the regularity of8 of the remaining 22 quadratic forms. More recently, R. Lemke Oliver [13]establishes the regularity of the last 14 quadratic forms under the generalizedRiemann Hypothesis. Watson’s result has been generalized by different authorsto definite ternary quadratic forms over other rings of arithmetic interest [3, 5],to higher dimensional representations of positive definite quadratic forms inmore variables [6], and to positive definite ternary quadratic forms which satisfyother regularity conditions [4].There are regular quadratic polynomials that are not quadratic forms. Awell-known example is the sum of three triangular numbers x ( x + 1)2 + x ( x + 1)2 + x ( x + 1)2 , which is universal (i.e. representing all positive integers) and hence regular.Given positive integers a, b, c , we follow the terminology in [7] and call thepolynomial ∆( a, b, c ) := a x ( x + 1)2 + b x ( x + 1)2 + c x ( x + 1)2a triangular form. It is primitive if gcd( a, b, c ) = 1. There are seven universalternary triangular forms–hence all are regular– and they were found by Liouvillein 1863 [14]. An example of a regular ternary triangular form which is notuniversal is ∆(1 , , Example 1.1.
It is easy to see that ∆(1 , ,
3) does not represent 8. Hence it isnot universal.A positive integer n is represented by ∆(1 , ,
3) if and only if 8 n + 5 isrepresented by h ( x ) = x + x + 3 x with the extra conditions that x ≡ x ≡ ≡ h ( x ) be the quadratic form x + x + 12 x . If r ( n ) is thenumber of representations of n by ∆(1 , , r ( n ) = r (8 n + 5) − r (8 n + 5)where, for i = 1 , r i (8 n +5) is the number of representations of 8 n +5 by h i ( x ).Note that both h ( x ) and h ( x ) have class number 1, and so the Minkowski-Siegel mass formula [12, Theorem 6.8.1] implies that both r (8 n +5) and r (8 n +5) can be expressed as products of local densities. For each odd prime p , h ( x )and h ( x ) are equivalent over Z p and hence the local densities α p (8 n + 5 , h )and α p (8 n + 5 , h ) are the same. It then follows from [12, Theorem 6.8.1] that r (8 n + 5) r (8 n + 5) = α (8 n + 5 , h )2 − α (8 n + 5 , h ) . The 2-adic densities α (8 n + 5 , h ) and α (8 n + 5 , h ) can be computed by [12,Proposition 5.6.1], and both of them can be shown to be equal to 16. Therefore, r (8 n + 5) = 2 r (8 n + 5) and hence r ( n ) = r (8 n + 5).Now, suppose that n is represented by ∆(1 , ,
3) over Z p for every prime p .Then, since h ( x ) has class number 1, 8 n + 5 is represented by h ( x ). Thismeans that r (8 n + 5) is not zero, whence r ( n ) is also not zero. Thus, n isrepresented by ∆(1 , , , ,
3) is regular.It is shown in [8] that there are only finitely many primitive ternary regulartriangular forms. In this paper, we will extend Watson’s finiteness results toternary quadratic polynomials with positive definite quadratic parts.A quadratic polynomial is called nondegenerate if its quadratic part is non-degenerate. Let f ( x ) be a nondegenerate quadratic polynomial. If Q ( x ) is itsquadratic part and B is the bilinear form corresponding to Q , then there existsa unique v ∈ Q n such that 2 B ( v , x ) is the linear part of f ( x ). The conductor of f ( x ), introduced in [10], is the smallest positive integer c such that c v ∈ Z n .The polynomial is integral if f ( a ) ∈ Z for all a ∈ Z n , and is called primitive ifthe ideal generated by the set { f ( a ) : a ∈ Z n } is Z . If Q is positive definite,then f ( x ) attains an absolute minimum on Z n . We then call f ( x ) positive if itsabsolute minimum is nonnegative.Another quadratic polynomial g ( x ) is said to be equivalent to f ( x ) if thereexist T ∈ GL n ( Z ) and u ∈ Z n such that g ( x ) = f ( x T + u ) . This defines an equivalence relation on the set of quadratic polynomials, andit is clear that the conductor of a quadratic polynomial is a class invariant.It is also clear that regularity is preserved under this notion of equivalence.However, simply changing the constant term of a regular quadratic polynomial3ill produce infinitely many inequivalent regular quadratic polynomials of thesame number of variables. Therefore, in order to obtain any finiteness resultsanalogous to Watson’s, we need to confine our attention to a special family ofquadratic polynomials. Following the terminology introduced in [8], we call aquadratic polynomial f ( x ) complete if it takes the form f ( x ) = Q ( x ) + 2 B ( v , x ) + Q ( v ) = Q ( x + v ) . By adjusting the constant term and multiplying by a suitable rational number,any nondegenerate quadratic polynomial can be changed to a primitive completequadratic polynomial. The main result of this paper is
Theorem 1.2.
Let c be a fixed positive integer. There are only finitely manyequivalence classes of positive primitive ternary regular complete quadratic poly-nomials with conductor c . Theorem 1.2 extends Watson’s finiteness result on regular ternary quadraticforms because a quadratic form is equivalent to a complete quadratic polynomialof conductor 1, and each equivalence class of complete quadratic polynomials ofconductor 1 contains a unique equivalence class of quadratic forms.The paper is organized as follows. Section 2 contains some preliminary re-sults on representations of quadratic polynomials in general. In Section 3, wediscuss various estimates on the number of integers in an interval which satisfycertain arithmetic conditions. A set of regularity preserving transformationson quadratic polynomials will be introduced in Section 4. These transforma-tions and their properties will be best described using the language of quadraticspaces, lattices, and cosets. It is this language that we will adopt for the rest ofthe paper. In Sections 5 and 6 we will present the proof of Theorem 1.2. Finally,in Section 7, we will discuss a particular family of quadratic polynomials calledpolygonal forms, and explain how Theorem 1.2 implies the finiteness result ofregular ternary triangular forms in [8] mentioned earlier.
2. Preliminaries
Let f ( x ) = Q ( x ) + 2 B ( v , x ) + m be an integral quadratic polynomial. Thenorm ideal of Q , denoted n , is the ideal of Z generated by the set of integersrepresented by Q ( x ). The set b := { B ( v , x ) : x ∈ Z n } is an ideal of Z . It isnot hard to check that both n and b are inside Z as a result of the integralityof f ( x ), and that n = Z if and only if b = Z . We denote the polynomial Q ( x ) + 2 B ( v , x ) by f ( x ), and let n ( f ) be the ideal of Z generated by theintegers represented by f ( x ). Lemma 2.1. If g ( x ) is equivalent to f ( x ) , then n ( f ) = n ( g ) . roof. It is clear that n ( f ) = n ( g ) if g ( x ) = f ( x T ) for any T ∈ GL n ( Z ).Therefore, we may assume that g ( x ) = f ( x + u ) for some u ∈ Z n . A simplecalculation shows that g ( x ) = f ( x + u ) − f ( u ), whence n ( g ) ⊆ n ( f ). Thereverse inclusion is obtained by observing that f ( x ) = g ( x − u ). (cid:3) Lemma 2.2. If c is the conductor of f ( x ) , then c Q ( v ) ∈ n ( f ) . If, in addi-tion, f ( x ) is primitive and complete, then c ∈ n ( f ) . Proof.
Since c v and 2 c v are in Z n , both Q ( v )( c + 2 c ) and Q ( v )(4 c + 4 c ) arein n ( f ). Thus, 4 c Q ( v ) ∈ n ( f ). If f ( x ) is primitive and complete, then Q ( v )is the constant term of f ( x ) and is relatively prime to n ( f ). This implies thesecond assertion. (cid:3) Let p be a prime and I be an ideal of Z p . We say that f ( x ) represents acoset modulo I if over Z p , f ( x ) represents r + I for some r ∈ Z p . Note that forevery a ∈ Z np , f ( x ) represents every p -adic integer that is represented by theone variable polynomial Q ( a ) x + 2 B ( v , a ) x over Z p . Lemma 2.3.
Let p be a prime. If f ( x ) is primitive, then f ( x ) represents a cosetmodulo p k Z p , where ≤ k ≤ ord p ( n ( f )) + 2 δ ,p and δ ,p is the Kroneckerdelta. Proof.
We may assume that f ( x ) = f ( x ) and n ( f ) = Z . Suppose first that p is an odd prime. If p | b , then there exists a ∈ Z np such that 2 B ( v , a ) ∈ p Z p and Q ( a ) ∈ Z × p . By the Local Square Theorem [16, 63:1] and [16, 63:8], thepolynomial Q ( a ) x + 2 B ( v , a ) x represents the coset Q ( a ) + p Z p . If, on the otherhand, 2 B ( v , b ) ∈ Z × p for some b ∈ Z np , then by the Local Square Theorem and[16, 63:8] again the polynomial Q ( b ) x + 2 B ( v , b ) x represents all of p Z p .The argument when p = 2 is along the same line and uses both the LocalSquare Theorem and [16, 63:8] in the same manner. If b = Z , then thereexists a ∈ Z n such that both 2 Q ( a ) and 4 B ( v , a ) are in Z × . In this case,the polynomial Q ( a ) x + 2 B ( v , a ) x represents every element in Z . Supposethat b = Z . Then, 2 B ( v , b ) ∈ Z × for some b ∈ Z n , and the polynomial Q ( b ) x + 2 B ( v , b ) x represents 2 Z .We are left with the case when 2 | b . In this case, there is a vector c ∈ Z n such that 2 B ( v , c ) is divisible by 2 but Q ( c ) is in Z × . We further dividethe discussion into three subcases: ord (2 B ( v , c )) = 1, ord (2 B ( v , c )) = 2,or ord (2 B ( v , c )) ≥
3. Let us look at the subcase ord (2 B ( v , c )) = 2. If ǫ ≡ Q ( c ) + B ( v , c ) Q ( c ) − mod 8, then B ( v , c ) + Q ( c ) ǫ ≡ Q ( c ) + 2 B ( v , c ) ≡ Q ( c ) mod 8Therefore, B ( v , c ) + Q ( c ) ǫ is the square of a 2-adic unit, and hence the polyno-mial Q ( c ) x + 2 B ( v , c ) x represents ǫ over Z . This shows that f ( x ) representsa coset modulo 8 Z . The other two subcases are treated similarly and we leavetheir proofs to the readers. (cid:3) emark 2.4. If f ( x ) is primitive and complete, then by Lemma 2.2 the integer k in Lemma 2.3 is bounded above by a constant depending only on c . Definition 2.5.
A positive quadratic polynomial f ( x ) is called Minkowski re-duced , or simply reduced , if its quadratic part Q ( x ) is Minkowski reduced and f ( x ) itself attains its minimum at the zero vector.Since every positive definite quadratic form is equivalent to a Minkowskireduced quadratic form [2, Chapter 12, Theorem 1.1], it follows easily that everypositive quadratic polynomial is equivalent to a reduced quadratic polynomial(see [8, Lemma 2.2] for the case of ternary quadratic polynomials).If f ( x ) = Q ( x ) + 2 B ( v , x ) + m is a reduced quadratic polynomial and { e , . . . , e n } is the standard basis of Z n , then Q ( x ) is Minkowski reduced,2 | B ( v , e i ) | ≤ Q ( e i ) for i = 1 , . . . , n , and m is the smallest integer representedby f ( x ). In the special case when f ( x ) is ternary, Q ( e ) ≤ Q ( e ) ≤ Q ( e ) arethe successive minima of Q ( x ) [17, Page 285]. Proposition 2.6.
Suppose that f ( x ) is a positive reduced ternary quadraticpolynomial. Let µ , µ , µ be the successive minima of the quadratic part of f ( x ) , and a = ( a , a , a ) be a vector in Z .(a) If | a | ≥ , then f ( a ) ≥ µ .(b) If | a | ≤ and | a | ≥ , then f ( a ) ≥ µ .(c) If | a | ≤ and | a | ≤ , then f ( a ) ≥ µ ( a − | a | ) .(d) If | a | ≤ , | a | ≤ , and | a | ≥ , then f ( a ) ≥ µ .Consequently, f ( a ) ≥ min (cid:26) µ , µ , µ (cid:27) unless | a | ≤ , | a | ≤ , and | a | ≤ . Proof.
Let f ( x ) be the quadratic polynomial obtained by taking away theconstant term from f ( x ). Then f ( x ) is also reduced and f ( x ) ≥ f ( x ) for all x ∈ Z . Therefore, we may assume that the minimum of f ( x ) is 0. The prooffor this special case can be easily extracted from the proof of [8, Theorem 1.1](see [8, Page 35] in particular). (cid:3)
3. Some Technical Lemmas
As is in the proof of the finiteness of regular ternary triangular forms [8,Theorem 1.2], we need estimates of the number of integers in an interval thatsatisfy various local conditions. See [2, Chapter 12] for the definition of Minkowski reduced quadratic forms. emma 3.1. [8, Lemma 3.4] Let T be a finite set of primes and a be an integernot divisible by any prime in T . For any integer d , the number of integers inthe set { d, a + d, . . . , ( n − a + d } that are not divisible by any prime in T is atleast n ˜ p − p + t − − t + 1 , where t = | T | and ˜ p is the smallest prime in T . Let χ , . . . , χ r be quadratic characters modulo k , . . . , k r , respectively, Γ bethe least common multiple of k , . . . , k r , and u , . . . , u r be values taken from theset {± } . Given a nonnegative number s and a positive number H , let S s ( H ) bethe number of integers n in the interval ( s, s + H ) which satisfy the conditions χ i ( n ) = u i for i = 1 , . . . , r and gcd( n, X ) = 1 , (3.1)where X is a positive integer relatively prime to Γ.Let I be a sequence of parameters. An inequality of the form A ≪ I B willmean that there exists a constant κ , depending only on the parameters in I ,such that | A | ≤ κB . Alternatively, we will write A = B + O I ( C ) if A − B ≪ I C .If I is empty, then we will simply use ≪ and O instead, and the implied constant κ in this case will be an absolute constant.The following proposition is essentially [8, Proposition 3.6]. Proposition 3.2.
Suppose that χ , . . . , χ r are independent. Let k be a fixedpositive integer and h = min { H : S s ( H ) > k } . Then S s ( H ) = 2 − r φ (Γ X )Γ X H + O ǫ (cid:16) H Γ + ǫ X ǫ (cid:17) , (3.2) and h ≪ ǫ,k Γ + ǫ X ǫ , (3.3) where φ is Euler’s phi-function. We will need another similar result which estimates S ′ s ( H ), the number ofintegers in ( s, s + H ) which satisfy the following stronger conditions χ i ( n ) = u i for i = 1 , . . . , r and n ≡ τ mod X, (3.4)where τ is a fixed integer relatively prime to X . In the statement of [8, Proposition 3.6], it requires r ≤ ω (Γ)+1, where ω (Γ) is the numberof distinct prime divisors of Γ. However, as the referee suggested to us, this inequality holdsas a consequence of the independence of the characters. roposition 3.3. Suppose that χ , . . . , χ r are independent. Let k be a fixedpositive integer and h = min { H : S ′ s ( H ) > k } . Then S ′ s ( H ) = 2 − r φ (Γ)Γ X H + O ǫ (cid:16) H (Γ X ) + ǫ (cid:17) , (3.5) and h ≪ ǫ,k Γ + ǫ X + ǫ , (3.6) where φ is Euler’s phi-function. Proof.
By the orthogonality of Dirichlet characters, we can express S ′ s ( H ) as S ′ s ( H ) = X s 12 (1 + u j χ j ( n )) 1 φ ( X ) X ψ mod X ψ ( τ ) ψ ( n )= 12 r φ ( X ) X ψ mod X ψ ( τ ) X R ⊆{ ,...,r } Y j ∈ R u j X s We note that although Polya’s estimate of character sum sufficesin our present argument, we choose to use Burgess’ in order to obtain a sharperestimate on the error terms, so that we have a better idea on how much isneeded to improve on the other estimates in order to obtain better results.8 . Watson Transformations We will describe a family of regularity preserving transformations on com-plete quadratic polynomials. The definition and properties of these transfor-mations are best explained in the geometric language of quadratic spaces andlattices, and this is the language we choose to conduct all our subsequent dis-cussions. The books [12] and [16] are standard references for quadratic spacesand lattices. Any other unexplained notations and terminologies used later inthis paper can be found in either of them.Let R be a PID, and ( V, Q ) be a nondegenerate quadratic space over thefield of fractions of R . If L is an R -lattice on V and A is a symmetric matrix, weshall write “ L ∼ = A ” if A is the Gram matrix for L with respect to some basis of L . The discriminant of L , denoted d ( L ), is defined to be the determinant of oneof its Gram matrices. An n × n diagonal matrix with a , . . . , a n as its diagonalentries is written as h a , . . . , a n i .An R -coset on V is a set L + v , where L is an R -lattice on V and v is avector in V . We define the discriminant of L + v to be d ( L ), the discriminantof L . The conductor of L + v is the fractional ideal { a ∈ R : a v ∈ L } . Inthe case R = Z , the conductor has a positive generator and we will abuse theterminology and call this number the conductor of L + v . The R -coset L + v is integral if the fractional ideal generated by Q ( L + v ), denoted n ( L + v ), iscontained in R ; and is primitive if n ( L + v ) = R . Two R -cosets L + v and M + w on V and W respectively are said to be isometric , written L + v ∼ = M + w , ifthere exists an isometry σ : V −→ W such that σ ( L + v ) = M + w . This is thesame as requiring σ ( L ) = M and σ ( v ) ∈ M + w . It is clear that the conductorof an R -coset is a class invariant.A Z -coset is called positive if the underlying quadratic space is positive defi-nite. We say that a rational number a is represented by a Z -coset L + v if thereexists a ∈ L such that Q ( a + v ) = a . For each prime p , the representation of a p -adic number by a Z p -coset is defined in the obvious way. A rational number a is represented by the genus of a Z -coset L + v if it is represented by V ∞ andby L p + v for every prime p . The Z -coset L + v is called regular if it representsall rational numbers that are represented by its genus. The readers are referredto [8] for more discussion on representations of numbers by Z -cosets in general.Let L + v be a Z -coset on V . Fix a basis { e , . . . , e n } of L . For any( x , . . . , x n ) ∈ Z n , we have Q ( x e + · · · + x n e n + v ) = n X i =1 n X j =1 B ( e i , e j ) x i x j + n X ℓ =1 B ( v , e ℓ ) x ℓ + Q ( v )which is an n -ary complete quadratic polynomial. Conversely, given an n -arycomplete quadratic polynomial f ( x ) = Q ( x )+ 2 B ( v , x )+ Q ( v ), the set Z n + v is9 Z -coset on the quadratic space Q n equipped with the quadratic form Q . Thuswe have a correspondence between Z -cosets and complete quadratic polynomi-als. Under this correspondence, primitive regular complete quadratic polyno-mials correspond to primitive regular Z -cosets, and the conductor of a completequadratic polynomial will be the same as the conductor of the associated Z -coset. One can readily check that this correspondence leads to a one-to-onecorrespondence between isometry classes of Z -cosets and equivalence classes ofcomplete quadratic polynomials. Definition 4.1. Let L be a Z -lattice. For any integer m , letΛ m ( L ) = { x ∈ L : Q ( x + z ) ≡ Q ( z ) mod m for every z ∈ L } , and for any prime p , letΛ m ( L p ) = { x ∈ L p : Q ( x + z ) ≡ Q ( z ) mod m for every z ∈ L p } . Lemma 4.2. [4, Lemma 2.2] Let L be a Z -lattice, m an integer, and p a prime.Then the following hold.(a) Λ m ( L ) is a sublattice of L and Λ m ( L p ) is a sublattice of L p .(b) Λ m ( L p ) = Λ m ( L ) p .(c) Λ m ( L p ) = L p whenever p ∤ m .(d) n (Λ m ( L )) ⊆ m Z and n (Λ m ( L p )) ⊆ m Z p .(e) If n ( L ) ⊆ Z , then pL ⊆ Λ p ( L ) and pL p ⊆ Λ m ( L p ) .(f ) If N splits L p and n ( N ) ⊆ p Z p , then N ⊆ Λ p ( L p ) . Lemma 4.3. Let L be a Z -lattice and p be an odd prime. Suppose that L p = M ⊥ N where M is unimodular and n ( N ) ⊆ p Z p . Then Λ p ( L p ) = pM ⊥ N .If, in addition, M is anisotropic, then Λ p ( L p ) = { x ∈ L p : Q ( x ) ∈ p Z p } . Proof. The first assertion is [4, Lemma 2.3]. For the second assertion, itfollows immediately from the definition of Λ p ( L p ) that Λ p ( L p ) is a subset of { x ∈ L p : Q ( x ) ∈ p Z p } . Conversely, suppose that x ∈ L p and Q ( x ) ∈ p Z p .Write x = x + x , where x ∈ M and x ∈ N . Assume on the contrary that x pM . Then x is a maximal vector in M , and by [16, 83:17] there exists z ∈ M such that B ( x , z ) = 1. The binary sublattice of M spanned by x and z has discriminant in − Z × p , and hence it is isotropic. This contradicts thehypothesis, thus x ∈ pM and x ∈ pM ⊥ N = Λ p ( L p ). (cid:3) Suppose that L is a Z -lattice on a nondegenerate quadratic space V . Let p be an odd prime such that p ∤ n ( L ). By Lemma 4.3, p n ( L ) ⊆ n (Λ p ( L )) ⊆ p n ( L ) . 10e denote by λ p the mapping that sends L to the following lattice on the scaledspace V p or V p : λ p ( L ) = ( Λ p ( L ) p if n (Λ p ( L )) = p n ( L ) , Λ p ( L ) p if n (Λ p ( L )) = p n ( L ) . (4.1)Collectively, these λ p are what we will refer to as Watson transformations. Notethat n ( λ p ( L )) = n ( L ). Lemma 4.4. [4, Lemma 2.5] Suppose that L is a ternary Z -lattice on a non-degenerate quadratic space. If p is an odd prime such that p | d ( L ) , then d ( λ p ( L )) = p t d ( L ) for some t ∈ { , , } . Definition 4.5. An integral Z -coset L + v is said to behave well at a prime p if L p has a unimodular Jordan component of rank at least 2.By [16, 92:1b], L p represents all p -adic units if L + v behaves well at p . Proposition 4.6. Let L + v be a primitive regular ternary Z -coset with con-ductor c , and p be an odd prime which does not divide c . If L + v does notbehave well at p , then there exists w in the quadratic space underlying λ p ( L ) such that λ p ( L ) + w is a primitive regular Z -coset with conductor c . Proof. Let j be the order of p modulo c . We claim thatΛ p ( L ) q + p j v = L q + v if q | c, Λ p ( L ) q if q = p,L q if q ∤ pc. (4.2)The first and the third cases in (4.2) are straightforward. As for the case p = q ,since L + v does not behave well at p , it follows from Lemma 4.3 that Λ p ( L ) p = { x ∈ L p : Q ( x ) ∈ p Z p } . Therefore, p j v ∈ Λ p ( L ) p , and hence Λ p ( L ) p + p j v =Λ p ( L ) p .Suppose that a is represented by the genus of Λ p ( L )+ p j v . By (4.2), a is alsorepresented by the genus of L + v . Since L + v is regular, a is in fact representedby L + v , which means that there exists x ∈ L such that Q ( x + v ) = a . By(4.2) again, x + v is contained in Λ p ( L ) q + p j v for every q = p . At p , since a is represented by Λ p ( L ) p + p j v = Λ p ( L ) p , p must divide a by Lemma 4.3.Thus, p | Q ( x + v ) and x + v must be in Λ p ( L ) p , by Lemma 4.3 one more time.Altogether we have shown that x + v is in Λ p ( L ) q + p j v for every prime q . So, x + v is in Λ p ( L ) + p j v , which proves that Λ p ( L ) + p j v is regular.It is clear that Λ p ( L ) + p j v has conductor c . Since the conductor and theregularity of a Z -coset are preserved upon scaling of the underlying quadratic When L + v is a Z -lattice, our definition of “behaves well” is slightly different from theone used in [4]. λ p ( L ) + p j v is also regular and has conductor c . It remains to show that λ p ( L ) + p j v is primitive. The quadratic form on λ p ( L ) is p i Q , where i = 1 or2, see (4.1). By (4.2), n (Λ p ( L ) q + p j v ) = n ( L q + v ) = Z q if q | c, n (Λ p ( L ) p ) = p i Z p if p = q, n ( L q ) = Z q if q ∤ pc. Therefore, n ( λ p ( L ) + p j v ) = Z , which is what we need to show. (cid:3) 5. Bounding the Discriminant I Given a positive Z -coset L + v , we can always choose v such that Q ( v ) = min { Q ( x + v ) : x ∈ L } . (5.1)If, in addition, { e , . . . , e n } is a Minkowski reduced basis of L , then the polyno-mial Q ( x e + · · · + x n e n + v ) will be a Minkowski reduced quadratic polynomial.From now on, unless stated otherwise, we always assume that (5.1) holds whenwe present a positive Z -coset in the form L + v . Lemma 5.1. There are only finitely many isometry classes of integral Z -cosetsof a fixed rank and discriminant. Proof. Let n and k be fixed integers, and let L be an integral lattice of rank n and discriminant k . If v ∈ Q L and L + v is integral, then 2 B ( v , x ) ∈ Z forall x ∈ L and hence 2 v is in L , the dual of L . Since L /L is a finite group ofsize k , there are only finitely many possible integral Z -cosets of the form L + v .The lemma is now clear since it is well-known that there are only finitely manyisometry classes of integral lattices of rank n and discriminant k . (cid:3) Let L + v be a positive Z -coset of rank n . The successive minima µ ≤ · · · ≤ µ n of L satisfy the inequality [9, Prop 2.3] d ( L ) ≤ µ · · · µ n . (5.2)Let c be the conductor of L + v . For every prime p dividing 2 c , Lemma2.3 shows that L p + v represents a coset r p + p k p Z p , where k p is a nonnegativeinteger bounded above by a constant depending only on c . Set a = a ( L + v ) := Y p | c p k p , r = r ( L + v ) := min { b ∈ N : b ≡ r p mod a for all p | c } . (5.3)Note that both a and r are bounded above by a constant depending only on c .12 roposition 5.2. Let L + v be a primitive regular positive ternary Z -coset ofconductor c . If L + v behaves well at all primes p not dividing c , then d ( L ) isbounded above by a constant depending only on c . Proof. Let T be the set of odd primes p such that p ∤ c and L p does notrepresent all p -adic integers. Then T is a finite set. Let t be the size of T , T bethe product of all primes in T , and ˜ p be the smallest prime in T . Since ˜ p > ω := ˜ p + t − p − ≤ t + 1 . Let a and r be the integers defined as in (5.3), and G be the set of all positiveintegers in the congruence class of r mod a that are relatively prime to T .If p | c , then by Lemma 2.3 L p + v represents all integers in G . If p ∤ c and p T , then certainly L p + v represents all integers in G . Suppose that p ∈ T .Then L p + v = L p behaves well, which means that L p + v represents every p -adicunit. Since L + v is regular, we see that L + v represents all positive integersin G . We shall use these integers to obtain an upper bound for the productof the successive minima of L , and compare this upper bound with d ( L ) by(5.2). Let { e , e , e } be a Minkowski basis of L . Then every a ∈ Z is a linearcombination a e + a e + a e .Let η be the smallest positive integer such that η > (cid:16) · · (2 (15 + p 225 + aη + r ) + 1) + 2 t − (cid:17) ω. (5.4)By means of contradiction, let us suppose that µ > r + aη . If Q ( a + v ) ≤ r + aη , then Proposition 2.6 (a), (b) and (c) imply that | a | ≤ | a | ≤ Q ( a + v ) ≥ µ ( a − | a | ). Therefore, if | a | > 15 + √ 225 + r + aη , then Q ( a + v ) > r + aη . This shows that the number of integers smaller than r + aη which are represented by L + v is at most43 · · (2 (15 + p 225 + r + aη ) + 1) . However, by Proposition 3.1, the number of integers in { r, r + a, . . . , r + a ( η − } which are represented by L + v is at least η ˜ p − p + t − − t + 1which can be shown to be larger that 43 · · (2 (15 + √ 225 + r + aη ) + 1) using(5.4). This is a contradiction, and hence we must have µ ≤ 23 ( r + aη ) . A straightforward calculation shows that η ≪ a t t ; thus µ ≪ a t t . (5.5)13et M be the sublattice spanned by e and e . Then d ( M ) ≤ µ µ . Foreach integer 1 ≤ j ≤ Q ( y e + y e + ( j − e + v ) is a positive binaryintegral quadratic polynomial in variables y = ( y , y ) which takes the form h j ( y ) = q ( y ) + 2 b ( w j , y ) + m j where q is the quadratic form on M and b is the bilinear form associated to q .We will find a positive integer in G which is not represented by any one of these17 binary quadratic polynomials. By Proposition 2.6 (a), this integer will leadto an upper bound on µ .Let D be the product of those primes in T that do not divide d ( M ). Forthe sake of convenience, we set ℓ to be 1. By Proposition 3.2, for 1 ≤ i ≤ n i such that (cid:16) − d ( M ) n i (cid:17) = − n i is relatively prime to2 aT ℓ ...ℓ i − (when i = 1, this condition becomes n ∤ aT ). The multiplicativeproperty of the Jacobi symbol then guarantees the existence of a prime divisor ℓ i of n i such that (cid:16) − d ( M ) ℓ i (cid:17) = − ℓ i ∤ aT ℓ . . . ℓ i − , and ℓ i ≪ ǫ d ( M ) + ǫ ( D aℓ · · · ℓ i − ) ǫ ≪ ǫ,a d ( M ) + ǫ D ǫ ( ℓ · · · ℓ i − ) ǫ . Let κ be the product ℓ · · · ℓ . Then, κ ≪ ǫ,a d ( M ) + ǫ D ǫ Y i =1 ( ℓ · · · ℓ i − ) ǫ ≤ d ( M ) + ǫ D ǫ κ ǫ and hence κ ≪ ǫ,a d ( M ) − ǫ ) + ǫ − ǫ D ǫ − ǫ . We now choose ǫ to be . This leads to κ ≪ a d ( M ) + D ≪ a ( t t ) T . For each 1 ≤ j ≤ M ℓ j is a binary unimodular Z ℓ j -lattice. Since b ( w j , y ) ∈ Z ℓ j for all y ∈ M ℓ j , w j is in M ℓ j and hence q ( w j ) ∈ Z ℓ j . By the ChineseRemainder Theorem, there exists m ≤ κ such that am ≡ ℓ j + m j − q ( w j ) − r mod ℓ j for 1 ≤ j ≤ . Then, for every integer λ and every 1 ≤ j ≤ 17, ord ℓ j (( a ( m + λκ )+ r )+ q ( w j ) − m j ) = 1, and hence ( a ( m + λκ )+ r )+ q ( w j ) − m j is not represented by q ( y + w j )over Z ℓ j . In other words, a ( m + λκ ) + r is not represented by h j ( y ). On theother hand, by Lemma 3.1 there must be a positive integer f ≤ ( t + 1)2 t suchthat aκ f + am + r is relatively prime to T . Thus, the integer a ( m + κ f ) + r isrepresented by Q ( x + v ) but not by h j ( y ) for any j . It follows from Proposition2.6(a) that there exists a ∈ L such that a ( m + κ f ) + r = Q ( a + v ) ≥ µ , µ ≤ 23 ( a ( m + κ f ) + r ) ≪ a ( t t ) T . (5.6)Finally, using the inequality (5.2) and combining (5.5) and (5.6), we have T ≤ d ( L ) ≤ µ µ µ ≤ µ µ ≪ a ( t t ) T . Since T grows at least as fast as t !, the above inequalities show that t , and hence T as well, must be bounded above by a constant depending only on a . Thismeans that d ( L ) is also bounded above by a constant depending only on a . Thisproves the proposition, since a is bounded above by a constant depending onlyon c . (cid:3) 6. Bounding the Discriminant II Let L + v be a primitive regular positive ternary Z -coset of conductor c .We can apply Proposition 4.6 repeatedly at suitable odd primes and eventuallyobtain a primitive regular positive ternary lattice K + w which behaves well atall odd primes not dividing c . Moreover, the conductor of K + w is still c and d ( K ) is a divisor of d ( L ). Let L be the set of prime divisors of d ( L ) which donot divide d ( K ). By Proposition 5.2, every prime divisor of d ( L ) that does notbelong to L is bounded by a constant depending only on c . Proposition 6.1. All the primes in L are bounded above by a constant depend-ing only on c . Proof. Let ℓ be a prime in L which does not divide 2 c . Without loss ofgenerality, we may assume that L + v does not behave well at ℓ but does soat all odd primes not dividing ℓc . Moreover, since successive applications ofProposition 4.6 at ℓ to L + v results in a Z -coset which behaves well at ℓ , we canfurther assume that L ℓ is isometric to h α, βℓ , γℓ i , where α, β, γ ∈ Z × ℓ . Let a and r be the positive integers as defined in (5.3), and I be the product of primedivisors of d ( L ) which do not divide 2 ℓc . It is easy to see that L + v representsall positive integers congruent to r mod a that are relatively prime to I . Let b be the gcd of a and r .By applying Proposition 3.3 to the quadratic residue character mod ℓ , taking ǫ = and X to be the product ab I , we see that the number of positive integersless than m which are represented by L + v is bφ ( ℓ )2 aℓI m + O c ( m ℓ ) . (6.1)Therefore, there exists a positive constant N , depending only on c , such thatthe number of positive integers less than m which are represented by L + v isat least bφ ( ℓ )2 aℓI m − N m ℓ . (6.2)15ow, suppose that m < µ , and let a ∈ L such that Q ( a + v ) ≤ m . We write a as a e + a e + a e , where { e , e , e } is a Minkowski reduced basis of L .Then, by Proposition 2.6, we must have | a | ≤ , | a | ≤ , and Q ( a + v ) ≥ µ ( a − | a | ) . Thus, if | a | > 15 + √ 225 + m , then Q ( a + v ) > m which would be impossible.Therefore, the number of positive integers less than m which are represented by L + v is at most 43 · · (2(15 + √ 225 + m ) + 1) . (6.3)Combining (6.2) and (6.3) together, we obtain the inequality bφ ( ℓ )2 aℓI m − N m ℓ ≤ · · (2(15 + √ 225 + m ) + 1) . Using the inequality ℓφ ( ℓ ) ≪ ℓ , we deduce that whenever m < µ , m ≪ c ℓ . This implies that µ itself also satisfies µ ≪ c ℓ .Since L ℓ ∼ = h α, βℓ , γℓ i , we must have ℓ ≤ µ µ ≤ µ ≪ c ℓ and hence ℓ is bounded above by a constant depending only on c . (cid:3) We are ready to prove Theorem 1.2, which is now restated in the languageof Z -cosets. Theorem 6.2. Let c be a fixed positive integer. There are only finitely manyisometry classes of positive primitive ternary regular Z -cosets with conductor c . Proof. Let L + v be a positive primitive ternary regular Z -coset with conductor c . In what follows, when we say that a numerical quantity is bounded, it will beunderstood that the said quantity is bounded above by a constant dependingonly on c . By virtue of Lemma 5.1 and inequality (5.2), the theorem will beproved once we show that the successive minima µ , µ , µ of L are all bounded.Let t be the number of prime divisors of d ( L ) which do not divide 2 c , andlet T be the product of all these primes. Propositions 5.2 and 6.1 show that T and t are bounded. Let a and r be the integers as defined by (5.3).As is in the proof of Proposition 6.1, it follows from Proposition 3.3 that thenumber of integers less than m represented by L + v is2 − t φ ( T ) T a m + O c ( m ) . (6.4)At the same time, if m < µ , the same number must be at most43 · · (2(15 + √ 225 + m ) + 1) . (6.5)16ombine (6.4) and (6.5) together and proceed as in the proof of Proposition 6.1,we deduce that µ is bounded. This, of course, implies that µ is also bounded.To bound µ , we proceed as in the proof of Proposition 5.2 but keep inmind that t, T, µ , µ , a , and r are all bounded. Let { e , e , e } be a Minkowskibasis of L . Let M be the sublattice spanned by e and e . As in the proof ofProposition 5.2, there are 17 primes ℓ , . . . , ℓ such that M ℓ j is anisotropic for1 ≤ j ≤ 17. Let κ be the product of these 17 primes, which are bounded. Thereexists a positive integer m ≤ κ such that for any integer λ , a ( m + λκ ) + r isnot represented by any of the 17 binary quadratic polynomials Q ( y e + y e +( j − e + v ).Let d be the gcd of aκ and am + r , which is relatively prime to T . ByProposition 3.3, there exists a bounded positive integer n such that dn is repre-sented by L p for all p | T and dn ≡ am + r mod aκ . Then (3.6) guarantees that dn is a bounded integer represented by L + v , and is of the form a ( m + λκ ) + r for some integer λ . It follows from Proposition 2.6(a) that µ ≤ dn , whichmeans that µ is bounded. (cid:3) 7. Polygonal Forms In this section, we will explain how either Theorem 1.2 or Theorem 6.2implies the finiteness result of regular ternary triangular forms proved in [8].But instead of focusing on triangular forms, we will broaden our discussion toinclude a wider class of quadratic polynomials.Let m ≥ m -gonal numbersis the set of integers represented by the integral quadratic polynomial( m − x − ( m − x . For examples, when m = 4 these numbers are precisely the squares of integers,and the case m = 3 gives us the triangular numbers. Given any positive integers a , . . . , a n , we call the polynomial∆ m ( a , . . . , a n ) := n X i =1 a i (cid:18) ( m − x i − ( m − x i (cid:19) (7.1)an m -gonal form (in n variables). When m = 3, this is exactly a triangularform. After completing the squares, (7.1) becomes∆ m ( a , . . . , a n ) = n X i =1 a i m − (cid:18) x i − m − m − (cid:19) − ( m − m − n X i =1 a i . It follows that the conductor of ∆ m ( a , . . . , a n ) is m − 2) if m is odd ,m − ( m ) = 1 , m − if ord ( m ) > . m , the conductor of ∆ m ( a , . . . , a n ) is the same for anychoices of a , . . . , a n .Let L be the Z -lattice which is isometric to h m − a , . . . , m − a n i with respect to some basis { e , . . . , e n } , and v be the vector − m − m − ( e + · · · + e n ). Then an integer k is represented by ∆ m ( a , . . . , a n ) if and only if L + v represents 8( m − k + ( m − ( a + · · · + a n ). In particular, ∆ m ( a , . . . , a n )is regular if and only if L + v is regular. Equivalent m -gonal forms will lead toisometric Z -cosets under this correspondence. However, we have the followingsimple lemma about equivalent m -gonal forms. Lemma 7.1. If ∆ m and ∆ ′ m are equivalent m -gonal forms, then up to a per-mutation of the variables ∆ m = ∆ ′ m . Proof. Suppose that ∆ m and ∆ ′ m are equivalent m -gonal forms in n variables.Let Q and Q ′ be the quadratic part of ∆ m and ∆ ′ m respectively. Then Q and Q ′ are equivalent quadratic forms, and hence Q and Q ′ have the samesuccessive minima. However, both Q and Q ′ are diagonal quadratic forms, anda diagonal quadratic form is determined, up to a permutation of the variables, byits successive minima. Therefore, after a suitable permutation of the variables,we may assume that Q = Q ′ . Since an m -gonal form is completely determinedby the coefficients of its quadratic part, we may conclude that ∆ m and ∆ ′ m mustbe equal. (cid:3) As a corollary of Lemma 7.1 and Theorem 6.2, we obtain the followingfiniteness result for regular ternary m -gonal forms which includes the case oftriangular forms as a special case. Corollary 7.2. 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