Multiplicative Properties of Integral Binary Quadratic Forms

Transcription

1 Southern Illinois University Carbondale OpenSIUC Articles and Preprints Department o Mathematics 2008 Multiplicative Properties o Integral Binary Quadratic Forms A. G. Earnest Southern Illinois University Carbondale, aearnest@math.siu.edu Robert W. Fitzgerald Southern Illinois University Carbondale, ritzg@math.siu.edu Follow this and additional works at: Part o the Number Theory Commons To appear in Contemporary Mathematics. Recommended Citation Earnest, A. G. and Fitzgerald, Robert W. "Multiplicative Properties o Integral Binary Quadratic Forms." ( Jan 2008). This Article is brought to you or ree and open access by the Department o Mathematics at OpenSIUC. It has been accepted or inclusion in Articles and Preprints by an authorized administrator o OpenSIUC. For more inormation, please contact opensiuc@lib.siu.edu.

2 Contemporary Mathematics Multiplicative Properties o Integral Binary Quadratic Forms A.G. Earnest and Robert W. Fitzgerald Abstract. In this paper, the integral binary quadratic orms or which the set o represented values is closed under k-old products, or even positive integers k, will be characterized. This property will be seen to distinguish the elements o odd order in the orm class group o a ixed discriminant. Further, it will be shown that this closure under k-old products can always be expressed by a k- linear mapping rom (Z 2 ) k to Z 2.Inthecasek = 2, this resolves a conjecture o Aicardi and Timorin. 1. Preliminaries Throughout this paper, the term orm will always reer to a nondegenerate integral binary quadratic orm ax bx 1x 2 + cx 2 2, which will be denoted simply by (a, b, c). For a orm, letd() denote the set o values represented by. The discriminant o =(a, b, c) isδ = b 2 4ac 0. It will be assumed here that all orms under consideration are either positive deinite (i Δ < 0) or indeinite (i Δ > 0). Two orms and g are equivalent, denoted g, i there is an integral transormation o determinant +1 taking one orm to the other. For a orm, [] will denote the set o all orms equivalent to. Aorm(a, b, c) is said to be primitive i g.c.d.(a, b, c) = 1. Classical Gaussian composition induces a binary operation on the equivalence classes o primitive orms o a ixed discriminant. For our purposes, the salient eature o the composition operation is that or primitive orms, g o the same discriminant Δ, there exist primitive orms ˆ,ĝ and h o discriminant Δ, and a bilinear mapping σ : Z 2 Z 2 Z 2 such that there is an identity o the type (1.1) ˆ(x)ĝ(y) =h(σ(x, y)), or all x, y Z 2. In this case, we will write [][g] =[h]. Under this operation, the set o equivalence classes o primitive orms o a ixed discriminant Δ is a inite abelian group, called the orm class group o discriminant Δ, which will be denoted by C Δ. The identity element o C Δ is the class id Δ consisting o the orms that represent 1. I =(a, b, c), then [] 1 =[ op ], where op =(a, b, c). A detailed description o the composition operation can be ound, or example, in [6]. A resh 2000 Mathematics Subject Classiication. Primary 11E16; Secondary 11E12, 11R29. 1 c 2008 American Mathematical Society

3 2 A.G. EARNEST AND ROBERT W. FITZGERALD perspective is given in the pioneering work o Bhargava [5], which has opened up new directions or broad generalizations o the classical theory. For a orm, the notation D([]) will denote the set D(g) or any g []. I and g are primitive orms that represent the integers k and l, respectively, then it can be seen rom (1.1) that the orms in the equivalence class [][g] representthe product kl; thatis, (1.2) D()D(g) D([][g]). Let be a primitive orm. Note that D( op ) = D(), since op (x 1,x 2 ) = (x 1, x 2 ). So D()D()D() = D()D( op )D() = D([])D([] 1 )D([]) D([]), where the inal containment ollows rom (1.2). So (1.3) D()D()D() D() or all primitive orms. That is, the three-old product o integers represented by is again an integer represented by. That this property extends to all, not necessarily primitive, orms can be seen by writing = c 0 where 0 is primitive and applying (1.3) to 0. This property was observed by Arnold [4], who reers to it as the tri-group property. In act, this property appears in an earlier paper o Goins [8], where it is derived rom a triple product ormula or certain 2 2 matrices. Moreover, it is shown in both [4] and[8] that there exists a 3-linear mapping σ : Z 2 Z 2 Z 2 Z 2 such that (1.4) (x)(y)(z) =(σ(x, y, z)) or x, y, z Z Background The classical identity (2.1) (x dx2 2 )(y2 1 + dy2 2 )=(x 1y 1 + dx 2 y 2 ) 2 + d(x 1 y 2 x 2 y 1 ) 2 shows that certain orms (in this case, those o the type =(1, 0,d)) have the property that their represented value set D() is closed under products (that is, D() orms a multiplicative semigroup). Arnold initiated the systematic study o orms with this property, which he reerred to as perect orms, in [4]. In subsequent papers [1], [2] and[3], Aicardi and Timorin have investigated several related conditions that produce such orms. In [7], we have shown that the primitive orms or which D() is closed under products are precisely those or which [] 3 =1in C Δ. The results or primitive orms are summarized in the ollowing statement. Proposition 2.1. Let be a primitive orm o discriminant Δ. The ollowing are equivalent: (1) D() is closed under products. (2) [] 3 =1in C Δ. (3) There exist α, β, γ, δ Z such that =(α 2 γδ, αγ βδ, γ 2 αβ). (4) There exists a bilinear mapping σ : Z 2 Z 2 Z 2 such that (x)(y) =(σ(x, y)) or all x, y Z 2.

4 MULTIPLICATIVE PROPERTIES OF INTEGRAL BINARY QUADRATIC FORMS 3 Sketch o proo. The equivalence o (1) and (2) appears in Corollary 2.4 o [7]. (2) (3) can be deduced by direct computation using the characterization o composition given by Bhargava [5]. (3) (4) ollows by considering the mapping σ : Z 2 Z 2 Z 2 deined coordinatewise by the equations σ(x, y) 1 = αx 1 y 1 + γx 1 y 2 + γx 2 y 1 + βx 2 y 2 and σ(x, y) 2 = δx 1 y 1 αx 1 y 2 αx 2 y 1 γx 2 y 2. (4) (1) is clear. For convenience, we will reer to a (not necessarily primitive) orm as being multiplicative, parametrizable or normed i the condition (1), (3) or (4) o Proposition 2.1, respectively, is satisied or. As the argument or (3) (4) above does not depend on primitivity, the ollowing implications hold in general: (2.2) parametrizable normed multiplicative. Proposition 2.1 shows that the converses o both o these implications are also true when we restrict to primitive orms. We will see in section 4 that the converse o the second implication is always true (see Theorem 4.2), as conjectured by Aicardi and Timorin [3]. However, an example appearing in section 5 shows that the converse o the irst implication is not true in general. In the rest o this paper, a general orm will be written as = c 0,where c is the g.c.d. o the coeicients o and 0 is primitive. The main result o [7] is: Theorem 2.2. is multiplicative i and only i c D( 0 ) D([ 0 ] 3 ). From this result, we obtain the equivalence o the three conditions in (2.2) or the case o diagonal orms. Corollary 2.3. Let be a diagonal orm. The ollowing are equivalent: (1) is multiplicative. (2) c D( 0 ). (3) is parametrizable. (4) is normed. Proo. (1) (2): Since 0 is diagonal, op 0 = 0 and so [ 0 ] 3 =[ 0 ][ op 0 ][ 0]= [ 0 ][ 0 ] 1 [ 0 ] = [ 0 ]. (2) (3): Let 0 = (a, 0,c). Since c D( 0 ), there exist u, v Z such that c = au 2 + cv 2. Taking α = au, β = cu, γ = cv, δ = av produces the desired parametrization o. The remaining implications are clear. In the remaining three sections o this paper, we will consider each o the properties in (2.2) in more detail. The discussion o multiplicative and normed orms will be set in the more general context o k-old products or arbitrary nonnegative even integers k. 3. Multiplicative orms Throughout this section, k and l will denote nonnegative integers. Deinition. Aorm is k-multiplicative i a 1,a 2,...,a k D() = a 1 a 2 a k D(). When k = 0 we take the empty product to be 1. Thus 0-multiplicative simply means 1 D().

5 4 A.G. EARNEST AND ROBERT W. FITZGERALD Proposition 3.1. (1) I is k-multiplicative then is (k +2)-multiplicative. (2) Every orm is k-multiplicative or each odd k. Proo. (1) For any a 1,a 2,...,a k+2 D(), a 1 a 2 a k D(), since is k- multiplicative. Then (a 1 a 2 a k )a k+1 a k+2 is a product o three elements o D() and so is in D() by (1.3). (2) Each is 1-multiplicative by deinition. Apply (1). Thus, in the remainder o this paper we will only be interested in the property k-multiplicative when k is even. The main theorem characterizing orms with this property is the ollowing, which generalizes Theorem 2.3 o our previous paper [7]. Theorem 3.2. Let k be even. The ollowing are equivalent: (1) is k-multiplicative. (2) There exists a prime p with p D( 0 ) and c k pk D(). (3) c k 1 D([ 0 ] l+1 ), or some even l, 0 l k. The main step in the proo o this theorem is contained in the ollowing lemma. Lemma 3.3. Let k be even. Suppose 0 is a primitive orm, p is a prime, and d Z with p D( 0 ) and dp k D( 0 ). Then d D([ 0 ] l+1 ) or some even l, 0 l k. For the proo o this lemma, it is convenient to recall the key lemma (Lemma 2.2) o our previous paper [7]. Lemma 3.4. Let g and h be primitive integral binary quadratic orms o the same discriminant Δ, letp be an odd prime and n an integer. I p D(g) and np D(h), then either n D([g][h]) or n D([g op ][h]). Proo o Lemma 3.3. The result is clear i k = 0 so suppose k>0. We can assume that k is the least positive, even integer with dp k D( 0 ). Claim: dp k j D([ 0 ] j+1 ), or 0 j k. We prove this by induction; the case j = 0 is our hypothesis. Say j > 0 and suppose dp k j D([ 0 ] j+1 ). Lemma 3.4, with g = 0,[h] =[ 0 ] j+1 and n = dp k j 1,givesdp k j 1 D([ 0 ] j+2 ) D([ 0 ] j ). I dp k (j+1) D([ 0 ] j+2 )then we have completed the induction argument and we are done. So suppose dp k j 1 D([ 0 ] j ). Let m be the least positive integer such that dp k+m 2j 1 D([ 0 ] m ). Note that this occurs i m = j. And i m = 1 then we have contradicted the minimality o k (as j > 0). Hence 1 <m j. Lemma 3.4, with g = 0,[h] =[ 0 ] m and n = dp k+m 2j 2,gives dp k+m 2j 2 D([ 0 ] m+1 ) D([ 0 ] m 1 ). Now dp k+(m 1) 2j 1 D([ 0 ] m 1 ) contradicts the minimality o m. have: Hence we dp k (j+1) = dp j m+1 p k+m 2j 2 D([ 0 ] j m+1 )D([ 0 ] m+1 ) D([ 0 ] j+2 ), which completes the induction proo o the Claim. Taking j = k in the Claim gives d D([ 0 ] k+1 ).

6 MULTIPLICATIVE PROPERTIES OF INTEGRAL BINARY QUADRATIC FORMS 5 Proo o Theorem 3.2. (1) (2) is clear: 0 represents a prime p since 0 is primitive; take each a i = c p in the deinition. (2) (3) is Lemma 3.3, as c k 1 p k D( 0 ). For (3) (1), let c a i D() or1 i k. Let 2s = k l. Then k a i =(a 1 a 2 a 3 ) (a 3s 2 a 3s 1 a 3s )a 3s+1 a 3s+2 a k i=1 is in D([ 0 ] s+(k 3s) )=D([ 0 ] l )=D([ 0 ] l ), where we have used (1.3) to conclude that each product o three a i s is again in D( 0 ). Hence, by (3), and so c k 1 This completes the proo. k a i D([ 0 ] l+1 )D([ 0 ] l ) D( 0 ) i=1 k (c a i ) c D( 0 )=D(). i=1 When is primitive (and so c = 1), condition (3) says [] l+1 =1. Wethus get: Corollary 3.5. Let be a primitive orm o discriminant Δ and let k be even. The ollowing are equivalent: (1) is k-multiplicative. (2) There is a prime p D() with p k D(). (3) The order o [] C Δ is odd and at most k +1. Deinition. Let k be even. A orm is strictly k-multiplicative i is k- multiplicative but not l-multiplicative or any even l, 0 l<k. Corollary 3.6. Let be a primitive orm o discriminant Δ and let k be even. The ollowing are equivalent: (1) is strictly k-multiplicative. (2) There is a prime p D() such that p k D() but p l / D() or even l, 0 l<k. (3) The order o [] C Δ is k Normed orms Throughout this section, n will denote a positive integer. To simpliy notation, let V = Z Z. Amapσ : V n V is n-linear i it is linear in each coordinate. Deinition. Aorm is n-normed i there exists n-linear σ : V n V such that (v 1 )(v 2 ) (v n )=(σ(v 1,v 2,...,v n )), or all v 1,v 2,...,v n V. Note that a orm is 2-normed by this deinition i and only i it is normed, in the terminology introduced in section 2. For example, the identity (2.1) shows that orms o the type (1, 0,d) are 2-normed.

7 6 A.G. EARNEST AND ROBERT W. FITZGERALD Lemma 4.1. (1) Every orm is 3-normed. (2) Suppose 1, 2,..., n,gare primitive orms o discriminant Δ. I n i=1 [ i]= [g] in C Δ then there exists n-linear σ : V n V such that 1 (v 1 ) 2 (v 2 ) n (v n )=g(σ( 1,v 2,...,v n )), or all v 1,v 2,...v n V. Proo. (1) This ollows rom (1.4). (2) We use induction on n. When n =1wehave 1 g. So there is M SL 2 (Z) such that 1 (v) =g(mv) or all v Z (viewed as a column vector). Set σ(v) =Mv. Let h be a primitive orm such that n 1 i=1 [ i]=[h]. By induction, there is (n 1)-linear τ such that 1 (v 1 ) 2 (v 2 ) n 1 (v n 1 )=h(τ(v 1,v 2,...,v n 1 )). We have [ n ][h] =[g] inc Δ. By (1.1) there exist orms n [ n ], h [h] and g [g] and a bilinear mapping γ : V 2 V such that n(v n )h (w) =g (γ(v n,w)). And there exist isometries β i : V V such that (v) = n(β 1 (v)), h(v) =h (β 2 (v)) and g(v) =g (β 3 (v)), or all v V.Then n (v n )h(w) =g(β 3 (γ(β 1 (v n ),β 2 (w)))). Let ν : V 2 V be given by ν(v, w) =β 3 (γ(β 1 (v),β 2 (w))). Clearly ν is bilinear. We obtain: 1 (v 1 ) 2 (v 2 ) n 1 (v n 1 ) n (v n )=(v n )h(τ(v 1,v 2,...,v n 1 )) = g(ν(v n,τ(v 1,v 2,...,v n 1 )). Clearly σ(v 1,v 2,...,v n )=ν(v n,τ(v 1,...,v n 1 )) is n-linear. Theorem 4.2. Let k 2 be an even integer. A orm is k-multiplicative i it is k-normed. Proo. Clearlyk-normedimpliesk-multiplicative. So suppose isk-multiplicative. By Theorem 3.2, c k 1 D([ 0 ] l+1 ), or some even l k. Pick g [ 0 ] l+1 and suppose g(u) =c k 1,whereu V.Writek l =2s. Now by Lemma 4.1 (1), there exists 3-linear β such that 0 (v 1 ) 0 (v 2 ) 0 (v 3 )= 0 (β(v 1,v 2,v 3 )). For v =(x, y) V,letv =(x, y). Now [g op ][ 0 ] s [ 0 ] k 3s =[ 0 ] (l+1) [ 0 ] l =[ op 0 ], in C Δ. Hence, by Lemma 4.1 (2), there exists (k 2s + 1)-linear τ such that We have: g op (z) s 0 (w j ) j=1 k i=3s+1 c k 1 (v 1 )(v 2 ) (v k )=g op (u ) (v i )= op 0 (τ(z, w 1,...w s,v 3s+1,...v k )). s 0 (β(v 3j 2,v 3j 1,v 3j )) j=1 k i=3s+1 0 (v i ) = 0 (τ(u,β(v 1,v 2,v 3 ),...,β(v 3s 2,v 3s 1,v 3s ),v 3s+1,...,v k ) ).

8 MULTIPLICATIVE PROPERTIES OF INTEGRAL BINARY QUADRATIC FORMS 7 Let σ map (v 1,v 2,...,v k )to τ(u,β(v 1,v 2,v 3 ),...,β(v 3s 2,v 3s 1,v 3s ),v 3s+1,...,v k ). Clearly σ is k-linear. We have c k 1 0 (v 1 ) 0 (v k )= 0 (σ(v 1,...,v k )). Multiply by c to get: (v 1 )(v 2 ) (v k )=(σ(v 1,v 2,...,v k )), showing that is k-normed. Remark. Specialized to the case k = 2, Theorem 4.2 establishes the truth o the Conjecture 0.1 o [3]. 5. Parametrizable orms Aicardi and Timorin [3] characterize all the orms and bilinear pairings σ or which the identity (x)(y) =(σ(x, y)) holds or all x, y V. These all into our types, which are enumerated in Theorem 1.1 o [3]. An examination o this result shows that in the irst three cases the orms are o the type rg where r D(g),andintheremainingcasetheorm is parametrizable. From this we conclude that i a orm is multiplicative but not parametrizable, then it must be o the type rg with r D(g). We are thus led to urther investigate the parametrizability o orms o the type r or r D(). For orms o this type, we give criteria or parametrizability in terms o the solutions o (u, v) =ra 2. Proposition 5.1. Let =(a, b, c) and r D(). Then r is parametrizable i and only i there exist α, δ Z such that (α, δ) =ra 2 and either: (1) δ 0, δ (α 2 ra), andδ 2 (α 3 raα + rbδ); or (2) δ =0, α rb, andα 3 (r 2 b 2 rcα 2 ). Proo. Suppose that r is parametrizable. So there exist α, β, γ, δ Z such that: (5.1) ra = α 2 γδ, rb = αγ βδ and rc = γ 2 αβ. Assume irst that δ 0. Solvingorγ in the irst equation o (5.1) gives γ = α2 ra, δ and it ollows that δ (α 2 ra). The second equation gives β = αγ rb δ = α(α2 ra) rbδ δ 2 = α3 raα rbδ δ 2, giving δ 2 (α 3 raα rbδ). The third equation o (5.1) then becomes rc =( α2 ra ) 2 α( α3 raα rbδ ), δ δ 2 rom which it ollows that ra 2 = aα 2 = bαδ + cδ 2 = (α, δ). For the converse, assume that (α, δ) =ra 2 and use the above expressions or β and γ. It is straightorward to veriy that the equations in (5.1) hold.

9 8 A.G. EARNEST AND ROBERT W. FITZGERALD Now suppose that (5.1) holds with δ = 0. The irst equation o (5.1) gives (α, δ) =(α, 0) = aα 2 = ra 2. The second equation o (5.1) becomes rb = αγ and so α rb. Substituting this expression or γ into the third equation o (5.1) and solving or β yields β = r2 b 2 rcα 2, α 3 and hence α 3 (r 2 b 2 rcα 2 ), as claimed. The converse again ollows by direct substitution o the above expressions or β and γ into (5.1) and using the condition that (α, 0) = ra 2. This proposition makes it easy to analyze the ollowing example, which shows that the converse o the irst implication in (2.2) does not hold in general. Example. The orm (4, 2, 12) is not parametrizable. To see this, apply Proposition 5.1 with r =2and =(2, 1, 6). The only representations o ra 2 =8 by the orm are (±2, 0). Condition (2) o Proposition 5.1 is not satisied or either o these, since α 3 = ±8 andr 2 b 2 rcα 2 = 44. When examining the multiples o a ixed primitve orm by represented values r, it generally happens that r is parametrizable or some values o r but not others. However, it can never be the case that or a given primitive orm there exist no values o r or which r is parametrizable. Corollary 5.2. For any orm, there exist inintely many r D() such that r is parametrizable. Proo. By replacing with an equivalent orm i necessary, we can assume that =(a, b, c) witha 0. Taker 0 = (a, a). It is easily checked that the conditions o Proposition 5.1 (1) are satisied. So r 0 is parametrizable. For any integers s 0,r 0 s 2 is also parametrizable (replace each parameter ρ by sρ). On the other hand, the ollowing result shows that it can happen that r is parametrizable or all r D(). Corollary 5.3. I =(a, b, c) and a b, thenr is parametrizable or every r D(). Proo. For r = (u, v), take α = au and δ = av. Case 1: v 0. Inthiscase α 2 ra = a 2 u 2 a(au 2 + buv + cv 2 )= av(bu + cv) = δ(bu + cv). So δ (α 2 ra). Further, α 3 raα + rbδ =(au) 3 (au 2 + buv + cv 2 )a(au)+(au 2 + buv = cv 2 )b(av) = a 2 cuv + abuv 2 + abcv 3. The last expression in the previous line is divisible by δ 2 = a 2 v 2 since a b; hence, δ 2 α 3 raα + rbδ and the conditions o Proposition 5.1 (1) are satisied. Case 2: v =0. Inthiscase,r = (u, 0) = aα 2. So rb =(aα 2 )b is divisible by α, and r 2 b 2 rcα 2 =(aα 2 ) 2 b 2 (aα 2 )cα 2 = α 4 (a 2 αb 2 ac) is divisible by α 3. Hence, the conditions o Proposition 5.1 (2) are satisied.

10 MULTIPLICATIVE PROPERTIES OF INTEGRAL BINARY QUADRATIC FORMS 9 Reerences [1] F. Aicardi, On the number o perect binary quadratic orms, Experiment. Math. 13 (2004), [2] F. Aicardi, On trigroups and semigroups o binary quadratic orms values and o their associated linear operators, Moscow Math. J. 6 (2006), [3] F. Aicardi and V. Timorin, On binary quadratic orms with semigroup property Proc. Steklov Inst. (Dedicated to the 70th birthday o V.I. Arnold) 258 (2007), [4] V.I.Arnold,Arithmetics o binary quadratic orms, symmetry o their continued ractions and geometry o their de Sitter world Bull. Braz. Math. Soc. 34 (2003), [5] M. Bhargava, Higher composition laws. I. A new view on Gauss composition, and quadratic generalizations Ann. o Math. 159 (2004), [6] D.A.Cox,Primes o the orm x 2 +ny 2. Fermat, class ield theory and complex multiplication John Wiley & Sons, New York, [7] A.G. Earnest and R.W. Fitzgerald, Represented value sets o integral binary quadratic orms Proc.Amer.Math.Soc.135 (2007), [8] E. Goins, A ternary algebra with applications to binary quadratic orms Contemp. Math. 284 (2001), Department o Mathematics, Southern Illinois University Carbondale Current address: Department o Mathematics, Mailcode 4408, Southern Illinois University, 1245 Lincoln Drive, Carbondale, Illinois address: aearnest@math.siu.edu Department o Mathematics, Southern Illinois University Carbondale Current address: Department o Mathematics, Mailcode 4408, Southern Illinois University, 1245 Lincoln Drive, Carbondale, Illinois address: ritzg@math.siu.edu