COMPOSITION OF BINARY QUADRATIC FORMS: FROM GAUSS TO BHARGAVA. 1. Introduction

Transcription

1 COMPOSITION OF BINARY QUADRATIC FORMS: FROM GAUSS TO BHARGAVA FRANÇOIS SÉGUIN Abstract. In is 004 article [], Barava introuces a new way to unerstan te composition law o interal binary quaratic orms trou wat e calls cubes o inteers. Te oal o tis article is to introuce te reaer to Barava s cubes an tis new composition law, as well as to relate it to te composition law as it is known classically. Trou a series o exercises, we will see a istorical exposition o te subject, rom Gauss to Barava, an see ow te ierent ormulations o te composition laws are equivalent. 1. Introuction Binary quaratic orms were alreay stuie in te seventeent an eiteent centuries. Oriinally te main questions ormulate in terms o binary quaratic orms were about representation o numbers. Given a binary quaratic orm x, y = ax + bxy + cy, we can ask wic inteers n can be written as n = x 0, y 0 or some x 0, y 0, in wic case is sai to represent n. Given an inteer n, we can also ask wic binary quaratic orms coul represent n. Finally, iven suc a representation, ow many oter representations can we in. It is interestin to note tat tese very questions will eventually come to play a role in subjects suc as Diopantine equations, quaratic reciprocity an even class iel teory. Matematicians like Fermat an Euler trie to classiy all te possible binary quaratic orms, but te biest results came rom Gauss in 1801 in is Disquisitiones Aritmeticae. Gauss spells out te unerlyin structure tat tose binary quaratic orms ave, structure tat will later came to be known as a roup. Wat is now known as Gauss composition law or binary quaratic orms as been moernize usin alebraic number teory trou te works o Diriclet amonst oters. Tis construction plays a critical role in number teory. In particular, it is one o te primary tools to unerstan class roups o quaratic number iels. Recently, Manjul Barava ormulate a bran new way to approac te composition law or binary quaratic orms. In is tesis an ten a series o articles 010 Matematics Subject Classiication. 11E16. Key wors an prases. Binary quaratic orms, Composition laws. 1

2 [], [3], [4], [5], Barava uses a eometrical approac to escribe te composition law, ivin insit into a new possible way to unerstan it. Barava also eneralizes te composition law to oter objects, establisin a corresponence wit structures in ier eree number iels. In tis article, we will explore ow te composition law came to be unerstoo by Gauss, Diriclet an inally Barava, as well as see ow tese ierent ormulations are really equivalent. We will o so trou a series o exercises esine to elp te reaer to ain a eeper unerstanin o te subject.. Preliminaries We eine an interal binary quaratic orm as a omoenous eree polynomial in two variables, i.e. o te orm F x, y = ax + bxy + cy were a, b, c Z. Since any binary quaratic orm is completely etermine by its tree coeicients a, b an c, we sometimes enote te above polynomial by a, b, c. Te iscriminant o a binary quaratic orm a, b, c is eine as Disca, b, c = b 4ac. Also, we say tat a binary quaratic orm is primitive i te tree coeicients are coprime, i.e. ca, b, c = 1 an positive einite i te binary quaratic orm takes only positive values, i.e. ax + bxy + cy > 0 or all x, y 0. Exercise 1. Sow tat te binary quaratic orm a, b, c is positive einite i an only i Disca, b, c < 0 an a, b > 0. From now on, we will use te term binary quaratic orms to mean interal primitive positive einite binary quaratic orms. Recall tat te roup SL Z is eine as { } a b SL Z := : a, b, c, Z, a bc = 1. c We eine te action o SL Z on a binary quaratic orm as ax + bxy + cy α β = aαx + βy + bαx + βyγx + δy + cγx + δy. γ δ Note tat tis is simply a cane o variable. An alternative way o expressin tis action is, or Qx, y a binary quaratic orm an M SL Z, Qx, y M = Qx, y were x x y = M. y

3 Exercise. Sow tat or any matrix M, Disc a, b, c M = etm Disca, b, c. From tis exercise, we can conclue tat te action o SL Z preserves te iscriminant. Exercise 3. Sow tat any binary quaratic orm tat is equivalent to a primitive binary quaratic orm, is primitive. We now ix a certain inteer D an consier all te possible binary quaratic orms o iscriminant D. On tis set, we eine te ollowin equivalence relation. I Q 1 an Q are binary quaratic orms o iscriminant D, ten Q 1 Q Q 1 M = Q or some M SL Z. We call te set o equivalence classes uner tis relation G D, i.e. G D = {a, b, c : a, b, c Z, Disca, b, c = D} /. We enote [a, b, c] te equivalence class containin a, b, c in G D. We say tat a binary quaratic orm a, b, c is reuce i b a c. Exercise 4. Sow tat tere are initely many reuce binary quaratic orms o a ixe iscriminant D. Ten sow tat every binary quaratic orm is equivalent to a unique reuce binary quaratic orm, wit te exception o a, b, a a, b, a an a, a, c a, a, c were te reuce orm is not unique. Conclue tat or any D, G D is inite. 3. Gauss s composition law We recall te ollowin ientity attribute to 7t century Inian matematician Bramaupta see [1]. Exercise 5. Sow tat or any inteers x 1, y 1, x, y, D, x 1 + Dy1 x + Dy = x1 x Dy 1 y + D x 1 y + x y 1. We can summarize tis previous ientity by sayin tat te numbers o te orm x +Dy are close uner multiplication. In 1801, Gauss aske in is Disquisitiones Aritmeticae weter it was possible to eneralize tis to numbers o a more eneral orm, namely ax + bxy + cy. He comes up wit te answer: yes! 3

4 Teorem 3.1 Gauss. Let a 1 x 1+b 1 x 1 y 1 +c 1 y1 an a x +b x y +c y be binary quaratic orms o iscriminant D. Ten, tere exists an explicit transormation cane o variables x 1 x X p 0 p 1 p p 3 x 1 y = Y q 0 q 1 q q 3 y 1 x y 1 y an inteers A, B an C suc tat a1 x 1 + b 1 x 1 y 1 + c 1 y1 a x + b x y + c y = AX + BXY + CY. Moreover, B 4AC = D. From tis, it is easy to conclue te ollowin. Exercise 6. Usin Teorem 3.1, sow tat G D is a inite abelian roup. It is wort notin tat te moern notion o a roup i not exist wen Gauss wrote is Disquisitiones. However, it is clear tat, witout usin our moern terms, tis is really wat e was ater. 4. Diriclet s unite orms We say tat two binary quaratic orms a 1, b 1, c 1 an a, b, c o iscriminant D are unite i c a 1, a, b 1 + b = 1. Exercise 7. Sow tat b 1 + b is always even. Proposition 4.1. I a 1, b 1, c 1 an a, b, c are unite orms, ten tere exist inteers B an C suc tat an a 1, b 1, c 1 a 1, B, a C a, b, c a, B, a 1 C. Exercise 8. I a 1, b 1, c 1 an a, b, c are unite orms, sow usin Proposition 4.1 an Teorem 3.1 tat in te roup G D, [a 1, b 1, c 1 ] [a, b, c ] = [a 1 a, B, C]. Exercise 8 allows us to compute some roup operations in a more eicient way tan wat is iven by Teorem 3.1. Exercise 9. Usin exercise 8, sow tat 4

5 [ ] 1, 0, D GD = [ ] 1, 1, 1 D 4 i D 0 mo 4 i D 1 mo 4 [a, b, c] 1 = [a, b, c] = [c, b, a]. Diriclet went on to use te notion o ieals in quaratic number iels to obtain te moern ormulation o te composition law, relyin on te ollowin teorem. Recall tat te narrow class roup o a number iel K, enote Cl + K, is te set o all te interal ieals o K moulo te principal ieals o positive norm. Teorem 4.. Te roup G D is isomorpic to te narrow class roup o K = Q D. More speciically, tere is an explicit isomorpism tat allows us to compute compositions o binary quaratic orms. To eac binary quaratic orm, we associate an ieal o O K in te ollowin way: a, b, c Φ az + b D Z. Conversly, or every ieal o O K, we associate a binary quaratic orm αx + βyαx + βy NA Ψ αz + βz = A were conjuation is eine by senin D to D. Exercise Sow tat Φ maps equivalent binary quaratic orms in G D to narrowly equivalent ieals in Cl + K. Sow tat Ψ maps narrowly equivalent ieals in Cl + K to equivalent binary quaratic orms in G D. 3 Sow tat Φ an Ψ are inverses o eac oter. Uner tis corresponence above, te composition o binary quaratic orms in G D correspons to te multiplication o ieals in Cl + K. Tereore, iven two binary quaratic orms o iscriminant D, say Q 1 an Q, we can compute Q 1 Q by 1 Finin ΦQ 1 an ΦQ, Finin a -elements basis or te ieal ΦQ 1 ΦQ as a Z-moule, say [α, β] one always exists since O K is a Deekin omain, 3 Computin Ψ[α, β], an inin a reuce representative i necessary. 5

6 Exercise 11. Let a 1, b 1, c 1 an a, b, c be unite orms. aloritm above tat Sow usin te [a 1, b 1, c 1 ] [a, b, c ] = [a 1 a, B, C] were B an C are te inteers rom Proposition Barava s cube Consier te ollowin cube o inteers, tat is a cube wit an inteer at every corner. e a b c Fiure 1. Barava s cube o inteers In Fiure 1, a, b, c,, e,,, are all inteers. We can cut te cube to obtain two squares in tree ways. 1 Front-Back: e a b c We ten eine te ollowin two matrices usin te two squares we et. a b e M 1 := N 1 :=. c We o te same or te next two cuts, rotatin te wole cube so tat a is always te top let entry o te irst matrix. 6

7 Let-Rit: e a b M := a e c N := b. 3 Up-Down: c e a b M 3 := a b e N 3 := c. c Now, we eine an action o te roup Γ = SL Z SL Z SL Z on te space C o all cubes o inteers. r s I γ Γ an is its it actor, 1 i 3, ten its action on a cube t u replaces M i, N i wit rm i + sn i, tm i + un i. In oter wors, eac actor o γ perorms a ace operation on te cube, similar to wat we woul o on a matrix wit row an column operations. Te irst actor perorms a ace operation on te ront an back aces, te secon actor on te let an rit aces, an te last on te up an own aces. Exercise 1. Sow tat te action o eac actor o γ Γ commutes wit eac oter. Given a cube C, we eine tree binary quaratic orms as or 1 i 3. Q C i = etm i x N i y Example 5.1. I we take C to be te cube o Fiure 1, ten Q C ax ey bx y 1 = et, cx y x y wic inee ives a binary quaratic orm. We say tat a cube C is projective i Q C 1, Q C, Q C 3 orms. are primitive binary quaratic Exercise 13. Sow tat or any cube C, Disc Q C 1 = Disc Q C = Disc Q C 3. 7

8 It tereore makes sense to eine DiscC or a cube C as te iscriminant o its associate binary quaratic orms. We now examine te action o Γ on tese newly eine binary quaratic orms. Exercise 14. Sow tat {1} SL Z SL Z acts trivially on Q C 1. Exercise 15. Sow tat γ = M 1 1 Γ acts on Q C 1 in te usual way, tat is Q C γ 1 = Q C 1 M. By symmetry, te above two exercises ol or te it actor o Γ an Q C i, 1 i 3. In particular, note tat we can conclue te ollowin corollary. Corollary 5.1. DiscC is invariant uner te action o Γ. From tere, we can impose a roup structure on te set o primitive binary quaratic orms o iscriminant D by eclarin tat or any triplet Q A 1, Q A, Q A 3 arisin rom a cube A o iscriminant D, Q A 1 + Q A + Q A 3 = 0. In oter wors we mo out tis relation on te ree abelian roup enerate by all binary quaratic orms o iscriminant D. Note tat we obtain te SL Z equivalence o binary quaratic orms or ree rom tis einition. Inee, i γ = M 1 1 Γ, Q A 1 + Q A + Q A 3 = 0 = Q A γ 1 + Q A γ + Q A γ 3 = Q A 1 M + Q A + Q A 3 Q A 1 = Q A 1 M. As it turns out, Barava prove te ollowin teorem Teorem 5. [, Tm 1]. Tere exists a projective cube A o iscriminant D wit Q A 1, Q A, Q A 3 i an only i [ ] [ ] [ ] Q A 1 Q A 1 Q A 1 = 1GD an tat cube is unique up to Γ-equivalence. We can ten sow tat tis composition law arees wit our previously eine composition. Start wit a projective cube C. e a b c 8

9 Since C is projective, we can sow tat ca, b, c,, e,,, = 1. As suc, we can in a Γ-equivalent cube suc tat a = 1. We ten use tis entry to clear entries b, c an e usin ace operations aain. We tereore ave te equivalent cube C Te tree binary quaratic orms associate to C are Te cube law tells us tat Q 1 = x + xy + y Q = x + xy + y Q 3 = x + xy + y [Q 1 ] [Q ] = [Q 3 ] 1. Let us now use Diriclet s unite orm to arrive at te same conclusion. Recall tat rom Exercise 9, Also, rom Exercise 8, we et [Q 3 ] 1 = [,, ]. [Q 1 ] [Q ] = [,, ] [,, ] = [,, ] = [Q 3 ] 1 by lettin a 1 =, a =, B = an C =. Tus, we conclue tat te two composition laws are equivalent. 6. Concluin remarks One o te most remarkable an surprisin aspect o tis new approac to te composition law mit be ow it eneralizes to ier compositions. Altou it was only briely mentione in te rest o tis article, Barava escribes in [] ow e eneralizes te cube law presente ere to retrieve some alebraic structure in ier eree number iels. Te corresponence between binary quaratic orms an te narrow class roup o quaratic iels was alreay well known, but ettin a anle on oter types o alebraic structures or ier eree is very important in alebraic number teory. In particular, see [], [3], [4], [5] or eneralizations. For a more involve introuction to te topic, we reer te reaer to [1]. Article 9

10 [11] also provies a oo introuction o te results arisin rom tis new teory o ier compositions. 7. Acknowlements I woul like to tank Proessor M. Ram Murty an Sii Patak or teir elpul comments an suestions on a previous version o tis article. 8. Appenix - Hints an solutions or exercises Exercise 1. I x, y = ax + bxy + cy o iscriminant D, ten 4ax, y = ax + by Dy. Exercise. Note tat or x, y = ax + bxy + cy o iscriminant D, a b/ x x, y = x y b/ c y an Tereore, x, y M = a b/ Discx, y = 4 b/ c. x an computin te eterminant we are one. y M T a b/ x M b/ c y Exercise 3. Consier te set o inteers tat can be written as Qx, y or a binary quaratic orm Q an x, y Z. Q is non-primitive i an only i tose inteers are all multiple o some N consier x, y = 1, 0, 0, 1 an 1, 1 or te reverse implication. However, since equivalence o binary quaratic orms is iven by a simple invertible cane o variables, any equivalent binary quaratic orms represent te same inteers. Exercise 4. Recall tat two enerators or SL Z are T = an S = Usin T n, we can sow tat a, b, c a, b + an, c 8.1 Note tat c is uniquely etermine rom te irst two entries by te act tat te iscriminant is ixe. Here, c = an + bn + c. Usin S, we ave a, b, c c, b, a

11 Usin 8.1 we can in a representative wit b a, an usin 8. we can insure tat a c. Uniqueness ollows rom te act tat T an S are enerators or SL Z, an any transormation can be expresse in terms o tese two. Exercise 5. Note tat Nx + y D = x + Dy. Tereore, x 1 + Dy1 x + Dy = N x 1 + y 1 D N x + y D = N x 1 x Dy 1 y + x 1 y + x y 1 D by te multiplicativity o te norm, an so = x 1 x Dy 1 y + D x 1 y + x y 1. Exercise 6. We nee to sow tat te composition law eine tis way is well x 1 x 1 eine on G D. Inee, te two canes o variables y 1 = M 1 an y 1 = M x y x y correspon to te cane o variable x 1x x 1 x x 1y y 1x = M x 1 y 1 M y 1 x y 1y y 1 y so te matrix p 0 p 1 p p 3 p 0 p 1 p p 3 q 0 q 1 q q 3 = M 1 M q 0 q 1 q q 3 will yiel te correct cane o variable or te multiplication o Q 1 x 1, y 1 an Q x, y. Exercise 7. D = b 1 4a 1 c 1 = b 4a c an so b 1 b mo 4 an b 1 b mo. Exercise 8. Consier te matrix C 0 a 1 a B in Teorem 3.1. Exercise 9. 1 We consier te case D 0 mo 4. Consier a, b, c o iscriminant D, we want to compute [a, b, c] [ ] 1, 0, D 4. Note tat b is even, say b = n. Usin notation rom te solution o Exercise 3, we can use T n to ave [ ] 1, 0, D 4 = [1, n, c ] = [1, b, c ]. Te last coeicient c is entirely etermine by 11

12 te irst two entries, an ere c = b D 4. Also, by te same reasonin c = b D 4a an so c = ac. So we ave [ [a, b, c] 1, 0, D ] = [a, b, c] [1, b, ac] = [a, b, c] 4 by Exercise 8. Case D 1 mo 4 is similar. We want to compute [a, b, c][c, b, a]. By exercise 8 wit B = b an C = 1, we et [a, b, c][c, b, a] = [ac, b, 1] = [1, b, ac] an applyin T n or a suitable n will retrieve te ientity rom te previous part. Exercise 10. See [8, Teorem 6.0] or te complete proo. Exercise 11. From Proposition 4.1, a 1, b 1, c 1 a 1, B, a C an a, b, c a, B, a 1 C or some B an C. Clearly ca 1, a, B = 1 see Exercise 3. Ten, we compute A = Φ a 1, B, a C Φ a, B, a 1 C B = [a D B D 1 a, a 1, a as a Z-basis o A. replace by B B D, B + D B ] D 4 Also, writin D = B 4a 1 a C, te last enerator can be. Finally, since ca 1, a, B = 1, we can in a linear combination o tem tat equals 1. As suc, we can easily see [ a 1 a, B ] D B [a D B D B ] D 1 a, a 1, a, B an since te reverse inclusion is trivial, we ave [ [a 1, B, a C][a, B, a 1 C] = Ψ a 1 a, B ] D = [a 1 a, B, C]. Exercise 1. Tis can be viewe simply as te analoue o row an column operation on matrices commutin wit eac oter. A moment s relection soul make tis clear. Exercise 13. From example 5.1, we can in a ormula or te iscriminant o as Q C 1 DiscQ C 1 = c +e +b +a ae+c+b bc+ce+be+bc 1

13 Ten, we notice tat we can retrieve M an N rom M 1 an N 1 by permutin te elements a, b, c,, e,,, as ollows: a b c e a c e b or in cycle notation ab c e. Actually, we notice tat we can also retrieve M 3 an N 3 rom M an N usin te same permutation. Finally, note tat te ormula above or te iscriminant is invariant uner tis permutation. Exercise 14. Let γ {1} SL Z SL Z act on te cube C. Suppose G is te secon actor o γ an G 3 te tir. Ten note tat G acts on M 1 an N 1 by column operation. Speciically, M 1 an N 1 become M 1 G an N 1 G respectively. On te oter an, G 3 act on M 1 G an N 1 G by row operations, an tey become G 3 M 1 G an G 3 M 1 G an G 3 N 1 G respectively. Tereore, we ave tat Q C γ 1 = et G 3 M 1 G x G 3 N 1 G y = etg 3 et M 1 x N 1 y etg = Q C 1. r s Exercise 15. Tis time, let γ = 1 1 act on C. Ten, t u Q C γ 1 x, y = et rm 1 + sn 1 x tm 1 + un 1 y = et M 1 rx ty N 1 uy sx = Q C 1 rx ty, uy sx = Q C r t 1. s u Reerences [1] Belabas, Karim. Paramétrisation e structures alébriques et ensité e iscriminants [ après Barava]. Seminaire Bourbaki Vol Astérisque No Exp. No. 935, ix, [] Barava, Manjul. Hier composition laws I: A new view on Gauss composition, an quaratic eneralizations. Annals o Matematics, : [3] Barava, Manjul. Hier composition laws II: On cubic analoues o Gauss composition. Annals o Matematics, : [4] Barava, Manjul. Hier composition laws III: Te parametrization o quartic rins. Annals o Matematics, : [5] Barava, Manjul. Hier composition laws IV: Te parametrization o quintic rins. Annals o Matematics, : [6] Barava, Manjul. Hier Composition Laws, P. D. Tesis, Princeton University, June 001. [7] Bucmann, Joannes; Vollmer, Ulric. Binary Quaratic Forms : An Aloritmic Approac. Berlin: Spriner-Verla, 007. [8] Buell, D.A. Binary Quaratic Forms : Classical Teory an Moern Computations. New York: Spriner-Verla, [9] Diriclet, P.G.L. Zalenteorie, 4t. eition, Viewe Brunswick,

14 [10] Gauss, C.F. Disquisitiones Aritmeticae, 1801 [11] Sankar, Arul; Wan, Xiaoen. Laws o composition an aritmetic statistics: From Gauss to Barava. Te Matematics Stuent, 84 nos , [1] Weil, Anré. Number Teory: An approac trou istory rom Hammurapi to Leenre. Basel: Birkäuser, 001. Department o Matematics, Queen s University, Kinston, Ontario K7L 3N6, Canaa. aress: rancois.seuin@queensu.ca 14