REGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction

Transcription

1 REGULAR POSITIVE TERNARY QUADRATIC FORMS BYEONG-KWEON OH Abstract. A postve defnte quadratc form f s sad to be regular f t globally represents all ntegers that are represented by the genus of f. In 997 Jagy Kaplansky and Schemann provded a lst of 9 canddates of prmtve postve defnte regular ternary quadratc forms and stated that there are no others. All but of 9 are already verfed to be regular (cf. [8]). In ths artcle we prove that among canddates the ternary form L() (for the defnton see Table 4.) s regular for every = and.. Introducton A postve defnte ntegral quadratc form f s called regular f f represents all ntegers that are represented by the genus of f. Regular quadratc forms were frst studed systematcally by Dckson n [4] where the term regular was coned. Jones and Pall n [9] classfed all prmtve postve defnte dagonal regular ternary quadratc forms. In the last chapter of hs doctoral thess [5] Watson showed by arthmetc arguments that there are only fntely many equvalence classes of prmtve postve defnte regular ternary forms. More generally a postve defnte ntegral quadratc form f s called n-regular f f represents all quadratc forms of rank n that are represented by the genus of f. It was proved n [] that there are only fntely many postve defnte prmtve n-regular forms of rank n + for n. See also [] on the structure theorem of n-regular forms for hgher rank cases. The problem of enumeratng the equvalence classes of the prmtve postve defnte regular ternary quadratc forms was recently resurrected by Kaplansky and hs collaborators [8]. They provded a lst of 9 canddates of prmtve postve defnte regular ternary forms and stated that there are no others. All but of 9 are already verfed to be regular. In fact ther algorthm reles on the complete lst of those regular ternary quadratc forms wth square free dscrmnant [7] and a method of descent set forth by Watson n [5]. Ths method of descent nvolves a collecton of transformatons whch change a regular ternary form to another one wth smaller Mathematcs Subject Classfcaton. Prmary E E. Key words and phrases. regular ternary quadratc forms.

2 BYEONG-KWEON OH dscrmnant and smpler local structure and t s ths method whch enables Watson to obtan the explct dscrmnant bounds for regular ternary quadratc forms. There are 794 prmtve postve defnte ternary quadratc forms havng class number and those forms are regular. If a postve ternary form f has class number bgger than as far as the author knows there s no general method of determnng the set of all ntegers that are represented by f. In 99 Duke and Schulze-Pllot proved n [5] that for any postve defnte ternary form f there s a constant C dependng only on f such that every nteger a greater than C s represented by f f a s prmtvely represented by the spnor genus of f. However there s no known effectve method of computng the constant C explctly. There are some methods on provng regularty of a partcular ternary form f havng class number greater than. One method s usng some other form havng class number related wth f and some specfc modularty dependng on the form f (cf. [4] [7] [8] [9] and [7]). Another method s to prove that the spnor class number of f s one and there are no spnor exceptonal ntegers (cf. [6] [] and []). These two methods provde the proof of the regularty of 9 (794 + ) = 97 ternary forms. Note that the second method s not avalable for provng regularty of the remanng canddates. In ths paper we show that the ternary form L() (for the defnton of each form see Table 4.) s regular for every = and. Our method s qute smlar to the former one explaned above. However we use a ternary lattce representng the canddate whereas the tradtonal method uses a genus mate that s a lattce n the genus of the canddate or a sublattce of the canddate. We also use one more fact that the number of representatons of a by f s always fnte for any nteger a and any postve defnte quadratc form f. The term lattce wll always refer to an ntegral Z-lattce on an n- dmensonal postve defnte quadratc space over Q. The scale and the norm deal of a lattce L are denoted by s(l) and n(l) respectvely. Let L = Zx + Zx + + Zx n be a Z-lattce of rank n. We wrte L (B(x x j )). The rght hand sde matrx s called a matrx presentaton of L. Throughout ths paper we always assume that every Z-lattce L s postve defnte and s prmtve n the sense that s(l) = Z. In partcular the Z-lattce L() denotes one of canddates of regular ternary forms whch

3 REGULAR POSITIVE TERNARY QUADRATIC FORMS are defned n Table 4. of Secton 4. A Z-lattce L s called odd f n(l) = Z even otherwse. For any Z-lattce L Q(gen(L)) (Q(L)) denotes the set of all ntegers that are represented by the genus of L (L tself respectvely). In partcular we call an nteger a elgble f a Q(gen(L)) followng Kaplansky. Any unexplaned notatons and termnologes can be found n [] or [4].. General Tools Let L be a Z-lattce. For any postve nteger m defne Λ m (L) = {x L : Q(x + z) Q(z) (mod m) for all z L}. The Z-lattce λ m (L) denotes the prmtve lattce obtaned from Λ m (L) by scalng L Q by a sutable ratonal number. For the propertes of ths transformaton see [] or [6]. Lemma.. Let p be a prme and L be a Z-lattce. If p s odd and a unmodular component of L p s ansotropc or p = and L s odd or L ( ) 4α for some α Z then Q(L) δpz = Q(Λ δp (L)) and Q(gen(L)) δpz = Q(gen(Λ δp (L))) where δ = f p = and L s even δ = otherwse. Proof. The proof s qute straghtforward. See for example []. Under the same assumpton as above the lemma mples the followng: If L s regular then λ δp (L) s also regular and conversely f λ δp (L) s regular then (Q(gen(L)) Q(L)) δpz =. For each =... one may easly show that λ δp (L()) s regular or λ δp (L()) = L(j) for some j where p s any prme satsfyng the condton gven n the lemma. For example λ (L(7)) 7 7 whch s a regular form and λ (L(8)) L(4) λ (L()) L(4) and λ (L(4)) L(). Hence f L(8) s regular then both L() and L(4) are also regular. From now on we wll use the matrx presentaton of each Z-lattce. Let M and N be any quadratc forms of rank m and n respectvely and l be

4 4 BYEONG-KWEON OH any postve nteger. We denote by R(N M) the set of all representatons from N to M that s R(N M) = {T M mn (Z) T t MT = N}. Let r be any nonnegatve nteger less than l. We defne R l (r N) = {x M n (Z/lZ) x t Nx r (mod l)}. For any subset S M n (Z) we defne S l = {x l = (φ(x ) φ(x )... φ(x n )) t x = (x x... x n ) t S} where φ : Z Z/lZ s a natural projecton map. The followng smple observaton s the startng pont of our method. Lemma.. Let a be a postve nteger such that a = x t Nx for some x M n (Z). If there s a T R(l N M) such that T x lm m (Z) then a s represented by M. Proof. Note that ( ) t ( ) l T x M l T x = l xt (T t MT )x = x t Nx = a. The lemma follows drectly from ths. We defne E M l (r N) = {x R l (r N) T R(l N M) T x lm m (Z/lZ)}. Every computaton such as El M (r N) for some M N r and l was done by the computer program MAPLE. The followng theorem s very useful n showng that every elgble nteger of M n a certan arthmetc progresson s represented by a partcular quadratc form M. Theorem.. For any nteger a Q(N) such that a r (mod l) f ( ) R(a N) l E M l (r N) then a s represented by M. In partcular f E M l (r N) = then Q(N) {a Z a r (mod l)} Q(M). Proof. Assume that there s an x R(a N) such that x l El M (r N). Then there s a T R(l N M) such that T x lm m (Z). Hence the theorem follows from Lemma..

5 REGULAR POSITIVE TERNARY QUADRATIC FORMS 5. Regular ternary forms In ths secton we show that all eght forms marked wth bold face n Table 4. are regular. Note that λ (L(8)) = L() λ (L()) = L(8) and λ 5 (L(9)) = L() λ 5 (L()) = L(9). Hence f L(8) and L(9) are regular L() and L() are also regular. Therefore t s enough to show that L() s regular for = Theorem.. The ternary form L(7) s regular. Proof. Let M = L(7) = 7 8 N = 7. 7 Note that dm = 4 7 and dn = 4 7. Furthermore M and N. One may easly show that λ (M) = N and M s represented by N. By a drect computaton we have {( ) ( ) ( ) ( )} R(9N M) = and R ( N) = R ( N) =. Furthermore R(9N N) contans the followng 4 sometres: ) ) ) ( 4 ( 4 ( ( 4 4 In fact R(9N N) = (see Table 4.) but we only need these 4 representatons. We denote by S the -th matrx gven above for = Let a be any elgble nteger of M. Snce M s represented by N and h(n) = (cf. [8]) a s represented by N. Let x = (x x x ) t be a vector such that x t Nx = a. Assume that a (mod ). Snce the unmodular component of M s ansotropc one may easly show that M represents a by Lemma.. Assume that a (mod ). In ths case one may easly show that E M ( N) =. Hence M represents a by Theorem.. ).

6 6 BYEONG-KWEON OH Fnally assume that a (mod ). If ( ) does hold then Theorem. gves the desred concluson that a s represented by M. So t s only necessary to consder further the case that ( ) does not hold; that s (.) R(a N) E M ( N). Note that E M ( N) = {( ± ±) t ( ± ) t }. Hence we may further assume that (x x x ) ( ± ±) or ( ± ) (mod ). Assume that x = (x x x ) t ( ± ±) t (mod ). Snce S S R(9N N) and S x S x M (Z) S x S x R(a N). Hence from the assumpton (.) we have x 4x x x x 4x (mod ). If we let x = s and x x = t for s t Z then s x 4t s x t (mod ). Therefore t s + x (mod ). From ths follows x x x x (mod ) and x x x x (mod ). Ths mples that S x ( ) t (mod ) or ( ) t (mod ) for =. Defne a matrx T such that 5 9 T = S S = From the above observaton we have T n x R(a N) for every nonnegatve nteger n. From the fact that R(a N) s fnte t then follows that there exst postve ntegers n > m for whch T n x = T m x that s T m (T n m I)x =. Note that there s a transton matrx P such that T = P λ P λ where λ λ are complex roots of 9t + t + 9 =. Hence dm(ker(t n m I)) =. Furthermore snce ( ) t = ker(t I) ker(t n m I) we have x ker(t n m I) = ( ) t.

7 REGULAR POSITIVE TERNARY QUADRATIC FORMS 7 If x = ( k k k) t one may easly verfy that a = x t Nx = 5k = (k k k)m(k k k) t. Now assume that x = (x x x ) t ( ± ) t (mod ). In ths case we may apply the smlar argument to the above by just replacng S and S by S and S 4 respectvely. Ths completes the proof. For) the quadratc form L(8) we take l = 8 and N = λ 4 (L(8)) = whch s a regular form. Note that ( L(8) 7 ( ) 8. 8 Therefore we may only consder an elgble nteger a of L(8) that s congruent to 7 modulo 8. Snce there are too many sometres for example R(64N N) = 88 we do not wrte them down here. In ths case one may easly show that E L(8) If we choose S = 8 (7 N) = {(± ±4 ±) t (± ±4 ±5) t }. ) ) S = ( ( R(64N N) then S x 8M (Z/8Z) for any = and x E L(8) 8 (7 N). Hence f we apply the same method descrbed above n ths stuaton we may easly show that a s represented by L(8). For the quadratc form L() we take l = 8 ( ) ( ) 4 8 and N = λ 4 (L()) = 5. Note that L() and E( L() ( N) ) = {(± ±6 ) ±7) t (± ± ±5) t }. In ths case we may use R(64N N). ( Snce all computatons are qute smlar to the case L(7) we only provde a table contanng all parameters needed for computatons for the remanng quadratc forms (cf. Table 4.). Remark.. The method descrbed n ths artcle could also be effectve even f M s not regular. For example one may show that every elgble nteger ( ) of the form 6n + 5 s represented ( ) by the Ramanujan( form) M = 4 by takng l = and N =. In case when M = 6 one 7 may also show that every elgble nteger) of the form 6n + 4 s represented by M by takng l = and N = (cf. Lemma 8. of []). In both ( cases h(m) = and N s contaned n the genus of M.

8 8 BYEONG-KWEON OH 4. Tables In the followng table the regularty of 8 ternary forms marked wth bold face was proved n ths artcle. 4 L() A L() A L() A L(4) A L(5) 5 A L(6) 4 6A L(7) 6 4A L(8) 8 9 A L(9) 8 6 A L() 5A L() 6 6 A L() 8 A L() 6 6A L(4) 6 4A L(5) A L(6) 6 A L(7) 8 A L(8) 5 A L(9) A L() 4A L() A L() 6 A. 6 9 (4.) canddates of regular ternary forms

9 REGULAR POSITIVE TERNARY QUADRATIC FORMS 9 d(l()) N L() ±R(9N L()) ±R(9N N) L() ±R ( N) ±R ( N) L(6) L() L(7) L(9) E L() ( N) E L() ( N) S S R(9N N) S S 4 R(9N N) 5 4 λ (L(6)) = 6 4 h h h 5 λ (L()) = h h h 4 4 h h 6 h h ( ± ±) (± ± ) h h h h h h h h ( ±) (± ± ) λ (L(7)) = h h h h h h h h h 4 ( ± ±) ( ± ) λ (L(9)) = h h h h h 4 4 h h h ( ± ±) ( ± ) for ( ± ±) 5 h (4.) Some data for regular ternary forms for ( ± ) 5

10 BYEONG-KWEON OH Acknowledgements. Ths work was supported by the Korea Research Foundaton Grant (KRF-8-4-C4) funded by the Korean Government. References [] J. Bochnak and B.-K. Oh Almost regular quaternary quadratc forms Ann. Inst. Fourer (Grenoble) 58(8) [] W. K. Chan and B.-K Oh Fnteness theorems for postve defnte n-regular quadratc forms Trans. Amer. Math. Soc. 55 () [] W. K. Chan and B.-K. Oh Postve ternary quadratc forms wth fntely many exceptons Proc. Amer. Math. Soc. (4) [4] L. E. Dckson Ternary quadratc forms and congruences Ann. of Math. 8 (97) 4. [5] W. Duke and R. Schulze-Pllot Representaton of ntegers by postve ternary quadratc forms and equdstrbuton of lattce ponts on ellpsods Invent. Math. 99 (99) [6] J. S. Hsa Regular postve ternary quadratc forms Mathematka. 8 (98) 8. [7] W. C. Jagy Fve regular or nearly-regular ternary q uadratc forms Acta Arth. 77 (996) [8] W. C. Jagy I. Kaplansky and A. Schemann There are 9 regular ternary forms Mathematka 44 (997) 4. [9] B. W. Jones and G. Pall Regular and sem-regular postve ternary quadratc forms Acta Math. 7 (94) [] I. Kaplansky A second genus of regular ternary forms Mathematka. 4 (995) [] I. Kaplansky A thrd genus of regular ternary forms 995. Unpublshed preprnt. [] Y. Ktaoka Arthmetc of quadratc forms Cambrdge Unversty Press 99. [] B.-K. Oh Postve defnte n-regular quadratc forms Invent. Math. 7 (7) [4] O. T. O Meara Introducton to quadratc forms Sprnger Verlag New York 96. [5] G. L. Watson Some problems n the theory of numbers Ph.D. Thess Unversty of London 95.

11 REGULAR POSITIVE TERNARY QUADRATIC FORMS [6] G. L. Watson Transformatons of a quadratc form whch do not ncrease the class-number Proc. London Math. Soc. () (96) [7] G. L. Watson Regular postve ternary quadratc forms J. London Math. Soc. (976) 97-. Department of Mathematcal Scences Seoul Natonal Unversty Seoul Korea E-mal address: bkoh@math.snu.ac.kr