CONGRUENCES MODULO 4 OF CALIBERS OF REAL QUADRATIC FIELDS. 1. Definitions and results

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1 Ann. Sci. Math. Québec 35 No (0) CONGRUENCES MODULO 4 OF CALIBERS OF REAL QUADRATIC FIELDS MASANOBU KANEKO AND KEITA MORI Dedicated to rofessor Paulo Ribenboim on the occasion of his 80th birthday. RÉSUMÉ. Nous résentons des conjectures sur les congruences modulo 4 des calibres (au sens restreint) d ordres quadratiques réels our certains discriminants et nous donnons des reuves dans quelques cas. ABSTRACT. We resent conjectures on congruences modulo 4 of the calibers (in the narrow sense) of real quadratic orders of articular discriminants and rove some artial results on them.. Definitions and results A real quadratic number w is reduced if it satisfies w > and < w < 0 where w is the algebraic conjugate of w over the rationals Q. It is well known that w is reduced if and only if its usual continued fraction exansion is urely eriodic. Consider the set of all reduced quadratic numbers of a given discriminant D: Q(D) := w disc(w) = D w > < w < 0 }. Here a real quadratic irrationality w is of discriminant D denoted disc(w) = D if w satisfies aw + bw + c = 0 a b c Z a > 0 GCD(a b c) = b 4ac = D. The set Q(D) is finite and its cardinality is denoted by κ(d). When D is a fundamental discriminant i.e. the discriminant of the real quadratic field Q( D) the number κ(d) is referred to as the caliber of Q( D) (see []). Any quadratic number of discriminant D is equivalent under the action of GL (Z) (via the linear fractional transformation) to an element in Q(D). We write w w if the two numbers w and w are GL (Z)-equivalent. It is known that w w if and only if their eriods of continued fraction exansions are cyclically equivalent. Let R(D) be the set of GL (Z)-equivalence classes of Q(D) and h(d) be its cardinality: R(D) = Q(D)/ h(d) = R(D). The number h(d) is nothing but the wide class number of discriminant D. Reçu le février 009 et sous forme définitive le 4 avril 009.

2 86 CONGRUENCES MODULO 4 OF CALIBERS OF REAL QUADRATIC FIELDS We also consider the corresonding notions for SL (Z)-equivalence. We call a real quadratic number w m-reduced if w > and 0 < w <. A number is m-reduced if and only if its minus continued fraction exansion is urely eriodic (see [5 3]). Let Q + (D) be the set of all m-reduced numbers of a given discriminant D: Q + (D) := w disc(w) = D w > 0 < w < }. This is also a finite set and its cardinality will be denoted by κ + (D). We refer to this number as the m-caliber of (not necessarily fundamental) discriminant D. Consider the equivalence under the action of SL (Z). Two numbers w and w are strictly equivalent written w w if the two are related with each other by a transformation in SL (Z). Any quadratic number of discriminant D is strictly equivalent to an element in Q + (D) and two elements in Q + (D) are strictly equivalent if and only if the eriods of their minus continued fraction exansions are cyclically equivalent. Let R + (D) be the set of SL (Z)-equivalence classes of Q + (D) and h + (D) be its cardinality (the narrow class number): R + (D) = Q + (D)/ h + (D) = R + (D). Throughout the aer we denote by a rime number congruent to modulo 4 and by q a rime number larger than which is also congruent to modulo 4. Integers x y x q y q are uniquely defined by the decomosition = x + y (0 < x < y ) q = x q + y q (0 < x q < y q ). We denote by ε D the fundamental unit of discriminant D and by N(ε D ) its norm. We conducted numerical exeriments on the -orders of the m-calibers κ + (8) and κ + (q) which are easily seen to be even integers and observed the following. Conjectures. () We have κ + (8) ( ) x (mod 4). () Assume < q and suose x x q (mod ). Then we have q κ + (q) ( ) x (mod 4). For the narrow class numbers h + (8) and h + (q) (which are always even) the following two roerties are known (cf. []): (i) h + (8) is divisible by 4 if and only if (mod 8) and q (ii) h + (q) is divisible by 4 if and only if =. The above conjectures say that the( divisibility ) by 4 of κ + (8) and κ + (q) deends on q the arity of x or x q not only on in the case () which may be of considerable interest. Secial case of the conjecture and Theorem. were discussed in Umeno [4].

3 M. Kaneko and K. Mori 87 In this aer we rove () in the case where N(ε 8 ) = and some artial result for () also in the case where N(ε q ) =. Theorem.. Let be a rime number such that (mod 4) and suose N(ε 8 ) =. Then the class modulo 4 of the m-caliber κ + (8) is given by κ + (8) ( ) x (mod 4). As far as Conjecture () is concerned we are not able to rove it even under the condition N(ε q ) = but we obtain the following artial result. Notice that corresonding to the two decomositions of q into sums of two squares q = (x x q + y y q ) + (x y q y x q ) = (x y q + y x q ) + (x x q y y q ) there exist exactly two quadratic numbers α β Q(q) whose eriods of usual continued fraction exansions are alindromic. We denote by l(α) and l(β) their eriod lengths. Theorem.. Besides the assumtions < q and x x q (mod ) we further suose N(ε q ) =. Then we have κ + l(α) l(β) x+ q (q) + sgn(x y q y x q )( ) (mod 4). Thus our conjecture in the case N(ε q ) = is that x y q y x q < 0 if and only if l(α) l(β) (mod 4).. Preliminaries As it is recalled in the revious section the elements α Q(D) and β Q + (D) have urely eriodic usual and minus continued fraction exansions and α = [a 0... a n ] = a 0 + a a n + a 0 + β = [[b 0... b m ]] = b 0 b... b m b 0 a +... b...

4 88 CONGRUENCES MODULO 4 OF CALIBERS OF REAL QUADRATIC FIELDS resectively. We define l(α) := n and l + (β) := m to be their minimum eriod lengths; moreover n S(α) := a i and S + (β) := j 0 j m b j 3 } i=0 are resectively called the sum of artial quotients and the number of artial quotients greater than or equal to 3 in the eriod. In order to establish our results we use the following result which may be standard but for which for the sake of convenience we reroduce the roof given in Suzuki [3]. Proosition.. We have κ + (D) = [α] R(D) S(α) and κ(d) = [β] R + (D) S + (β). Proof. We rove the first assertion the second being similarly roved (we shall not use the second formula). By definition we have κ + (D) = l + (β). [β] R + (D) We may assume that all reresentatives β are bigger than because at least one of the artial quotients b j in the exansion β = [[ b 0... b m ]] is bigger than and hence by a cyclic ermutation we obtain a number greater than equivalent to β. As is easily seen the ma (.) T : Q(D) α α + Q + (D) gives a bijection between the sets Q(D) and β Q + (D) β > }. If the usual continued fraction exansion of α Q(D) is α = [ a 0... a n ] then the minus continued fraction exansion of T (α) = α + is given by (see e.g. [5]) [[ a a }} a }} n +... ]] if n is even }} a a 3 a n [[ a a }} n a }} n +... ]] if n is odd. }} a a 0 a n If n is even then the images under T of [a i... a n a 0... a i ] which are equivalent to α slit into two classes (under the equivalence ) according to the arity of i. By the formula above we have and hence l +( T ([ a 0... a n ]) ) = a + a a n l +( T ([ a... a n a 0 ]) ) = a + a a n + a 0 l +( T ([ a 0... a n ]) ) + l +( T ([ a... a n a 0 ]) ) = S(α). If n is odd then all T ([ a i... a n a 0... a i ]) are equivalent and l + (T (α)) = S(α).

5 M. Kaneko and K. Mori 89 It is clear that the equivalence T (α ) T (α ) imlies α α. From this and the surjectivity of (.) we conclude the assertion κ + (D) = [α] R(D) S(α). The following lemma is reeatedly used in our roof. Lemma.. Let E = 0 0 and O = 0 be elements in GL (F ) of order and 3 resectively. Consider the roduct M = M M n of length n of k O s and (n k) E s (i.e. M i = E or O and the number of O s is k). Then we have the equivalence } 0 n k (mod ) M I = O = O =. Proof. Using relations E = I O 3 = I OEO = E (OE = EO ) any roduct of E s and O s reduces to one of the six elements I O O 0 E EO = OE =. 0 In this reduction rocess the arity of (total length of roduct) (number of O s) does not change and out of the above six elements this arity is even exactly when it is I O O. Lemma.3. Let α Q(D). (i) If D is odd then we have l(α) S(α) (mod ). (ii) If D is even and N(ε D ) = then S(α) is even. Proof. Let the quadratic equation satisfied by α be aα + bα + c = 0 with GCD(a b c) = and b 4ac = D. For the continued fraction exansion α = [ a 0 a... a n ] we set q a0 a an :=. r s Then we have α = α + q and thus rα + s rα + (s )α q = 0. Put d := GCD(r s q) (> 0). Then we have r = da s = db and q = dc. (i) Assume D is odd. Then b is odd. If d is odd then s is odd and so s is even hence qr is odd because s qr = ( ) n. Thus q 0 = O or = O r s 0.

6 90 CONGRUENCES MODULO 4 OF CALIBERS OF REAL QUADRATIC FIELDS If d is even then r q 0 s (mod ) whereuon q 0 = I (mod ). r s 0 By Lemma. we conclude n S(α) (mod ) because the arities of S(α) and of the number of odd a i are the same. (ii) Assume D is even. Then b should be even and so s (mod ). If s (mod ) and q r 0 (mod ) then d is even (otherwise a b c are all even) and s = db 0 (mod 4). In this case we have s qr 0 (mod 4). This contradicts s qr = which is equivalent to the condition N(ε D ) =. Therefore we have the ossibilities q r s ( 0 ) (mod ). By Lemma. we see S(α) n (mod ) and thus S(α) is even because n is odd. 3. Proofs of Theorems Proof of Theorem.. Corresonding to the decomosition 8 = ( (x + y ) ) + ( (y x ) ) there exist exactly two elements α β Q(8) which satisfy α = /α β = /β. These are the largest roots of the equations (3.) (x + y )α + (x y )α (x + y ) = 0 and (y x )β (x + y )β + x y = 0 resectively. By our assumtion on the norm of the fundamental unit (N(ε 8 ) = ) we may take α and β as art of the reresentatives of R(8) the other reresentatives being in the form γ /γ... γ t /γ t. From Lemma.3 (ii) we have S(γ i ) + S( /γ i ) = S(γ i ) 0 (mod 4). Hence by Proosition. it suffices to rove that 0 (mod 4) if x is even S(α) + S(β) (mod 4) if x is odd. Let α = [ a 0... a n a n a n... a 0 ] and β = [ b 0... b m b m b m... b 0 ] be their (alindromic) continued fraction exansions (by our assumtion N(ε 8 ) = their eriod lengths are odd) and set q a0 an P Q q an r := := r s 0 0 R S r s 0 q s ( q ) ( b0 bm P Q r s := ) ( q 0 0 R S := ) ( bm r ) r s 0 q s.

7 M. Kaneko and K. Mori 9 Then we have (3.) Rα + (S P )α Q = 0 R β + (S P )β Q = 0. Set d := GCD(R S P Q) and d = GCD(R S P Q ). Lemma 3.. (i) P + S = P + S. (ii) d = d. Proof. Let ε D be the fundamental unit of discriminant D. The classical construction of ε D from the continued fraction exansions shows that the equalities ε D = Rα + S = R β + S holds. Comuting ε D + ε D in two ways and using it as ε D + ε D = R(α + α ) + S = R P S R + S = P + S = R (β + β ) + S = R P S R + S = P + S we obtain equality (i). The second equality (ii) follows similarly from the comutation of (ε D ε D )/ D. (3.3) (3.4) (3.5) (3.6) From (3.) and (3.) we have By (3.3) and (3.4) we obtain R = Q = d(x + y ) R = Q = d(y x ) P S = d(y x ) P S = d(x + y ). (3.7) R R = Q Q = dx and by (3.5) (3.6) and Lemma 3. (i) we have P P = S S = dx. By Lemma.3 (ii) and its roof we have a n b m 0 (mod ) and ( P Q P Q ) 0 R S R S (mod ). 0 Then from (3.3) we have that d is odd and from (3.5) we deduce that P S (mod 4). Assume P 0 S (mod 4). By S = r a n + rs (mod 4) r must be odd and we have a n + s (mod 4). If a n (mod 4) then s is even and q is odd. In this case we have Q = ra n + qr + s = ra n + qr + ( ) n + + ( ) n (mod 4).

8 9 CONGRUENCES MODULO 4 OF CALIBERS OF REAL QUADRATIC FIELDS When Q (mod 4) is odd or even according as n is even or odd and we have resectively q q 0 (mod ) or (mod ). r s 0 r s 0 In either case we conclude by Lemma. that S(α) (mod 4). Similarly we have S(α) 0 (mod 4) when Q (mod 4). Also by the same argument we see that when a n 0 (mod 4) we have S(α) 0 (mod 4) or S(α) (mod 4) according as Q (mod 4) or Q (mod 4). In summary we have 0 (mod 4) if S(α) (mod 4) if P Q 0 (mod 4) R S P Q 0 (mod 4). R S Likewise we obtain by considering the different cases and using Lemma. that P Q 0 (mod 4) if (mod 4) R S 0 S(α) P Q (mod 4) if (mod 4). R S 0 All these show that we have 0 (mod 4) if Q (mod 4) S(α) (mod 4) if Q (mod 4). Exactly the same holds for S(β) that is to say 0 (mod 4) if Q (mod 4) S(β) (mod 4) if Q (mod 4). Now by (3.7) Q Q (mod 4) if and only if x is even. Hence we have S(α) + S(β) which is what we need to establish Theorem.. 0 (mod 4) if x is even (mod 4) if x is odd Proof of Theorem.. We roceed similarly and kee the same notation. Let α and β be the largest roots of the equations and x x q + y y q α x y q y x q α x x q + y y q y y q x x q β (x y q + y x q )β + x x q y y q = 0 = 0

9 M. Kaneko and K. Mori 93 resectively. These are the only elements in Q(q) satisfying α = /α β = /β corresonding to the decomositions q = (x x q + y y q ) + (x y q y x q ) = (x y q + y x q ) + (x x q y y q ). By the assumtion x x q (mod ) x y q y x q and x y q + y x q are odd. As in the roof of Theorem. we may take as reresentatives of R(q) α β and the other in the form γ /γ... γ t /γ t. By Lemma.3 (i) and the assumtion N(ε q ) = we have By the theorem of Rédei-Reichardt [] Thus that is Since S(γ i ) + S( /γ i ) = S(γ i ) (mod 4). R(q) 0 t i= (mod 4)( t is odd) q =. ( S(γi ) + S( /γ i) ) q (mod 4) = t i= ( S(γi ) + S( /γ i) ) q + κ + (q) = S(α) + S(β) + (mod 4). t ( S(γi ) + S( /γ i) ) by Proosition. to rove the theorem we need to show q ( S(α) + S(β) + sgn(x y q y x q )( ) (3.8) i= x+ l(α) l(β) l(α) l(β) x+ sgn(x y q y x q )( ) (mod 4). (We have used the simle fact that k ±k (mod 4) when k is even.) Let the continued fraction exansions of α and β be the same as in the roof of Theorem. and similarly construct the matrices q P Q etc. r s R S Then we have (3.9) (3.0) (3.) R = Q = d x x q + y y q R = Q = d y y q x x q P S = d x y q y x q ) (3.) P S = d(x y q + y x q )

10 94 CONGRUENCES MODULO 4 OF CALIBERS OF REAL QUADRATIC FIELDS and (because Lemma 3. also holds in this case) (3.3) (3.4) R R = Q Q = dx x q P P = S S = d min(x y q y x q ). Since D = q is odd by Lemma.3 (i) and Lemma. we have P Q 0 0 or or (mod ) R S 0 0 and a n b m (mod ). We need to carry out rather tedious case distinctions but here we describe how to deduce the conclusion only in the case when P Q 0 (mod ) R S 0 the other cases being similarly treated. In this case because and r are both odd and thus Q = ra n + qr + ( ) n 0 (mod ) P = a n + q a n + q (mod 4). Suose P (mod 4). If a n (mod 4) then q is even and by Lemma. we conclude S(α) (mod 4) (res. 3 (mod 4)) when n is odd (res. even). If a n 3 (mod 4) then q is odd and we similarly have S(α) (mod 4) (res. 3 (mod 4)) when n is odd (res. even). Likewise when P (mod 4) we have S(α) (mod 4) (res. 3 (mod 4)) when n is even (res. odd). Summing u we conclude ( ) n (mod 4) if P (mod 4) S(α) ( ) n (mod 4) if P (mod 4). Because QR 0 (mod 4) and P S QR = we have P S (mod 4) whereuon d necessarily satisfies d (mod 4) (note x y q y x q is odd). By (3.3) and the assumtion x x q (mod ) we have R R Q Q (mod 4). Hence we have ( P Q ) 0 R S (mod ) 0 and S(β) ( ) m (mod 4) if P (mod 4) ( ) m (mod 4) if P (mod 4). Suose x y q > y x q i.e. sgn(x y q y x q ) =. Then by (3.4) P P = dy x q y x q + ( ) x (mod 4) (recall the assumtion x x q (mod )). From the above we have ( ) n + ( ) m + ( ) n m (mod 4) if x is odd S(α) + S(β) ( ) n ( ) m ( ) n m (mod 4) if x is even.

11 M. Kaneko and K. Mori 95 We can combine these into the congruence S(α) + S(β) ( ) x+n m (mod 4). When x y q < y x q we have P P = dx y q x y q ( ) x (mod 4) and in the same way we obtain S(α) + S(β) + ( ) x+n m (mod 4). We therefore have S(α) + S(β) sgn(x y q y x q )( ) x+n m (mod 4). Since n m = (l(α) l(β))/ we have (3.8) and Theorem. is roved. Acknowledgement. The resent work was artially suorted by Grant-in-Aid for Scientific Research (B) No REFERENCES [] G. Lachaud On real quadratic fields Bull. Amer. Math. Soc. (N.S.) 7 (987) no [] L. Rédei and H. Reichardt Die Anzahl der durch vier teilbaren Invarianten der Klassengrue eines beliebigen quadratischen Zahlkörers J. Reine Angew. Math. 70 (934) no [3] K. Suzuki Reduced quadratic forms and continued fraction exansions (in Jaanese) Master s thesis Kyushu University March 008. [4] T. Umeno On the -divisibility of calibers of real quadratic fields (in Jaanese) Master s thesis Kyushu University March 006. [5] D. B. Zagier Zetafunktionen und quadratische Körer University Text Sringer-Verlag Berlin-New York 98. viii+44. M. KANEKO FACULTY OF MATH. KYUSHU U. 744 MOTOOKA NISHI-KU FUKUOKA JAPAN mkaneko@math.kyushu-u.ac.j K. MORI URESHINO SPECIAL NEEDS EDUCATION SCHOOL GOCHOUDAKOU 877- SHIOTA URE- SHINO SAGA JAPAN mori-keita@st.saga-ed.j