On the 4-rank of the tame kerne K O in positive definite terms P. E. Conner and Jurgen Hurrebrink Abstract: The paper is about the structure of the tame kerne K O for certain quadratic number fieds. There has been recent progress in making expicit the 4-rank of the tame kerne of quadratic number fieds, [11], [1], and even in obtaining resuts about the 8-rank, [10], [13], [17]. The emphasis of this paper is to determine the 4-rank of the tame kerne in definite terms. Our characterizations are in terms of positive definite binary quadratic forms X +3Y, X +py, X + py over Z. The resuts make numerica computations readiy avaiabe, and the characterizations might generate some interest in density resuts concerning the 4-rank of tame kernes. Mathematics Subject Cassification: 11R70, 11R9, 19C30, 19F99.
P. E. Conner and J. Hurrebrink 1. Introduction. The tame kerne of an agebraic number fied with ring O of integers is the Minor K- group K O. This paper is on the 4-rank of the finite abeian group K O for quadratic number fieds whose discriminants have exacty two odd prime divisors p,. We investigate the case p 7mod8, 1mod8with = p p =+1. In this situation, the genera agorithms described in [10], [11], [1] require the soution of an indefinite norm equation which in principe coud take a ong time. The main point of our paper is to characterize the possibe vaues of the 4-rank of the tame kerne in positive definite terms. The determination of the 4-rank of K O essentiay reduces to checking which ones of the positive definite quadratic forms X +3Y, X +py,x + py represent a particuar power of over Z; compare Theorems 5. to 5.5. Sections and 3 are purey number theoretic. In section we anayze the unique unramified cycic degree 4 extension of 1Q p. In section 3 we then deveop the representation of a particuar power of by positive definite binary quadratic forms over Z, the main too of our approach. The connection to K-theory is being made in section 4 where quadratic symbos from this approach and from [10], [11], [1] become reated to each other. The main resuts then drop out in the four theorems in section 5. For fieds 1Q p, a version of resut 5.4 was stated in [9] via the approach from [8]; we are peased that in this paper aso the imaginary fieds 1Q p as we as the rea fieds 1Q p,1q p can be handed in a unified way. In section 6 we have made an effort to iustrate our resuts numericay. We are ed naturay to ask questions about densities concerning the 4-rank of K O for fieds considered in this paper and in more generaity. Ony itte do we know. Yet, this K -probem can be considered to be an anaog of the most cassica K 0 -probem of investigating the structure, in particuar -power-ranks, of idea cass groups of number fieds. Acknowedgement: We are most gratefu to Manfred Koster for the streamined approach in section 4 that improved an earier version of the paper. It is a peasure for us to
Onthetamekerne 3 give credit to LSU graduate student Robert Osburn for the numerica resuts in section 6.. An unramified cycic extension Let p be a prime, p 7mod8, and consider the imaginary quadratic fied K = 1Q p. The -primary subgroup of the idea cass group CK ofk is cycic of order divisibe by 4, see e.g. 18.6, 18.4, 19.6 in [6]. Thus the Hibert cass fied of K contains a unique unramified cycic degree 4 extension N over K. Let us determine it. We begin with the rea quadratic fied L =1Q with ring of integers O L = Z[ ], group of units O L, and fundamenta unit ε =1+ ofnorm 1. The fied L has narrow cass number h + L =1. Since p 7mod8, the prime p spits in L over 1Q and so has two extensions P p p and P p p to L. è.1è Lemma: If P p O L is an extension of p to L, then P p has a generator π = a + b O L, unique up to mutipication by the square of a unit in OL, for which N L/1Q π =a b = p and the dyadic prime D L O L is unramified in L 1 = L π over L. Proof: Since h + L = 1, there is a generator a + b O L for P p p such that a b = p. Ceary a is odd. Then p 1 1 b mod8andthusb is even. We now specify a 1mod4 if b 0mod4 a 3mod4 if b mod4. Then for π = a + b wehave,forsomeη O L, π =1+4η if b 0mod4 πε =1+4η if b mod4; just notice that a + b 3 + = 3a +4b +a +3b with3a +4b 1mod4 and a +3b 0mod4 if a 3mod4 and b mod4. Thus the dyadic prime D L O L is unramified in L 1 = L πoverl. If π 1 is a second such generator for P p p of norm p, thenwehaveπ 1 = πu for some u O L with N L/1Qu =+1. Either u is totay positive or totay negative. Since h + L =1
4 P. E. Conner and J. Hurrebrink we concude u = v or u = v for some v O L. However D L ramifies in L 1 over L, so the ony possibiity is π 1 = πv for some v O L. If P p p in.1 is repaced by its conjugate idea P p p, then we ceary can repace the generator π = a + b ofp p with π = a b. Because of N L/1Q π = p, the norma cosure of the degree 4 number fied L 1 = L π=1q, πover1q is given by the degree 8 number fied N =1Q, π, p over 1Q. Consider the diagram of reative quadratic extensions: / N =1Q, π, p 1Q, p L 1 =1Q, π L =1Q 1Q è.è Lemma: The quadratic extension N =1Q, π, p over 1Q, p is unramified. Proof: In the norma extension N over 1Q the ony ramified rationa primes are,p, and the infinite prime. The prime is ramified in L over 1Q, but p is not. The dyadic prime D L in L is not ramified in L 1 = L πoverl, by.1. Moreover, the ony primes of L ramified in L 1 over L are P p p and a rea infinite prime of L. In particuar, Pp p is not ramified in L 1 over L. Since p 1mod8, every dyadic prime in L 1 spits in N = L 1 p overl 1 and every extension of Pp p to L 1 ramifies in N over L 1. We have observed: in the norma extension N over 1Q the rationa primes and p both have ramification index e =. Now, the intermediate fied 1Q, p is totay compex and,p have ramification index e =in1q, p over1q. Thus N over 1Q, p is not ramified.
Onthetamekerne 5 Consider the tower of reative quadratic extensions: N =1Q, p, π 1Q, p K =1Q p 1Q The degree 4 number fied 1Q, p over1q is the unique unramified quadratic extension of K. Since -prim CK is cycic of order divisibe by 4, as noted above, we know that -prim C1Q, p is a nontrivia cycic group. Therefore 1Q, p aso admits a unique unramified quadratic extension, which, by., is N =1Q, p, π. We have obtained: è.3è Lemma: The unique unramified cycic degree 4 extension N over K = 1Q p is norma over 1Q and is given by N =1Q, p, π N = K, π=k p, π, with a generator π as in.1. 3. The rationa primes For K =1Q p with a prime p 7mod8, we have determined in the preceding section the unique unramified cycic degree 4 extension N of K, see.3. Reca N 1Q, p K 1Q. We are now concerned with a rationa primes 1mod8forwhich p =p =+1. Such primes spit competey in the degree 4 extension 1Q, p over1q. In this section
6 P. E. Conner and J. Hurrebrink we wi characterize, in terms of positive definite binary quadratic forms over Z, such primes that spit competey in the degree 8 extension N over 1Q. Let D be the unique dyadic prime in the ring O K = Z[ p] of integers of K. If n + m p O K were a generator of D, then we woud have n +pm = with n, m Z, which is impossibe. Thus the cass cd ofd in the idea cass groups CK of K is nontrivia. Because of D =O K we have: cd 1andcD =1inCK; that is, cd is the unique eement of order in CK. Let H denote the Hibert cass fied of K. So we have H N 1Q, p K with the idea cass group of K being canonicay isomorphic to the Gaois group of H over K : CK = GaH/K. By restriction there is an epimorphism GaH/K GaN/K andgan/k is cycic of order 4. Since -prim CK is cycic of order divisibe by 4, CK 4 is the unique subgroup of CK of index 4. Thus CK/CK 4 = GaN/K and anaogousy CK/CK = Ga1Q, p/k. We note that the endomorphisms CK CK givenbyc c hk/4 and c c hk/ where hk denotes the cass number of K, have kerne CK 4 and CK and image the unique cycic subgroup of CK of order 4 and, respectivey. Thus: è3.1è Remark: Let P be a prime idea of O K.Then: P spits competey in N over K if and ony if c P hk/4 =1in CK. P spits in 1Q, p over K if and ony if c P hk/ =1in CK; that is, if and ony if either c P hk/4 =cd 1or c P hk/4 =1in CK. Now et 1 mod 8 be prime with p =p =+1. We have O K = PP with a pair P, P of conjugate prime ideas in O K. Since spits competey in 1Q, p over1q, P spits in 1Q, p overk and we concude by 3.1:
Onthetamekerne 7 Either cp hk/4 =1or cp hk/4 = cd 1inCK. Moreover, cp = cp 1 and hence cp hk/4 = cp hk/4 in CK. Thus: è3.è Remark: For as above, et O K = PP. Then: spits competey in N over 1Q if and ony if P spits competey in N over K if and ony if cp hk/4 =1in CK. does not spit competey in N over 1Q if and ony if cp hk/4 = cd 1in CK. prove: In terms of the positive definite quadratic form X +py over Z we are going to è3.3è Lemma: Let 1 mod 8 be prime with p = p = +1. Then: spits competey in N over 1Q if and ony if hk/4 = n +pm for some n, m Z with m 0 mod. Proof: Let spit competey in N over 1Q; so c P hk/4 =1inCK, by 3., where O K = PP. Hence P hk/4 has a generator σ = n + m p O K with ord P σ = hk/4 ord p σ = 0 for a prime ideas P O K,P P. Then N K/1Q σ=n +pm = hk/4 with n, m Z and we want to show that m 0mod. Assume the contrary: m 0mod and hence n 0mod. Then η = m + n p OK and σ = η. So, ord P σ = ord P + ord P η =1+ ord P η 1, contradicting ord P σ =0. For the converse et hk/4 = n +pm for some n, m Zwithm 0mod. Then σ = n + m p ies in O K and satisfies N K/1Q σ= hk/4. So, = PP in O K with ord P σ + ord P σ = hk/4 ord P σ = 0 for a prime ideas P O K,P P, P. Assume that ord P σ>0and ord P σ>0. Then σ = n + m p ies in OK = Z[ p]; hence m ies in Z, contradicting m 0mod. By interchanging P and P, if necessary, we may write ord P σ = hk/4 ord P σ =0
8 P. E. Conner and J. Hurrebrink and thus σ O K generates P hk/4 ;thatis, cp hk/4 =1inCK and, by 3., spits competey in N over 1Q. over Z: Next we are going to prove in terms of the positive definite quadratic form X + py è3.4è Lemma: Let 1 mod 8 be prime with p =p =+1. Then: does not spit competey in N over 1Q if and ony if hk/4 =n + pm for some n, m Z with m 0 mod. Proof: Assume that does not spit competey in N over 1Q; so, by 3., cp hk/4 = cd inck. Hence DP hk/4 is principa and there exists a generator σ = r + m p in O K of DP hk/4 with ord D σ =1 ord P σ = hk/4 ord P σ = 0 for a prime ideas P O K,P D, P. Now N K/1Q σ =r +pm = hk/4 ; ceary r is even and, as in the first part of the proof of 3.3, we see that m 0mod. We set r =n with n Zandobtain: n + pm = hk/4 with n, m Z, m 0mod. For the converse et hk/4 =n + pm for some n, m Zwithm 0mod. Then hk/4 =n +pm and thus σ =n + m p O K satisfies N K/1Q σ = hk/4. We concude: ord D σ =1 ord P σ + ord P σ = hk/4 ord P σ = 0 for a prime ideas P O K,P D, P, P. Assuming that ord P σ>0and ord P σ>0we derive a contradiction as in the second part of the proof of 3.3. Thus we may write again ord P σ = hk/4 ord P σ =0 and hence σ O K generates DP hk/4 ;thatis, cp hk/4 = cd inck and, by 3., does not spit competey in N over 1Q.
Onthetamekerne 9 We wi reate both characterizations 3.3 and 3.4 to a Legendre symbo π invoving the generator π = a + b from.1. Reca: L =1Q, O L = Z[ ], P p O L is an extension of the prime p 7mod8toLand π O L generates P p. Let us deduce that for primes 1mod8 with = p p = 1 the symbo π is we-defined. First of a, π is unique up to mutipication by the square of a unit in OL.Moreover, the prime p spits in L over 1Q. If po L = P p P p and π = a + b isageneratorofp p as in.1, then a b is such a generator of Pp. Their product a + b a b equas p, hence is a square moduo. Finay, the prime spits in L over 1Q. If P O L is an extension of to L, then the residue fied O L /P is isomorphic to Z/ Z=IF, the finite fied with eements. Since is a square moduo we have α and α in IF with α =inif. So, we may identify π = a + b O L with a + bα IF, and for any two choices π, π we have π = π in IF /IF. Thus π = ±1 is we-defined π a + b = for any choice of a, b Z as in.1. Reca the incusions N = L 1 p L 1 = L π L 1Q. The prime spits in L over 1Q. Since p =+1, every extension of to L1 wi spit in N over L 1. Thus spits competey in N over 1Q if and ony if any extension P O L of to L spits in L 1 over L. Thatis, spits competey in N over 1Q if and ony if π =+1. Thus, in terms of the Legendre symbo π, we have the foowing reformuation of 3.3 and 3.4. è3.5è Proposition: Let 1 mod 8 be prime with hk/4 = n +pm for some n, m Z with m 0mod p = p =+1. Then: if and ony if =+1. π hk/4 =n + pm for some n, m Z with m 0mod if and ony if π = 1. 3.5. è3.6è Iustration: For p = 7 and 31 with = 193 we iustrate.1, 3.3, 3.4, For p =7withK =1Q 14 and hk =4 wechooseπ = 1 ±, and for p =31withK =1Q 6 and hk =8wechooseπ =1± 4 in.1. We have 193 = 3 +7 5, 193 = 131 + 31 18 ; so by 3.4 and 3.3,
10 P. E. Conner and J. Hurrebrink 193 does not spit competey in N = 1Q, 7, π, yet 193 spits competey in N =1Q, 31, π. Hence π 193 = 1 forp =7and π 193 =+1forp =31accordingto3.5,whichof course is consistent, for p =7, with π = 1 ± 89 = 88, 90 in IF 193 and, for p =31, with π =1± 89 = 14, 16 in IF 193. The above proposition 3.5, combined with the foowing addendum 3.7, wi serve in the next section as the main too in estabishing our goa of determining the 4-rank of tame kernes in positive definite terms. For primes 1 mod 8, the quadratic symbos They satisfy: Proof: 1± and ± are we-defined. è3.7è Addendum: Let 1 mod 8 be prime. Then: 1+ a = +1 if and ony if = x +3y for some x, y Z. b = +1 if and ony if 1 mod 16. + a Compare [1], page 70. b For an appropriate choice of a primitive 16th root of unity, we have + = +ξ 16 + ξ 16 = ξ 16 + ξ 16. The caim foows since spits competey in 1Q ξ 16 + ξ 16 if and ony if spits competey in 1Qξ16 if and ony if 1 mod 16. 4. Characterizations in terms of symbos In this section we wi determine expicity the 4-rank of the tame kerne of the fieds E =1Q p, 1Q p andf =1Q p, 1Q p in terms of the quadratic symbo π exhibited in section 3. Reca that we are considering primes p 7mod8, 1mod8 with p = p =+1. For any number fied, the -rank of the tame kerne is given by the so-caed Tate -rank
Onthetamekerne 11 formua in terms of the -rank of an idea cass group, cp. 6.1 and 6.3 in [14]. For quadratic number fieds, the -rank of the tame kerne has been made expicit in [5]. It is s + t for rea quadratic number fieds, s + t 1 for imaginary quadratic number fieds, where s is the number of eements in {±1, ±} which are norms from the given quadratic fied, and t is the number of odd primes which are ramified in the given quadratic fied. For our fieds E and F : s =1andt =,so 4.1 -rk K O E = 3 -rk K O F =. What are the possibe vaues of the 4-rank? Because of p 7 mod 8 we know that 4-rk K O E > 0, by []. Moreover, since E is rea, the Steinberg symbo { 1, 1} of order ink O E cannot be a square in K O E ; thus 0 < 4-rk K O E < -rk K O E and hence: 4. 4-rk K O E =1or. Because of p 7 mod 8, we aso know for the imaginary fieds F that { 1, 1} is not a square in K O F, by 5.10 in [10], Cor. 3.8 in [11]. So, 4-rk K O F < -rk K O F and hence: 4.3 4-rk K O F =0or1. Our task amounts to characterizing when the 4-rank is given by the arger one, say, of the two possibe vaues in each of 4. and 4.3. By the main resut in [10], this determination can be made by just computing the matrix rank over the fied IF of 3 3 matrices M E/1Q, M F/1Q invoving Hibert symbos; compare Lemma 5.1 in [10] and the exampes concerning quadratic fieds in section 5.1 of [10]. The 4-rank of K O E,K O F takes on the argest vaue possibe if and ony if the matrix rank of M E/1Q, M F/1Q, respectivey, takes on the smaest possibe vaue. In our cases the matrices are of the form M E/1Q = 0 0 0 α + β β α,m F/1Q 1 1 0 = 1 1 0 α + β β α 1 1 0
1 P. E. Conner and J. Hurrebrink for some α, βɛif. So, just by inspection: 4.4 4-rk K O E = if and ony if rank M E/1Q = 1 4-rk K O F = 1 if and ony if rank M F/1Q = 1, which in each case happens if and ony if α = 0. Let us be more specific. Since a odd prime divisors of d = ±p, ±p are congruent to ±1 mod 8 and hence d is a norm from L =1Q over 1Q, we have d = u w with u, wɛ Z. The entry α of M E/1Q, M F/1Q is in both cases reated to the Hibert symbo d, u + w by d, u + w = 1 α, thus, in both cases, the matrix rank is 1 if and ony if d, u+w =1. Now,u +w is easiy seen to be on -adic unit and hence d, u + w =, u + w. We have obtained in both cases of 4.4 that the matrix rank is 1 if and ony if, u+w =1; in other words, if and ony if the Legendre symbo u+w is +1. Thus, we have deduced: d è4.5è Proposition: For E,F = 1Q with d = ±p, ±p as above, d = u w with u, wɛ Z, we have: 4-rk K O E = 4-rk K O F = 1 if and ony if if and ony if u+w u+w =+1 =+1. In the formuation of 4.5, we now have reated the approach from [10] to the main resuts earier obtained by Qin for quadratic number fieds by an independent approach; specificay, compare [11], tabe II and [1], tabe I. We can finish this section by reating the Legendre symbo u+w the quadratic symbo π from section 3. The resut is: from this section to
Onthetamekerne 13 è4.6è Proposition: Let d = ±p, ±p be as above, d = u w with u, wɛ Z. Then the symbos u+w from 4.5 and π from.1, 3.5 are reated as foows: u+w = π if d = p u+w = π + if d =p u+w = π 1+ if d = p u+w = π if d = p. Proof: Reca that the norm of π from L =1Q over 1Q is p and the norm of u + w from L over 1Q isd. Let us write = N L/1Q α + β d = N L/1Q γ + δ with α, β, γ, δɛ Z. Then d = N L\1Q α + β γ + δ = N L\1Q αγ +βδ +αδ + βγ, yieding for some choice of u, w: u + w = βγ +βδ + αδ + βγ. The symbo u+w is 1 if and ony if u + w is a square moduo if and ony if u + w is a square in Z[ ]/α + β. Hence we may repace α by β in the expression for u + w. Moreover we note that = =+1since = α β and hence ceary every β prime divisor of β is a square moduo. So: β u + w u + wβ =. Now, u + wβ βγ +βδ βδ +βγ β mod α + β γ +δ δ +γ β mod α + β 1 γ δ β mod α + β which is a square moduo α + β if and ony if 1 + γ + δ is. Therefore: u + w = 1+ This identity wi yied our caim in each of the four cases. γ + δ.
14 P. E. Conner and J. Hurrebrink For d = p we have d =p = N L/1Q π, so γ + δ = 1+ π u + w π yieding =. For d =p we have d =p = N L/1Q π,,up to squares; so γ + δ = π,up to squares; u + w π + yieding =. For d = p we obtain accordingy from the first case u + w π 1+ = and for d = p we obtain as we from the second case u + w π + 1+ π = = as caimed., 5. Characterizations in positive definite terms The upshot of the preceding two sections wi be that for a fieds 1Q ±p,1q ±p under consideration we can determine the 4-rank of the tame kerne in terms of the binary positive definite quadratic forms X +3Y, X +py, X + py. In a cases, a combination of 3.5, 3.7, 4.5 and 4.6 wi yied the desired resuts. In order to state the characterizations in positive definite terms it wi be convenient to introduce the foowing terminoogy: è5.1è Deænition: For primes p 7 mod 8, 1 mod 8 with p =p =+1and K =1Q p we say: satisfies A + if and ony if = x +3y for some x, y Z satisfies A if and ony if x +3y for a x, y Z
Onthetamekerne 15 satisfies <,p> if and ony if hk/4 =n + pm for some n, m Z with m 0mod satisfies < 1, p>if and ony if hk/4 = n +pm for some n, m Z with m 0mod. Let us start with the rea quadratic fieds E =1Q p,1q p. We know by 4. that 4-rk K O E = 1 or and obtain: è5.è Theorem: Let p 7 mod 8, 1 mod 8 be primes with p =p =+1. If E =1Q p then: 4-rk K O E = 1 if and ony if satisfies <,p>; 4-rk K O E = if and ony if satisfies < 1, p>. Proof: By4.5wehave4-rkK O E = if and ony if u+w = +1; hence, by 4.6, if and ony if π = +1; hence, by 3.5, if and ony if satisfies < 1, p>. And, by the same references: 4-rk K O E = 1 if and ony if u+w = 1 if and ony if π = 1 if and ony if satisfies <,p>. è5.3è Theorem: Let p 7 mod 8, 1 mod 8 be primes with p =p =+1. If E =1Q p with 1mod16, then : 4-rk K O E =1 if and ony if satisfies <,p>; 4-rk K O E = if and ony if satisfies < 1, p>. If E =1Q p with 9mod16, then: 4-rk K O E =1 if and ony if satisfies < 1, p >; 4-rk K O E = if and ony if satisfies <,p>. Proof: Asabovewehave4-rkK O E = if and ony if u+w = +1; hence, by 4.6, if and ony if π 1+ = +1; hence if and ony if π =+1and 1+ =+1
16 P. E. Conner and J. Hurrebrink or π = 1 and 1+ = 1 ; hence, by 3.5 and 3.7, if and ony if satisfies < 1, p> and 1 mod 16 or satisfies <,p> and 9 mod 16. Anaogousy, 4-rk K O E = 1 if and ony if π =+1and 1+ = 1 or π = 1 and 1+ =+1, and the caim foows. We now turn to the imaginary quadratic fieds F =1Q p, 1Q p. In this case we have by 4. that 4-rk K O F = 0 or 1 and obtain: è5.4è Theorem: Let p 7 mod 8, 1 mod 8 be primes with p =p =+1. If F =1Q p then: 4-rk K O F =0 if and ony if satisfies A + and <,p> or A and < 1, p>; 4-rk K O F =1 if and ony if satisfies A + and < 1, p> or A and <,p>. Proof: This time, in view of 4.6, we just have to spe out what it means that π 1+ = +1. By 3.5 and 3.7 that is equivaent to satisfying A + and < 1, p> or A and <,p> which henceforth is equivaent to 4-rk K O F = 1. Thus aso the characterization of 4-rk K O F = 0 foows. è5.5è Theorem: Let p 7 mod 8, 1 mod 8 be primes with p =p =+1. If F =1Q p with 1mod16, then : 4-rk K O F =0if and ony if satisfies A + and <,p> or A and < 1, p>; 4-rk K O F =1if and ony if satisfies A + and < 1, p> or A and <,p>. If F =1Q p with 9mod16, then : 4-rk K O F =0if and ony if satisfies A + and < 1, p> or A and <,p>; 4-rk K O F =1if and ony if satisfies A + and <,p> or A and < 1, p>. Proof: This time, in compete anaogy with the above:
4-rk K O F = 1 if and ony if π ever number of the three symbos π Onthetamekerne 17 = π + 1+ = +1 if and ony if an,, and is equa to 1, a of which + 1+ have been characterized in the given positive definite terms. In view of the above four theorems, we have accompished our task. Theorems 5. through 5.5 have the obvious coroary: è5.6è Coroary: If 1mod16, then the 4-ranks of the tame kernes of the rea fieds 1Q p and1q p are equa and the 4-ranks of the tame kernes of the imaginary fieds 1Q p and1q p are equa as we. If 9mod16, then the absoute vaue of the difference between the 4-ranks of the tame kernes of the rea fieds 1Q p and1q p is 1 and, as we, the absoute vaue of the difference between the 4-ranks of the tame kernes of the imaginary fieds 1Q p and 1Q p is1. 6. Iustrations, Densities In view of the characterizations in positive definite terms in Theorems 5., 5.3, 5.4, 5.5 it is now a simpe task to efficienty determine the 4-rank of the tame kerne of any given fied E =1Q p, 1Q p F =1Q p, 1Q p with primes p 7mod8, 1mod8, satisfying ν = 4-rk K O 1Q p µ = 4-rk K O 1Q p p = p =+1. Let σ =4-rkK O 1Q p τ =4-rkK O 1Q p. Reca that we have ν, µ = 1 or whie σ, τ = 0 or 1. Let us reaize various combinations. Because of 5.6, we can expect at most eight different tupes ν, µ, σ, τ. è6.1è Iustration: Let us choose p = 7. The primes 1mod8with 7 =+1 are the primes 1, 9, 5 mod 56; the cass number hk ofk =1Q 14 is 4. It turns
18 P. E. Conner and J. Hurrebrink out that a eight combinations of ν, µ, σ, τ can occur. Here are primes that reaize the indicated types ν, µ, σ, τ. ν, µ, σ, τ 113 1 1 0 0 193 1 1 1 1 137 1 1 0 33 1 0 1 617 1 1 0 457 1 0 1 449 0 0 19 1 1. Let us expain how, for the ast prime = 19 in our ist, the tupe,, 1, 1 is obtained. We have 1mod7 16 and 19 = 79 + 1400 = 7 +14 10,so satisfies < 1, 7 >. Thus by 5. together with 5.3 or 5.6 it foows: 4-rk K O 1Q 7 =4-rkK O 1Q 7 =. Aso, 19 = 81 + 048 = 9 +3 8,so satisfies A +. Hence by 5.4 together with 5.5 or 5.6 we obtain: 4-rk K O 1Q 7 =4-rkK O 1Q 7 =1. For rea quadratic fieds E, the confirmation of the Birch-Tate conjecture provides us with the order of K O E in terms of the vaue of the Dedekind zeta function at 1. That, together with the knowedge of the 4-rank, sometimes determines the structure of the -primary subgroup of K O E. 7 For exampe, for E = 1Q with the first two primes = 113 and 193 in our ist one has #K O E = 6 3 9 and 7 373, respectivey, and -rk K O E =3, 4-rk K O E = 1 in both cases by 4.1 and 5.. So, the structure of the -primary subgroup of K O E isgivenby C C C 16 and C C C 3, respectivey. In genera, not much seems to be known about densities concerning particuar 4-ranks of tame kernes. In the specia case of quadratic number fieds whose discriminant is of
Onthetamekerne 19 sma absoute vaue or has ony very few prime divisors, resuts on the 4-rank of the tame kerne have been made expicit in ists and tabes e.g. in [3],[4],[7],[11],[1]. As a motivating exampe we summarize, compare 5.1 in [6]: è6.è Exampe: Let q be a prime. The structure of the -primary subgroup of the tame kerne of the rea quadratic fieds 1Q q is given by: q = : C C q 3, 5mod8: C C q 7mod8 : C C a with a q 1mod8 : C C C if q satisfies A C C C a with a ifq satisfies A +. It has been proven in [6], see 1.6 and p.199, that the sets {q 1mod8:q satisfies A + } and {q 1mod8:q satisfies A } each have a natura density of 1 in the set of a primes q 1mod8. Hence we may concude by Dirichet s Theorem on primes in arithmetic progressions that, in the set of a primes q, the subsets {q : the tame kerne of 1Q q has 4-rank 0}, {q : the tame kerne of 1Q q has 4-rank 1} have natura densities 1 4 + 1 4 + 1 8 = 5 8 and 1 4 + 1 8 = 3 8, respectivey. In short, for rea fieds 1Q q : 4-rank 0 appears with density 5 8, 4-rank 1 appears with density 3 8. The structure of the -primary subgroup of the tame kerne of imaginary quadratic fieds 1Q q isgivenby: q = : {1} q 3, 5mod8: {1} q 7mod8 : C q 1mod8 : C a with a.
0 P. E. Conner and J. Hurrebrink So, for imaginary fieds 1Q q : 4-rank 0 appears with density 3 4, 4-rank 1 appears with density 1 4. For quadratic fieds whose discriminants have two odd prime divisors p, we have concentrated on p 7mod8, 1mod8with = p p = +1. Concerning densities, our choice perhaps is a somewhat invoved case. Yet, in view of the characterizations given in 5. 5.5 in positive definite terms, it might be reasonabe to expect some density resuts in cases ike this one, as we. è6.3è Density considerations: Let us singe out Theorem 5. for comment. For a fixed prime p 7mod8,et p Ω={ 1mod8prime: = =+1}, p Ω 1 = {ɛω : satisfies <,p>}, Ω = {ɛω : satisfies < 1, p>}. Consider the fieds E =1Q p with ɛω. By 5. we have: 4-rk K O E = 1 if and ony if ɛω 1, 4-rk K O E = if and ony if ɛω. Numericay one obtains: For p =7,thereare9, 730 primes in Ω with <10 6 ;amongthem, there are 4, 866 primes that ie in Ω 1 and 4, 864 that ie in Ω approximate percentages of 50.01% and 49.99%. We might ask: For the fieds E =1Q p as above, do 4-rank 0 and 4-rank 1 each appear with natura density 1? An affirmative answer wi be given in Robert Osburn s dissertation. Let us comment on the tabe in 6.1. For a fixed prime p 7mod8and in Ω we consider the fieds 1Q ±p and 1Q ±p. The 4-ranks of their tame kernes are given by tupes ν, µ, σ, τ. Numericay one obtains: For p =7,amongthe9, 730 primes in Ω with <10 6,the eight possibe cases are reaized by 115, 113, 18, 110, 110, 18, 15, 101 primes, respectivey. Again, it wi be proved in Robert Osburn s dissertation that each of the eight possibe tupes ν, µ, σ, τ isted in 6.1 appears with natura density 1 8.
Onthetamekerne 1 It remains to be seen if density resuts for the 4-rank or even some higher -power ranks of the tame kerne of quadratic number fieds, in genera, wi be in reach. The above approach in positive definite terms makes numerica computations easiy accessibe. The characterizations of particuar 4-ranks in terms of quadratic symbos might be some indication of the existence of densities in some generaity. References [1] P. Barrucand and H. Cohn, Note on primes of type x +3y, cass number and residuacity, J. reine angew. Math. 38 1969 67-70. [] B. Brauckmann, The -Syow subgroup of the tame kerne of number fieds, Can. J. Math. 43 1991 55-64. [3] J. Browkin, Tabe of the structure of the group K O F for rea quadratic fieds F, private communication, 1994. [4] J. Browkin and H. Gang, Tame and wid kernes of quadratic imaginary number fieds, to appear in Math. Computations. [5] J. Browkin and A Schinze, On -Syow subgroups of K O F for quadratic fieds, J. reine angew. Math. 331 198, 104-113. [6] P. E. Conner and J. Hurrebrink, Cass Number Parity, Ser. Pure Math. 8, Word Sci., Singapore 1988. [7] P. E. Conner and J. Hurrebrink, Exampes of quadratic number fieds with K O containing no eements of order four, circuated notes, 1989. [8] P. E. Conner and J. Hurrebrink, The 4-rank of K O F, Can. J. Math. 41 1989 93-960. [9] P. E. Conner and J. Hurrebrink, On eementary abeian -Syow K of rings of integers of certain quadratic number fieds, Acta Arith. 73 1995 59-65. [10] J. Hurrebrink and M. Koster, Tame kernes under reative quadratic extensions and Hibert symbos, J. reine angew. Math. 499 1998, 145-188.
P. E. Conner and J. Hurrebrink [11] H. Qin, The -Syow subgroups of the tame kerne of imaginary quadratic fieds, Acta Arith. 69 1995 153-169. [1] H. Qin, The 4-rank of K O F for rea quadratic fieds, Acta Arith. 7 1995 33-333. [13] H. Qin, Tame kernes and Tate kernes of quadratic number fieds, to appear in J. reine angew. Math. [14] J. Tate, Reations between K and Gaois cohomoogy, Invent. Math. 36 1976 57-74. [15] A. Vazzana, On the -primary part of K of rings of integers in certain quadratic number fieds, Acta Arith. 80 1997 5-35. [16] A. Vazzana, The 4-rank of K O and reated grpahs in certain quadratic number fieds, Acta Arith. 81 1997 53-64. [17] A. Vazzana, 8-ranks of K of rings of integers in quadratic number fieds, J. Number Th. 76 1999 48-64. P. E. Conner Jurgen Hurrebrink Department of Mathematics Louisiana State University Department of Mathematics Louisiana State University Baton Rouge, La. 70803 Baton Rouge, La. 70803 USA USA e-mai: jurgen@math.su.edu