The class number one problem for the dihedral CM

The class number one problem for the dihedral CM-fields Yann LEFEUVRE and St´ephane LOUBOUTIN Universit´e de Caen, Campus 2 Math´ematique et M´ecanique, BP 5186 14032 Caen cedex, France [email protected] Algebraic number theory and Diophantine analysis (Graz, 1998), 249–275, de Gruyter, Berlin, 2000.

Abstract We delineate the determination by S. Louboutin, R. Okazaki and Y. Lefeuvre of all the normal CM-fields with Galois group any dihedral group which have class number one (there are 32 such fields). We refer the reader to [Lou11] for the much easier determination by S. Louboutin of all the normal CM-fields with Galois group any dicyclic group which have class number one (there are no such fields).

Contents 1 Introduction and sketch of the determination

2

2 Real dihedral fields Fps of degree 2ps

6

3 Dihedral CM-fields of degree 4m = 2r ≥ 8

10

4 Dihedral CM-fields of degree 4m = 2r p

12

5 Lower bounds on relative class numbers 13 5.1 Dihedral CM-fields of 2-power degrees . . . . . . . . . . . . . 14 5.2 Dihedral CM-fields of degree 2r p and 2r p2 . . . . . . . . . . . 16 6 Computation of relative class numbers

18

7 Tables

22 1

NON-ABELIAN NORMAL CM-FIELDS

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2

Introduction and sketch of the determination

It is known that there are only finitely many normal CM-fields with class number one (see [Odl], [Hof]). However, the proofs of this finitess result did not point to any easy to handle technique for determining all of them. In this survey, we present the reader with the various techniques we have developed for determining all the normal CM-fields with class number one whose Galois groups are dihedral groups. There are three problems to overcome: we must give good lower bounds on the relative class numbers of such CM-fields, we must explain how class field theory enables us to construct such CM-fields, and we must finally explain how we compute the relative class numbers of such CM-fields. If E is a number field, we let dE , wE , hE and ζE denote the absolute value of its discriminant, its number of complex roots of unity, its class number and its Dedekind zeta function, respectively. If E is a CM-field, we let h− E and QE ∈ {1, 2} denote its relative class number and its Hasse unit index, respectively (see [Wa]). Let N be a non-abelian dihedral CM-field of degree 2n > 4, i.e. N is a normal CM-field whose Galois group is the dihedral group of order 2n whose presentation is: D2n = ha, b : an = b2 = 1, b−1 ab = a−1 i. Since the complex conjugation c must be in the center Z(D2n ) of its Galois group (see [LOO]), then n = 2m is even, which yields Z(D4m ) = {1, am } and c = am . Since D4m /Z(D4m ) is isomorphic to D2m , the dihedral group of order 2m, the maximal totally real subfield N+ of N is a real dihedral field of degree 2m. Let L be the fixed field of the cyclic subgroup < a > generated by a. Hence, L is a real quadratic field. Let M denote the maximal abelian subfield of N, i.e. M is the fixed field of the derived subgroup D(D4m ) =< a2 >= {a2k ; 0 ≤ k ≤ m − 1} of D4m , and Gal(M/Q) is isomorphic to the quotient group D4m /D(D4m ). Therefore, M is bicyclic biquadratic, contains L and M is imaginary if m is odd, whereas M is real if m is even. Write 2n = 2r D with D ≥ 1 odd and r ≥ 2. We let N2 be the fixed field of the cyclic subgroup generated by aD , hence N2 is a dihedral CM-field of degree 2r which is cyclic over L. Notice that N2 is abelian if and only if 2r = 4. We have the following lattice of subfields: Q

2

L

2

M

2r−2 odd N2 N

In the same way, for any odd divisor d of D, we let Nd denote the subfield of N fixed by the cyclic subgroup < aD/d > of D2n and notice that Nd is

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a dihedral subfield of N of degree 2r d which is cyclic over L and contains N2 . We also let Fd denote the subfield of N fixed by the cyclic subgroup < a2m/d > of D2n and notice that Fd is a real dihedral subfield of N of degree 2d. Moreover, Nd = N2 Fd . Finally, let Kp denote any one of the p conjugate subfields of degree p of Fp . If p is prime and ps divides 2m, we have the following lattice of subfields: N

Nps = Fps N2

ps−1 + N+ ps = Fps N2

Np = Fp N2

Fp s

ps−1

ps−1

p + N+ p = Fp N2

N2

Fp

p 2

Kp

p

N+ 2 2r−2

p L 2 Q Proposition 1 (See [LOO, Th. 5]). Let k ⊆ K be two CM-fields. Assume − that [K : k] = m is odd. Then Qk = QK and h− k divides hK .

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− − − − − Therefore, h− N2 divides hNp , hNp divides hNps and hNps divides hN and we proceed as follows:

1. We determine the set N2 of all the dihedral CM-fields N2 of 2-power degrees 2r ≥ 4 with relative class number one. It will turn out that this set N2 is finite and contains 171 elements, 24 out of them being non-abelian, i.e. of degree 2r ≥ 8 (see Proposition 15 and Theorem 9). 2. To determine the set NP of all the dihedral CM-fields Np of degree 2r p, p any odd prime and r ≥ 2, with relative class number one, for each N2 ∈ N2 we determine all the Np ∈ NP containing N2 . It will turn out that this set NP is finite and contains 19 elements (see Theorem 10). 3. To determine the set NP 2 of all the dihedral CM-fields Np2 of degree 2r p2 , p any odd prime and r ≥ 2, with relative class number one, for each Np ∈ NP we determine all the Np2 ∈ NP 2 containing Np . It will turn out that this set NP 2 is empty (see Corollary 18). 4. To determine the set Nh− =1 of all the non-abelian dihedral CM-fields N with relative class number one, we notice that if N ∈ Nh− =1 is of degree 2r d then for any prime p dividing d the dihedral CM-subfield Np of N of degree 2r p is in NP . Using point 2, we get d = ps for some odd prime p and some s ≥ 0 (see Corollary 11). Using point 3 we get s = 1. Therefore, Nh− =1 = N2 ∪ NP contains 43 = 24 + 19 elements. 5. We determine the subset Nh=1 of Nh− =1 of all the non-abelian dihedral CM-fields N with class number one. It will turn out that this set Nh=1 contains 32 elements. We now give a more detailed account of this determination: 1. First, we characterize the dihedral CM-fields N2 of 2-power degrees 2r ≥ 8 with odd relative class number (see Theorem 8). In particular, N2 which must be the narrow Hilbert 2-class field of L is well determined by L. Second, we give lower bounds on the relative class numbers of the dihedral CM-fields N2 of 2-power degrees 2r ≥ 8 with odd relative class number (see Theorem 14). We deduce that dL > 4 · 106 implies h− N2 > 1. Finally, we compute the relative class numbers of 6 all the possible N2 for which h− N2 is odd and dL ≤ 4 · 10 (see Section 6). We obtain that there are 24 dihedral CM-fields of 2-power degrees 2r ≥ 8 with relative class number one (see Theorem 9).

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2. We determine all the imaginary biquadratic bicyclic number fields M4 with relative class number one. There are 147 such fields M4 (see Proposition 15). We notice that the Dedekind zeta functions of the 171 = 24 + 147 biquadratic bicyclic subfields M of the 171 dihedral CM-fields N2 of 2-power degrees with relative class number one satisfy ζM (β) ≤ 0, 0 < β < 1. Hence, if N is a dihedral CM-field of degree 4m with h− N = 1 then ζN (β) ≤ 0 for 0 < β < 1. 3. Let L denote the finite set of the 155 real quadratic subfields L associated with the 171 dihedral CM-fields N2 ∈ N2 of 2-power degree 2n = 2r ≥ 4 with relative class number one. Now, for each of these 155 possible real quadratic fields L ∈ L, we set out to determine all the dihedral CM-fields N cyclic over L such that h− N = 1. 4. Let F be a real dihedral field of degree 2ps cyclic over a real quadratic field L. We prove that there exists some positive integer fF/L such that the conductor of the cyclic extension F/L is equal to (fF/L ) (see Proposition 2). To simplify, fF/L is also called the conductor of F/L. Using point 2, we give lower bounds on the relative class numbers of the dihedral CM-fields N for which h− N2 = 1 (see Theorem 16). 5. We prove that if there exists a dihedral field Fp of degree 2p, cyclic over L and of conductor fFp /L > 1, then p divides some easy to compute positive integer iL (fFp /L ) (see Corollary 3). This necessary condition enables us to compute a lower bound FL (p) on the conductors of the real dihedral fields Fq of degree 2q ≥ 2p which are cyclic over L. Using point 4 and these FL (p), we prove that if h− Np = 1 then p ≤ 17, a very reasonable bound (see Corollary 18). 6. Now, for each of the 155 possible L ∈ L and for each p ≤ 17 we compute a bound BL (p) such that h− Np = 1 implies fFp /L ≤ BL (p) and we use class field theory for constructing all the real dihedral fields Fp of degree 2p and conductors fFp /L ≤ BL (p) (see Section 5.2). Finally, we compute the relative class numbers of all the possible Np = N2 Fp with fFp /L ≤ BL (p). We end up with a list of 19 dihedral CM-fields Np for which h− Np = 1 (see Theorem 10). 7. We notice that no two of the 19 CM-fields Np for which h− Np = 1 are cyclic over the same real quadratic subfield L (see Corollary 11). r s Therefore, if h− N = 1 then 2n = 4m = 2 d with either d = 1 or d = p . Finally, using point 4 and proceeding in much the same way as in

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point 6, we reduce the determination of all the Np2 ’s with relative class number one to the computation of the relative class numbers of finitely many Np2 ’s and prove that if h− Np = 1 and Np2 contains Np − then hN 2 > 1 (see Corollary 18). p

2

Real dihedral fields Fps of degree 2ps

This section is devoted to outlining the construction of the real dihedral fields Fps of degree 2ps and cyclic of degree ps over a given real quadratic field L. To this end, we use class field theory to build Fps from L. Let F be an integral ideal of a real quadratic field L. Let IL (F) denote the subgroup of the group IL of fractional ideals of L generated by the integral ideals relatively prime to F, and let PL (F) denote the subgroup of IL (F) generated by the principal ideals of the form (α) where α ∈ AL satisfies α ≡ 1 (mod F). According to class field theory, for any abelian extension F/L of conductor dividing F the kernel H = ker ΦF/L of the surjective Artin map ΦF/L : IL (F) −→ Gal(F/L) satisfies PL (F) ⊆ H ⊆ IL (F). In particular, there is a bijective correspondence between (1) the cyclic groups of order n generated by the primitive characters χ of order n on a ray class group ClL (F) = IL (F)/PL (F) and (2) the cyclic extensions F/L of degree n and conductor F. If χ is a primitive character of order n on a ray class group ClL (F), we let χ0 denote its associated modular character on (AL /F)∗ defined by χ0 (α) = χ((α)). This modular character χ0 is primitive of order dividing n. Finally, we let PL,Z (F) denote the subgroup of PL (F) generated by the principal ideals (α) where α ranges over the α ∈ AL which satisfy α ≡ a (mod F) for some integer a relatively prime to F. Proposition 2 (See[LPL]). Let p be an odd prime, s ≥ 1 and let L a real quadratic field be given. We let F denote a real dihedral field of degree 2ps , cyclic over L. The conductor FF/L of the cyclic extension F/L is invariant under the action of Gal(L/Q), ker ΦF/L contains PL,Z (FF/L ) and there exists a positive rational integer fF/L ≥ 1 such that FF/L = (fF/L ). Moreover, for a given positive rational integer f ≥ 1, there is a bijective correspondence between (1) the dihedral fields F of degree 2ps , cyclic over L and such that FF/L = (f ) and (2) the cyclic groups of order ps generated by the primitive characters of order ps on ClL ((f )) which are trivial on

NON-ABELIAN NORMAL CM-FIELDS

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PL,Z ((f )). In particular, if χ is any one of the characters of order ps on the ray class group ClL ((fF/L )) associated with F, then its associated modular character χ0 is a primitive character of p-power order on (AL /(fF/L ))∗ which is trivial on the image of (Z/fF/L Z)∗ in (AL /(fF/L ))∗ and on the fundamental unit L of L. Proof. Let a be a rational integer relatively prime to FF/L and let s σ ∈ D2p = Gal(F/Q). Then σΦF/L ((a))σ −1 = ΦF/L (σ(a)) = ΦF/L ((a)). Therefore, ΦF/L ((a)) is in the centre of D2ps which is trivial. Hence, (a) ∈ ker ΦF/L , χ is trivial on PL,Z (FF/L ) and χ0 is primitive and trivial on the image of Z. Since FF/L is invariant under the action of the Galois group Gal(L/Q), to get the desired result we must prove that for any prime ramified ideal Q of L dividing FF/L the exponent e of Q in FF/L is even. Let χQ denote the component of χ0 on (AL /Qe )∗ . Notice that χQ is primitive and trivial on the image of Z. Assume that e = 2f + 1 were odd. If α ≡ 1 (mod Q2f ), then α = 1+q f β for some β ∈ AL . Since the canonical map Z/qZ −→ AL /Q is bijective, there exists a ∈ Z such that β ≡ a (mod Q), which yields α ≡ 1+q f a (mod Q2f +1 ) and χQ (α) = χQ (1+q f a) = +1 for χQ is trivial on ImZ. A contradiction. • Corollary 3 (See[LPL]). Let p be an odd prime, let L be a real quadratic field and F be a real dihedral field of degree 2ps , cyclic over L. 1. (a) Let q 6= p be a prime. If there exists a primitive character of order ps on (AL /(q e ))∗ which is trivial on the image of (Z/q e Z)∗ in this group, then e = 1 and q ≡ χL (q) (mod ps ). (b) Assume that p ≥ 3 does not divide dL . If there exists a primitive character of order ps on (AL /(pe ))∗ which is trivial on the image of (Z/pe Z)∗ in this group, then e = s + 1. (c) Assume that p ≥ 3 divides dL . There exists a primitive character of order ps on (AL /(pe ))∗ which is trivial on the image of (Z/pe Z)∗ in this group if and only if i. p ≥ 5 and e = s, ii. p = 3, dL ≡ 3 (mod 9) and e = s, iii. p = 3, dL ≡ 6 (mod 9), s ≥ 2 and e = s + 1, iv. p = 3, dL ≡ 6 (mod 9), s = 1, and e = 1 or e = 2. 2. If f > 1 is the conductor of some real dihedral field of degree 2p > 6 then a

f =p

r Y i=1



qi

with a = 0 or a =

2 1

if p does not divide dL if p divides dL

NON-ABELIAN NORMAL CM-FIELDS

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where the qi ’s are pairwise distinct primes different from p satisfying qi ≡ χL (qi ) (mod p). In particular, we obtain f ≥ p. 3. Assume f > 1. Set Gf = (AL /(f ))∗ /(Z/f Z)∗ . Then the order nL (f ) = min{k ≥ 1; ∃a ∈ Z / kL ≡ a

(mod (f ))}

(1)

of L in this multiplicative group Gf divides the order φL (f ) = f

(1 − q −1 χL (q)).

Y

(2)

q|f

of this group Gf , and if there exists a real dihedral field Fps of degree 2ps , cyclic over L and such that FFps /L = (f ) then p divides def

iL (f ) = φL (f ) / nL (f ). Moreover, if the cyclic subextension Fp /L is ramified at at least one finite prime then ps divides iL (f ). Definition 4 Let L be given. We set fL (p) = min{f > 1 as in Corollary 3, point 2 and p divides iL (f )}

(3)

and FL (q) = min{fL (p); p prime and p ≥ q}.

(4)

Notice that since p > 3 implies fL (p) ≥ p, then q > 3 implies FL (q) = min{fL (p); p prime and q ≤ p ≤ fL (q)} which is efficiently computed by the following algorithm: Input: q and L Begin FL (q) := fL (q); p := q + 2; While p ≤ FL (q) do If p is prime then FL (q) := min(FL (q), fL (p)); p = p + 2; Return FL (q); End Remark 5 Let us explain how one can compute nL (f ). We let gL√ denote the unique √ rational integer with the √ same parity as dL such that dL − 2 < gL < dL and set ωL = (gL + dL )/2. Then L = ql−1 ωL + ql−2 where l ≥ 1 is the length of the purely periodic continued fractional expansion ωL

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= [a0 , · · · , al−1 ] of ωL and where q−2 = 1, q−1 = 0 and qk = ak qk−1 + qk−2 , k ≥ 0. Moreover, setting T = TL/Q (L ) = ql−1 gL +2ql−2 and N = NL/Q (L ) = ±1, we have kL = Ak L − N Ak−1 where A−1 = −N , A0 = 0 and Ak+1 = T Ak − N Ak−1 , k ≥ 0. Therefore, nL (f ) = min{k ≥ 1; f divides ql−1 Ak }. Theorem 6 (See [LPL]). Let p ≥ 3 be an odd prime, L be a real quadratic field and f > 1 be a rational integer. 1. Assume that p does not divide the class number hL of L. Then, there is a bijective correspondence between (1) the cyclic groups of order ps generated by the primitive modular characters of order ps on (AL /(f ))∗ which are trivial on the image of (Z/f Z)∗ in this group and on L and (2) the real dihedral fields Fps of degree 2ps containing L and such that fFps /L = f. 2. Assume that p divides the class number hL of L and that the p-Sylow subgroup of the ideal class group of L is cyclic of order p. Let Q be a prime ideal of norm q 6≡ 0, 1 (mod p) a prime whose ideal class generates this p-Sylow subgroup (notice that q will be relatively prime to the conductors of all the real dihedral fields Fps of degree 2ps containing L). Let αQ ∈ AL be such that Qp = (αQ ). Let Hp denote the Hilbert p-class field of L (notice that Hp is a real dihedral field of degree 2p containing L). Then, (a) If Fp is a real dihedral field of degree 2p containing L such that fFp /L = f , then the p − 1 primitive modular characters of order p on (AL /(f ))∗ associated with Fp are trivial on the image of (Z/f Z)∗ in this group, on L and on αQ . Conversely, for any primitive modular character χ0 of order p on (AL /(f ))∗ which is trivial on the image of (Z/f Z)∗ in this group, on L and on αQ there are p real dihedral fields Fp of degree 2p containing L such that fFp /L = f and whose p−1 associated primitive modular characters lie in the cyclic group generated by χ0 . (b) There is a bijective correspondence between (1) the cyclic groups of order p generated by the primitive modular characters of order p on (AL /(f ))∗ which are trivial on the image of (Z/f Z)∗ in this group, on L but not on αQ and (2) the real dihedral fields Fp2 of degree 2p2 containing Hp and such that fFp2 /L = f. Proof. We content ourselves with a proof of the second point in the simplest case where hL = p. Let χ denote any one of the characters of order ps on

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ClL ((f )) associated with Fps and whose associated modular character is equal to a given primitive modular character χ0 . Set ζχ = χ(Q) and notice that we must have ζχp = χ0 (αQ ). For any integral ideal I ∈ IL (f ) there exists a unique kI ∈ {0, 1, · · · , p − 1} such that IQkI = (βI ) is principal, and we must have χ(I) = χ0 (βI )ζχ−kI . (5) In case 2(a), ζχ must be a pth complex root of unity, which yields χ0 (αQ ) = 1. Conversely, assume that χ0 (αQ ) = 1 and choose for ζχ any one of the pth complex root of unity. Since the coresponding p characters χ defined by (5) generate pairwise distinct cyclic groups and all have their associated modular characters equal to χ0 , we get the desired result. In case 2(b), χ has order p2 while χ0 has order p. Therefore, ζχ must be a primitive p2 th complex root of unity, which yields χ0 (αQ ) = ζχp 6= 1. Conversely, assume that χ0 (αQ ) 6= 1 and choose for ζχ any one of the p primitive p2 th complex roots of unity such that ζχp = χ0 (αQ ). Since the corresponding p characters χ defined by (5) generate the same cyclic group (ζχ0 −(1+λp)kI I = ζχ1+λp yields χ0 (I) = χ0 (βI )ζχ−k = χ0 (βI )ζχ = (χ0 (βI )ζχ−kI )1+λp 0 = (χ(I))1+λp ) and all have their associated modular characters equal to χ0 , we get the desired result. •

3

Dihedral CM-fields of degree 4m = 2r ≥ 8

To begin with, we use the following Proposition to characterize the dihedral CM-fields of 2-power degrees with odd relative class number. Proposition 7 1. (See [LO1] and [Lou4]). Let t denote the number of prime ideals of N which are ramified in the quadratic extension N/N+ . Then, 2t−1 − t divides h− N , and if QN = 2, then 2 divides hN . Moreover, if QN = 2 − and 2 divides hN+ , then 2 divides hN . 2. Let N = N1 N2 be a CM-field which is a compositum of two CM-fields. + Assume that N+ 1 = N2 and that N1 and N2 are isomorphic. Then − 2 h− N = (QN /2)(hN1 /QN1 ) .

(6)

+ In that situation, if h− N is odd then QN = 2, N/N is unramified at all the finite places and hN is odd.

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Theorem 8 (See [Lou5]). N is a dihedral CM-field of 2-power degree 8l = 2r ≥ 8 with odd relative class number if and only if N is the narrow Hilbert 2-class field of a real quadratic field L such that 1. the norm of the fundamental unit of L is equal to +1, 2. and the 2-Sylow subgroup of the narrow ideal class group of L is cyclic. √ In that situation, QN = 2, hN is odd and L = Q( pq) for some positive prime p and q not equal to 3 modulo 4. Proof. Assume that N is a dihedral CM-field of 2-power degree 2n = 8l = 2r ≥ 8 with odd relative class number. Let K, N1 and N2 be the fixed fields of the subgroups {1, b, c = a2l , bc}, {1, b} and {1, bc}, respectively. Notice that N2 = al (N1 ) is isomorphic to N1 . According to the previous Proposition, the quadratic subextension N/N+ of the cyclic extension of 2-power degree N/L is unramified at all the finite places. Therefore, N/L is unramified at all the finite places. Moreover hN is odd. Therefore, the narrow genus field GM of M is included in N, hence is included in M which is the maximal abelian subfield of N. Therefore, the biquadratic bicyclic √ √ real field M is equal to its narrow genus field and therefore M = Q( p, q) for some distinct positive primes p and q not equal to 3 modulo 4. Since N/L is cyclic of degree 2r−1 and unramified at all the finite places then the 2-Sylow subgroup of the narrow ideal class group of L is not trivial and L √ is therefore equal to Q( pq), and according to genus theory the 2-Sylow subgroup of the narrow ideal class group of such an L is cyclic. • Theorem 9 (See [LO2]). There are 24 dihedral CM-fields of 2-power degrees 2n = 2r ≥ 8 with relative class numbers equal to one. More precisely, 1. There are 19 dihedral CM-fields of degree 8 with relative class numbers equal to one: the narrow Hilbert 2-class fields of the 19 real quadratic √ number fields L = Q( pq) with pq ∈ {2 · 17, 2 · 73, 2 · 89, 2 · 233, 2 · 281, 5 · 41, 5 · 61, 5 · 109, 5 · 149, 5 · 269, 5 · 389, 13 · 17, 13 · 29, 13 · 157, 13 · 181, 17 · 137,√17 · 257, 29 · 53, 73 · 97}. The narrow Hilbert 2-class field of L = Q( 5√· 269) has class number 3, the narrow Hilbert 2-class field of L = Q( 17 · 257) has class number 3, and the 17 remaining narrow Hilbert 2-class fields have class number one. 2. There are 5 dihedral CM-fields of degree 16 with relative class numbers equal to one: the narrow Hilbert 2-class fields of the 5 real quadratic √ number fields Q( pq) with pq ∈ {2 · 257, 5 · 101, 5 · 181, 13 · 53, 13 · 61}.

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√ The narrow Hilbert 2-class field of L = Q( 2 · 257) has class number 3, and the 4 remaining narrow Hilbert 2-class fields have class number one. 3. Dihedral CM-field of degree 2n = 2r > 16 have relative class numbers greater than one.

4

Dihedral CM-fields of degree 4m = 2r p

Theorem 10 There are 19 dihedral CM-fields Np in NP . More precisely, 1. (See [LOO]). There are 16 non-abelian normal CM-fields Np = Fp M of degree 4p = 12 with relative class number one: the 16 dihedral CM-fields Np = KM given in TABLE 1. Here, K = Q(αK ) (with PK (αK ) = 0) is a totally real non normal cubic number field and M is an imaginary bicyclic biquadratic number field with relative class √ number equal to 1 and such that L = Q( dK ) ⊆ M. Moreover, 9 out of these 16 fields Np have class number one. 2. (See [Lef]). There are 2 dihedral CM-fields Np = Fp M of degree 4p ≥ 20 with relative class number one: the composita Np = KM of degree 4p = 20 given in TABLE 2. Here, K = Q(αK ) (with PK (αK ) = 0) is a totally real non normal quintic number field and M is an imaginary bicyclic biquadratic √ number field with relative class number equal to 1 such that L = Q( dK ) ⊆ M, and such that KL is dihedral of degree 10. Only 1 of these 2 fields Np has class number one. 3. (See [Lef]). There is only one dihedral CM-field of degree 2r p, r ≥ 3 with relative class number √ one: the narrow Hilbert class field of the real quadratic field L = Q( 1345). [N : Q] = 8p = 24, N = KN2 q We have √ √ √ where N2 = Q( 5, 269, −(17 + 2 5)) is a dihedral octic CM-field and K is a non-normal cubic field defined by the cubic polynomial PK (X) = X 3 − 7X − 1. Moreover, N has class number one.

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TABLE 1 (p = 3) dL dL0 dL1 fFp /L PK (X) hN+ 5 −3 −15 18 X 3 − 12X − 14 1 12 −3 −4 15 X 3 − 15X − 20 1 21 −3 −7 6 X 3 − 6X − 2 1 24 −3 −8 9 X 3 − 9X − 6 1 3 37 −4 −148 2 X + X 2 − 3X − 1 1 57 −3 −19 5 X 3 − X 2 − 8X − 3 1 60 −3 −20 9 X 3 − 18X − 12 2 88 −8 −11 7 X 3 + X 2 − 16X − 8 3 105 −7 −15 9 X 3 − 27X − 51 2 120 −3 −40 11 X 3 − 33X − 22 2 456 −19 −24 3 X 3 − 18X − 16 2 469 −7 −67 1 X 3 + X 2 − 5X − 4 1 473 −11 −43 1 X 3 − 5X − 1 1 940 −4 −235 1 X 3 − 7X − 4 2 1304 −8 −163 1 X 3 − X 2 − 11X − 1 1 1708 −4 −427 1 X 3 − X 2 − 8X − 2 2 (note the misprints in [LOO, Table 1] for the case dK = 8505 where we should have written (D0 , D1 ) = (7, 15)). TABLE 2 (p = 5) dL dL0 dL1 fFp /L PK (X) hN+ 280 −8 −35 5 X 5 − 35X 3 − 30X 2 + 10X + 4 2 817 −19 −43 1 X 5 + X 4 − 6X 3 − 5X 2 + 3X + 1 1

Corollary 11 If the relative class number of a dihedral CM-field of degree 4m = 2r d (with d > 1 odd) is equal to one then d = ps is a prime power. Proof.

If at least two distinct odd primes p1 and p2 divide d then h− Np

1

= h− Np2 = 1 and Np1 and Np2 are cyclic over the same real quadratic subfield subfield L which, according to Theorem 10, cannot happen. •

5

Lower bounds on relative class numbers

Let N be any CM-field of degree 2n and maximal totally real subfield N+ . Then (see [Wa, Chapter 4]) h− N

Q N wN = (2π)n

s

dN Ress=1 (ζN ) . dN+ Ress=1 (ζN+ )

(7)

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To get lower bounds on relative class numbers, we use (7), lower bounds on Ress=1 (ζN ) and upper bounds on Ress=1 (ζN+ ): Proposition 12 1. (See [Lou8]). Let L be a real quadratic number field. Then, 1 Ress=1 (ζL ) ≤ (log dL + 0.05), 2

(8)

and if there exists β ∈]0, 1[ such that ζL (β) = 0, then we also have Ress=1 (ζL ) ≤

1−β log2 dL . 8

(9)

2. (See [Lou8], [Lou9]). Let L be a given real quadratic number field. There exists µL > 0 such that for any primitive Hecke character χ associated with an abelian extension N+ /L unramified at all the infinite places we have 

|L(1, χ)| ≤ Ress=1 (ζL )

1 log NL/Q (Fχ ) + 2µL . 2 

(10)

Moreover, if the finite part Fχ of the conductor of χ is trivial, then µL Ress=1 (ζL ) ≤

1 log2 dL . 8

(11)

3. (See [Lou1, Proposition A]). Let N be CM-field of degree 2n. Set βN = 1 − (2/ log dN ) and choose εN =

  2 1/2n exp −2πn/dN 5

1/2n

or εN = 1 − (2πne1/n /dN ).

Then β ∈ [βN , 1[ and ζN (β) ≤ 0 imply Ress=1 (ζN ) ≥

5.1

εN (1 − β). e

(12)

Dihedral CM-fields of 2-power degrees

Corollary 13 Let N be a dihedral CM-field of degree 8l. Assume that N/L is unramified at all finite places. Then Ress=1 (ζM ) ≤

1 (log dL + 0.1)3 , 32

(13)

NON-ABELIAN NORMAL CM-FIELDS

15

and if there exists β ∈ [ 21 , 1[ such that ζM (β) = 0, then we also have Ress=1 (ζM ) ≤

1−β (log dL + 0.05)4 . 128

(14)

Finally, according to (11), we have 1 log2 dL Ress=1 (ζN+ ) ≤ Ress=1 (ζM ) 8 

2l−2

.

(15)

Theorem 14 (See [LO2]). Let N be a dihedral CM-field of degree 8l. Assume that N/L is unramified at all finite places. Then, h− N

QN ≥ εN 2el



4dL 4 π (log dL + 0.1)4

l

.

(16)

In particular, if 8l = 2r ≥ 8 and dL ≥ 4 · 106 imply h− N > 1. Proof. We note that ζN (β) = ζM (β)

Y

L(β, χ, N/L)

χ

where the product is taken over the 4l − 2 non-quadratic and non-trivial characters of degree one of the abelian extension N/L. Since these characters come in conjugate pairs, we get (ζN /ζM )(β) ≥ 0 for β ∈]0, 1[. Note that dN 2l = d4l L and dN+ = dL . Now, on the one hand, if ζM (βN ) ≤ 0, then (12) yields Ress=1 (ζN ) ≥ 2εN /e log dN = εN /2el log dL , and by using (7), (13) and (15) we finally get h− N

QN εN ≥ 2el



4dL 4 π (log dL + 0.1)4

l

.

(17)

On the other hand, if ζM (βN ) > 0, then there exists β ∈ [βN , 1[ such that ζM (β) = 0. Then ζN (β) = 0 and by using (7), (12), (14) and (15) we get h− N

4QN εN ≥ e



4dL π 4 (log dL + 0.05)4

l

.

(18)

Since the right hand side of (18) is greater than the right hand side of (17), we get the desired lower bound on h− N. •

NON-ABELIAN NORMAL CM-FIELDS

5.2

16

Dihedral CM-fields of degree 2r p and 2r p2

Proposition 15 1. Let M4 = L0 L1 be an imaginary biquadratic bicyclic field, where L0 and L1 denote the two imaginary quadratic subfields of M. If M4 = Q(ζ8 ) then h− M4 = 1, and if M4 6= Q(ζ8 ) then h− M4 = QM4 hL0 hL1 /2. Hence, h− M4 = 1 implies hLi ∈ {1, 2} and we obtain that there are 147 imaginary biquadratic bicyclic fields M4 with relative class number one. 2. The Dedekind zeta functions ζM of the maximal abelian (i.e. of the biquadratic bicyclic) subfields M of the 171 = 24 + 147 dihedral CMfields N2 ∈ N2 of 2-power degree 2r ≥ 4 with relative class number one satisfy ζM (β) ≤ 0 for 0 < β < 1. 3. (See [LOO] and [Lou11]). Let Np be a dihedral CM-field of degree 4p and assume that 2 is totally ramified in M/Q. Then 2p−1 divides h− Np . Theorem 16 Let N be a dihedral CM-field of degree 4m and assume that the zeta function ζM of its maximal abelian (i.e. its bicyclic biquadratic) subfield M satisfies ζM (β) ≤ 0 for 0 < β < 1. Then ζN (β) ≤ 0 for 0 < β < 1, and according to (7), to point 3 of Proposition 12 and since QN = QM (see Proposition 1) and wN = wM , we obtain h− N

p

2QM wM dN /dN+ ≥ N . e(2π)2m (log dN )Ress=1 (ζN+ )

(19)

Since dN ≥ d2N+ , we also obtain h− N

√ 2 dN+ ≥ εN . e(2π)2m (log dN+ )Ress=1 (ζN+ )

(20)

Notice that according to Proposition 1 and to point 2 of Proposition 15, if N is a dihedral CM-field such that h− N = 1 then ζM (β) ≤ 0 for 0 < β < 1. Therefore, we may use Theorem 16 to obtain an upper bound on the discriminant of N. In particular, we get that there are only finitely many dihedral CM-field with relative class number one.

NON-ABELIAN NORMAL CM-FIELDS

17

Proposition 17 Let us fix one N2 ∈ N2 out of the 171 dihedral CM-fields of degree 2r ≥ 4 with relative class number one (see Theorem 9 and Proposition 15). Let N = N2 Fps range over the dihedral CM-fields of degree 4m = 2r ps containing N2 . Then, ζM (β) ≤ 0 for 0 < β < 1, N+ 2 /L is unramified, r−2

2(p−1)

dN+ = d2Fps , dFp = dpL fFp /L , dF

p2

2

2(p−1) 2(p2 −p) p2 /L

= dpL fFp /L fF

and Ress=1 (ζN+ ) ≤ (Ress=1 (ζL ))m (log fFps /L + 2µL )m−1 , In particular set f = fFp /L and assume that s = 1. Then, εN

4πm 2 = exp − 1/4m 5 dN

!

2r π p 2 , ≥ exp − √ 5 dL f 2/3 



and using (20) we obtain h− N ≥ HL (f, p)

(21)

where def

HL (f, p) = and

2 5e2r−3 f 2r−2

r−2 log f + 2µL (CL (f ))2 p p log dL + 2(p − 1) log f

√

dL f 4π exp − √ f −2/3 . 4π 2 Ress=1 (ζL ) log f + 2µL dL Now, f 7→ HL (f, p) increases with f for f ≥ 5 and p 7→ HL (f, p) increases with p ≥ 3, provided that f is such that CL (f ) ≥ exp(1/2). 

def



CL (f ) =

Proof. We have f

∂ log HL ∂f

= 2r−2 (p − 1) −

2r−2 p − 1 2(p − 1) 2r+1 pπ √ − + log(dpL f 2(p−1) ) log f + 2µL 3f 2/3 dL

≥ 2r−2 (p − 1) −

2r−2 p − 1 1 − log f log f

=

2r−2 (p − 1) p (log f − ) log f p−1

which is greater than or equal to 0 for f ≥ exp(p/(p − 1)), hence for f ≥ exp(3/2), hence for f ≥ 5. In the same way, we have ∂ log HL log dL + 2 log f 1 = 2r−2 log CL (f ) − ≥ log CL (f ) − ∂p p log dL + 2(p − 1) log f p−1 is greater than or equal to 0 for CL (f ) ≥ exp(1/2). •

NON-ABELIAN NORMAL CM-FIELDS

18

Corollary 18 1. Let Np = N2 Fp be a dihedral CM-field of degree 4m = 2r p. Then, r = 2 and p ≥ 17, or r ≥ 3 and p ≥ 19 imply h− Np > 1. 2. Let Np of degree 2r p be any one of the 19 dihedral CM-fields in NP (see Theorem 10). Let Np2 = N2 Fp2 be a dihedral CM-field of degree 4m = 2r p2 . If Np2 contains Np then h− N 2 > 1. p

Proof. 1. Set p0 = 17 if r = 2 and p0 = 19 if r ≥ 3. For each of the 155 real quadratic fields L ∈ L we computed FL (p0 ) (see TABLES I and II). Now, L ∈ L and p ≥ 17 imply that p does not divide hL , hence imply f = fFp /L > 1. Now, L ∈ L and f ≥ FL (p0 ) imply CL (f ) ≥ exp(1/2) (use the data in TABLES I and II) and HL (f, p) ≥ HL (f, p0 ) ≥ HL (FL (p0 ), p0 ) > 1. 2. Use Theorem 16 and Proposition 17 to obtain lower bounds on h− Np2 . Use these lower bounds to build the finite list of Np2 ’s for which these − lower bounds on h− N 2 do not imply hN 2 > 1. Finally, compute the p

p

relative class numbers of the Np2 ’s of this short list (see TABLE IV). • Now, let N2 ∈ N2 be a given dihedral CM-field of degree 2r ≥ 4 with relative class number one (see Theorem 9 and point 1 of Proposition 15). Let also p ≤ p0 ∈ {17, 19} be a given odd prime. Using Proposition 17 we obtain an upper bound B(N2 , p) such that h− Np = 1 implies fFp /L ≤ B(N2 , p) for any dihedral CM-field Np = N2 Fp of degree 4m = 2r p containing N2 . (More precisely, if 2r = 4 then we use (19) instead of (20)). By constructing all the fields Np = N2 Fp of conductor fFp /L ≤ B(N2 , p) (see Section 2) and by computing their relative class numbers (see Section 6), we can prove Theorem 10.

6

Computation of relative class numbers

To begin with we notice that we need only compute relative class numbers of dihedral CM-fields Nps of degree 2n = 4m = 2r ps with 1 ≤ s ≤ 2. Theorem 19 Let Nps = N2 Fps be a dihedral CM-field of degree 4m = 2r ps . Let χps denote any one of the φ(2m) characters of order 2m associated

NON-ABELIAN NORMAL CM-FIELDS

19

with the cyclic extension Nps /L, let Fps denote the conductor of the cyclic extension Nps /L and set fps = NL/Q (Fps ). Then, 1. (See [LOO, Th. 5]). It holds QN

p2

= QNp = QN2

p2

= wNp = wN2 .

and wN

− − − Moreover, h− N2 divides hNp , and hNp divides hN 2 . p

2. (See [LP] and [Lou12]). Both − − 2 h− Np /hN2 = (hNp /N2 )

and − − h− N 2 /hNp = (hN

p2 /Np

p

)2

are perfect squares. Moreover, we have h− Np /N2

=

2r−2 Yp

p

dL fp L(1, χkp ) 4π 2

k=1, k odd gcd(k,p)=1

and h− N

p2 /Np

=

2 2r−2 Yp k=1, k odd gcd(k,p)=1

(22)

q

dL fp2

4π 2

L(1, χkp2 ).

− To compute the positive integers h− Np /N2 and hN

p2 /Np

(23)

, it suffices to com-

pute good enough numerical approximations of them. According to (22) and (23), is suffices to compute good enough numerical approximations of all the L(1, χkp )’s and L(1, χkp2 )’s. The following Lemma 20 and Theorem 21 explain how to perform such computations: Lemma 20 (See [Lou10]). Let γ = 0.577 215 · · · denote Euler’s constant and let B be positive. Set K1 (B) = 1 + 4

X n≥0

n X 1 1 γ + log B − − 2n + 2 k=1 k

!

B 2n+2 (2n + 2)(n!)2

(24)

NON-ABELIAN NORMAL CM-FIELDS

20

and K2 (B) = πB + 4

X n≥0

n X 1 1 γ + log B − − 2n + 1 k=1 k

!

B 2n+2 . (25) (2n + 1)(n!)2

Then B > 0 implies 0 ≤ K2 (B) ≤ K1 (B) ≤ 2e−B . Theorem 21 Let L(s, χ) =

X an (χ) n≥1

ns

be a Hecke L-series associated to a primitive character on a ray class group of a real quadratic number field L. Assume that χ is ramified at the infinite places ∞1 and ∞2 of L, let ∞1 ∞2 Fχ be the conductor of χ, set Aχ =

q

dL NL/Q (Fχ )/π 2

and F (s, χ) = Asχ Γ2 ((s + 1)/2)L(s, χ), let Wχ be the Artin root number associated to this L-series (the complex number of absolute value equal to 1 such that F (1 − s, χ) = Wχ F (s, χ)). ¯ Then, we have the following absolutely convergent series expansion L(1, χ) =

X an (χ) n≥1

n

K1 (n/Aχ ) + Wχ

X an (χ) n≥1

n

K2 (n/Aχ )

(26)

and if we let SM,χ denote the value obtained by disregarding in the series (26) the indices n > M , then |L(1, χ) − SM,χ | ≤ 4(log(M e) + 2)2 e−M/Aχ .

(27)

Finally, since for any non-abelian dihedral CM-field N of degree 4m the Artin root numbers Wχ of the φ(2m) characters of ordre 2m associated with the cyclic extensions N/L are equal to +1 (see [FQ]), then (22), (26) and (27) enable us to compute relative class numbers of dihedral CM-fields. It now only remains to explain how to compute the coefficients an (χ) which appear in (26). Since gcd(n1 , n2 ) = 1 implies an1 n2 (χ) = an1 (χ)an2 (χ) (that is, n 7→ an (χ) is multiplicative), we only have to explain how to compute alk (χ) for any prime l ≥ 2 and any k ≥ 1.

NON-ABELIAN NORMAL CM-FIELDS

21

Proposition 22 (See [Lou10]). Let L be real quadratic field. Let f ≥ 1 be a positive rational integer. Let χ be a character on the group IL ((f )) which is trivial on its subgroup PL,Z ((f )). Let l ≥ 2 be any rational prime. If l divides f , then alk (χ) = 0. Now, assume that l does not divide f . Then 1. If χL (l) = −1 then χ((l)) = +1 and 

alk (χ) =

0 1

if k is odd if k is even.

In particular, if χL (n) = −1 then an (χ) = 0. 2. If χL (l) = 0, then (l) = L2 in L, χ(L) = ±1 and alk (χ) = (χ(L))k . 3. If χL (l) = +1 and (l) = LL0 in L then χ(L0 ) = χ(L) and alk (χ) =

 k  (k + 1)(χ(L)) 

=(ζ k+1 )/=(ζ)

if χ(L) = ±1 if ζ = χ(L) 6= ±1.

NON-ABELIAN NORMAL CM-FIELDS

7

22

Tables

In TABLE I we list the 147 imaginary bicyclic biquadratic fields M = L1 L2 with relative class number one (see Proposition 15). Here, L1 , L2 and L denote the imaginary quadratic subfields and real quadratic subfield of M, respectively, and we give the invariants λL = Ress=1 (ζL ), and µL refered to in Proposition 12. In cases 6, 19, 20, 32, 45 and 51 the prime 2 is totally ramified in M/Q and consequently 2p−1 divides h− N for any dihedral CMfield N of degree 4p cyclic of degree p over L (see Proposition 15, point 3). The ten bold faced cases of indices 60, 61, 74, 78, 85, 94, 119, 122, 127 and 142 are the cases for which some odd prime p divides the class number hL of L, and for which we will have to use point 2 of Theorem 6 to construct the Fp ’s and Fp2 ’s containing L. In TABLE II, for the 24 dihedral CMfields of 2-power degrees 2n = 4m = 2r ≥ 8 with relative class number one refered to in Theorem 9, we list the invariants λL = Ress=1 (ζL ) and µL of their real quadratic subfields L over which they are cyclic (we refer the reader to [Lou9] for an efficient method for computing these invariants). The bold faced case of index 157 is the case for which some odd prime p divides the class number hL of L. TABLES I and II are used to prove point 1 of Corollary 18 In TABLE III, for each of the 11 = 10 + 1 bold faced cases in TABLES I and II for which some odd prime p divides hL , we give the relative class numbers h− Np of the dihedral CM-fields Np = Fp N2 of degree r 2n = 4m = 2 p and of least conductors fFp /L . Notice that, according to Theorem 6, for each choice of fFp /L > 1, there are p such CM-fields Np to consider. TABLE III is an excerpt of the larger TABLE we had to build to prove points 2 and 3 of Theorem 10. In TABLE IV, for each of the 19 non-abelian dihedral CM-fields Np of degree 2n = 4m = 2r p with relative class number one given (see Theorem 10), we give the relative class numbers r 2 h− Np2 of the dihedral CM-fields Np2 = Fp2 N2 of degree 2n = 4m = 2 p containing Hp and of least conductors fF 2 /L > 1. TABLE IV is an excerpt p

of the larger TABLE we had to build to prove point 2 of Corollary 18 (see [Lef] for this comprehensive TABLE). We should point out that the huge relative class numbers which appear at the bottom of this TABLE IV were computed thanks to an improvement of the method we have delineated in section 6 (see [Lou12]). Our computations were done by using Pr. Y. Kida’s UBASIC. In particular, we did not use PARI and we do not have to assume any Riemann hypothesis to warrant the results of these computations.

NON-ABELIAN NORMAL CM-FIELDS

index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

dL 5 5 5 5 8 8 8 12 13 13 17 17 21 24 28 29 33 37 40 40 41 44 56 57 60 60 61 76 77 88 89 104 105 105 120 120 129

hL 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 2 2 1 1 1 1 1 2 2 2 2 1

dL1 −3 −4 −7 −8 −3 −4 −11 −3 −4 −7 −3 −11 −3 −3 −4 −8 −3 −4 −4 −8 −3 −4 −7 −3 −3 −4 −7 −4 −7 −8 −3 −8 −3 −7 −3 −8 −3

TABLE I dL2 QM wM −15 1·6 −20 1·4 −35 1·2 −40 1·2 −24 1·6 −8 −88 1·2 −4 2 · 12 −52 1·4 −91 1·2 −51 1·6 −187 1·2 −7 2·6 −8 2·6 −7 2·4 −232 1·2 −11 2·6 −148 1·4 −40 −20 −123 1·6 −11 2·4 −8 2·2 −19 2·6 −20 1·6 −15 1·4 −427 1·2 −19 2·4 −11 2·2 −11 2·2 −267 1·6 −52 −35 1·6 −15 1·2 −40 1·6 −15 1·2 −43 2·6

23

λL ≤ 0.431 0.431 0.431 0.431 0.624

µL ≤ 0.102 0.102 0.102 0.102 0.141

FL (17) 1087 1087 1087 1087 239

0.624 0.761 0.663 0.663 1.017 1.017 0.684 0.936 1.047 0.612 1.333 0.820

0.141 0.189 0.222 0.222 0.217 0.217 0.331 0.300 0.310 0.463 0.307 0.431

239 607 443 443 1103 1103 1361 647 419 1409 151 1103

1.300 0.903 0.909 1.513 1.066 1.066 0.939 1.338 0.498 1.275 1.465

0.365 0.478 0.552 0.408 0.510 0.510 0.520 0.493 0.985 0.559 0.540

683 379 929 101 433 433 761 113 367 1301 101

1.721 1.721 1.128 1.128 1.836

0.531 0.531 0.708 0.708 0.563

229 229 461 461 307

NON-ABELIAN NORMAL CM-FIELDS

index 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

dL 133 140 140 152 172 201 209 232 264 264 268 280 280 296 301 344 345 357 364 364 456 460 469 473 489 536 561 561 572 645 652 696 705 728 737 805 817

TABLE I (continued) hL dL1 dL2 QM wM λL ≤ 1 −7 −19 2 · 2 0.894 2 −4 −35 1 · 4 0.838 2 −7 −20 1 · 2 0.838 1 −8 −19 2 · 2 0.699 1 −4 −43 2 · 4 1.350 1 −3 −67 2 · 6 1.954 1 −11 −19 2 · 2 1.583 −4 −232 2 −3 −88 1 · 6 1.199 2 −11 −24 1 · 2 1.199 1 −4 −67 2 · 4 1.404 2 −7 −40 1 · 2 1.487 2 −8 −35 1 · 2 1.487 −8 −148 1 −7 −43 2 · 2 1.157 1 −8 −43 2 · 2 1.073 2 −3 −115 1 · 6 2.049 2 −7 −51 1 · 2 0.623 2 −4 −91 1 · 4 1.689 2 −7 −52 1 · 2 1.689 2 −19 −24 1 · 2 1.429 2 −4 −115 1 · 4 1.440 3 −7 −67 2 · 2 1.157 3 −11 −43 2 · 2 1.424 1 −3 −163 2 · 6 2.121 1 −8 −67 2 · 2 1.088 2 −3 −187 1 · 6 2.341 2 −11 −51 1 · 2 2.341 2 −11 −52 1 · 2 0.532 2 −43 −15 1 · 2 0.763 1 −4 −163 2 · 4 1.463 2 −3 −232 1 · 6 1.209 2 −3 −235 1 · 6 1.969 2 −8 −91 1 · 2 0.592 1 −11 −67 2 · 2 1.477 2 −7 −115 1 · 2 1.026 5 −19 −43 2 · 2 2.285

24

µL ≤ 0.811 0.951 0.951 1.161 0.748 0.669 0.779

FL (17) 173 419 419 1063 191 229 509

0.958 0.958 0.878 0.857 0.857

271 271 743 911 911

0.923 1.085 0.815 1.531 0.863 0.863 1.011 1.071 1.090 1.170 0.919 1.244 0.895 0.895 2.328 1.566 1.183 1.345 1.106 2.252 1.313 1.415 1.093

223 443 1033 379 239 239 461 563 797 173 281 173 151 151 1289 733 1063 239 223 1877 227 1031 137

NON-ABELIAN NORMAL CM-FIELDS

index 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111

dL 860 861 920 940 988 1001 1005 1032 1141 1273 1304 1309 1340 1353 1505 1608 1612 1645 1672 1708 1720 1729 1793 1869 1880 2337 2345 2445 2552 2680 2812 2821 2881 3097 3224 3260 3416

TABLE I (continued) hL dL1 dL2 QM wM λL ≤ 2 −43 −20 1 · 2 0.611 2 −7 −123 1 · 2 0.946 2 −8 −115 1 · 2 0.687 6 −4 −235 1 · 4 1.770 2 −19 −52 1 · 2 1.534 2 −11 −91 1 · 2 1.842 2 −67 −15 1 · 2 0.947 2 −43 −24 1 · 2 0.778 1 −7 −163 2 · 2 1.242 1 −19 −67 2 · 2 2.268 3 −8 −163 2 · 2 1.077 2 −7 −187 1 · 2 1.291 2 −67 −20 1 · 2 0.776 2 −11 −123 1 · 2 1.987 2 −43 −35 1 · 2 1.752 2 −67 −24 1 · 2 0.668 2 −4 −403 1 · 4 1.406 2 −7 −235 1 · 2 1.006 2 −19 −88 1 · 2 1.089 6 −4 −427 1 · 4 1.400 2 −43 −40 1 · 2 1.501 2 −19 −91 1 · 2 3.091 1 −11 −163 2 · 2 1.601 2 −7 −267 1 · 2 0.940 2 −8 −235 1 · 2 0.750 2 −19 −123 1 · 2 1.740 2 −67 −35 1 · 2 1.423 2 −163 −15 1 · 2 0.857 2 −11 −232 1 · 2 0.899 2 −67 −40 1 · 2 1.257 2 −19 −148 1 · 2 1.106 2 −7 −403 1 · 2 1.374 1 −43 −67 2 · 2 3.399 1 −19 −163 2 · 2 2.188 2 −8 −403 1 · 2 1.151 2 −163 −20 1 · 2 0.887 2 −8 −427 1 · 2 1.011

25

µL ≤ FL (17) 2.334 191 1.430 257 2.123 739 1.153 761 1.296 67 1.237 347 1.455 113 2.138 103 1.363 229 1.278 703 1.634 151 1.373 83 2.076 151 1.346 137 1.421 47 2.739 569 1.615 227 1.757 73 2.012 619 1.629 373 1.572 43 1.142 373 1.577 37 1.755 599 2.391 271 1.731 347 1.925 103 2.006 659 2.053 409 2.057 137 2.243 19 1.606 281 1.250 59 1.714 101 1.942 487 2.215 59 2.138 103

NON-ABELIAN NORMAL CM-FIELDS

index 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147

TABLE I (continued) dL hL dL1 dL2 QM wM 3484 2 −67 −52 1·2 3784 2 −43 −88 1·2 3912 2 −163 −24 1·2 4408 2 −19 −232 1·2 4433 2 −11 −403 1·2 4697 2 −11 −427 1·2 4945 2 −43 −115 1·2 5073 6 −19 −267 1·2 5705 2 −163 −35 1·2 5896 2 −67 −88 1·2 6097 14 −67 −91 1·2 6364 2 −43 −148 1·2 6520 2 −163 −40 1·2 7009 1 −43 −163 2·2 7657 2 −19 −403 1·2 7705 10 −67 −115 1·2 8241 2 −67 −123 1·2 8313 2 −163 −51 1·2 8476 2 −163 −52 1·2 9976 2 −43 −232 1·2 10105 2 −43 −235 1·2 10921 1 −67 −163 2·2 11481 2 −43 −267 1·2 14344 2 −163 −88 1·2 14833 2 −163 −91 1·2 15745 2 −67 −235 1·2 18361 2 −43 −427 1·2 18745 2 −163 −115 1·2 24124 2 −163 −148 1·2 27001 2 −67 −403 1·2 28609 14 −67 −427 1·2 30481 2 −163 −187 1·2 37816 2 −163 −232 1·2 38305 2 −163 −235 1·2 43521 2 −163 −267 1·2 65689 2 −163 −403 1·2

26

λL ≤ 1.653 2.090 0.792 1.021 1.611 1.391 2.544 1.457 1.389 2.045 2.703 1.890 1.196 3.525 2.315 2.990 2.757 1.660 1.458 1.874 3.351 3.990 2.561 2.008 2.698 3.309 3.463 2.965 2.416 3.616 3.947 3.671 2.580 3.763 3.361 4.213

µL ≤ 1.793 1.551 2.777 2.663 1.951 2.207 1.757 2.421 2.407 1.761 1.729 1.908 2.620 1.559 2.049 1.726 1.772 2.489 2.453 2.132 1.720 1.598 2.023 2.188 2.125 1.937 1.960 2.127 2.127 2.102 1.981 2.061 2.219 2.114 2.197 2.244

FL (17) 229 577 1709 443 151 227 137 139 289 83 683 409 151 571 229 829 443 229 47 647 431 47 1301 307 173 173 191 83 361 139 107 821 419 101 103 367

NON-ABELIAN NORMAL CM-FIELDS

index 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171

2r 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 16 16 16 16 16

dL 136 205 221 305 377 545 584 712 745 1345 1537 1864 1945 2041 2248 2329 2353 4369 7081 505 689 793 905 2056

TABLE II hL λL ≤ 2 1.458 2 1.051 2 0.728 2 1.578 2 1.266 2 1.418 2 0.939 2 1.210 2 2.500 6 3.004 2 2.626 2 1.979 2 2.657 2 3.362 2 1.680 2 2.926 2 2.612 2 3.573 2 3.737 4 2.631 4 1.630 4 2.580 4 1.751 4 1.612

27

µL ≤ 0.629 0.852 1.063 0.914 1.193 1.246 1.446 1.427 0.994 1.060 1.213 1.335 1.307 1.133 1.513 1.303 1.390 1.359 1.497 0.822 1.213 0.982 1.232 1.595

FL (19) 1291 361 529 163 233 191 743 1229 113 379 191 373 281 277 443 1237 229 1063 59 557 1901 709 361 841

NON-ABELIAN NORMAL CM-FIELDS

dL1 −7 −11 −4 −8 −4 −19 −19 −67 −67

dL2 −67 −43 −235 −163 −427 −267 −43 −115 −91

dL 469 473 940 1304 1708 5073 817 7705 6097

−67 −427 28609

5

269

1345

TABLE III − − 2 fFp /L h− Np = (hNp /M ) with hNp /M = 62 24, 30, 45 85 27, 45, 108 91 36, 60, 72 53 27, 30, 51 227 66, 168, 270 55 36, 81, 99 79 820, 1345, 4225, 6505, 12980 55 6301, 6921, 24671, 33916, 34751 1189 765 104 081 4 066 653 227 10 008 078 059 13 699 296 569 13 721 986 264 16 694 290 249 16 782 950 947 7 167 49 494 697 69 531 448 153 177 143 308 444 857 562 737 259 636 146 917 662 575 151 3 101 4356, 5886, 9297

hL p 3 3 3 3 6 3 3 3 6 3 6 3 5 5 10 5 14 7

14

6

28

NON-ABELIAN NORMAL CM-FIELDS

dL

p

5 12 21 24 37 57 60 88 105 120 456 469 473 940 1304 1708 1345 280 817

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5

dL1

dL2

−3 −3 −3 −3 −4 −3 −3 −8 −7 −3 −19 −7 −11 −4 −8 −4 5 −8 −19

−15 −4 −7 −8 −148 −19 −20 −11 −15 −40 −24 −67 −43 −235 −163 −427 269 −35 −43

TABLE IV fFp /L fF 2 /L p

18 15 6 9 2 5 9 7 9 11 3 1 1 1 1 1 1 5 1

none none none none none none none none 27 none none 13 31 23 9 45 47 26139 275 2376 38443 55778 109 19852 03714 72807

29

h− N

p2

1632

7122 112892 29432 1712 249392 717362 570052 498752

NON-ABELIAN NORMAL CM-FIELDS

30

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