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30 E. BENJAMIN, F. LEMMERMEYER, AND C. SNYDER Put K1 = k1( √ πρ ) and K1 = k1( √ πρ ′ ); we claim that h2( K1) = 2. This is equivalent to h2( L1) = 1, where L1 = k1( √ π, √ ρ ′ ) is a quadratic unramified extension of K1. Put F1 = k1( √ π ) and apply the ambiguous class number formula to F1/k1 and L1/ F1: Since there is only one ramified prime in each of these two extensions, we find Am( F1/k1) = Am( L1/ F1) = 1; note that we have used the assumption that [π/ρ ′ ] = −1 in deducing that r ′ is inert in F1/k1. In our proof of Theorem 1 we have seen that there are the following possibilities when h2(K2) | 4: q2 Cl2(K2) q1 h2(K1) q2 Cl2( K2) h2(L) 2 (2) 1 2 2 (2) 2 m+1 2 (4) 1 4 ? (2, 2) 2 m+3 In order to decide whether q2 = 1 or q2 = 2, recall that we have h2(K1) = 4; thus K1 must be the field with 2-class number 2, and this implies h2( L) = 2 m+2 and q2 = 1. In particular we see that 4 | h2(K2) if and only if 4 | h2(K1) as long as K1 = k1( √ πρ ) with [π/ρ] = +1. The ambiguous class number formula shows that Cl2(K1) is cyclic, thus 4 | h2(K1) if and only if 2 | h2(L1), where L1 = K1( √ π ) is the quadratic unramified extension of K1. Applying the ambiguous class number formula to L1/F1, where F1 = k1( √ π ), we see that 2 | h2(L1) if and only if (E : H) = 1. Now E is generated by a root of unity (which always is a norm residue at primes dividing r ≡ 1 mod 4) and a fundamental unit ε. Therefore (E : H) = 1 if and only if {ε/R1} = {ε/R2} = +1, where rOF1 = R1R2 and where { · / · } denotes the quadratic residue symbol in F1. Since {ε/R1}{ε/R2} = [ε/r] = +1, we have proved that 4 | h2(K1) if and only if the prime ideal R1 above r splits in the quadratic extension F1( √ ε ). But if we fix p and q, this happens for exactly half of the values of r satisfying (p/r) = −1, (q/r) = +1. If d2 = 8 and p = 2, then 2Ok1 = 22′ , and we have to choose 2 h = (π) in such a way that k1( √ π )/k1 is unramified outside p. The residue symbols [α/2] are defined as Kronecker symbols via the splitting of 2 in the quadratic extension k1( √ α )/k1. With these modifications, the above arguments remain valid. References [1] E. Benjamin, F. Lemmermeyer and C. Snyder, Imaginary quadratic fields k with cyclic Cl2(k 1 ), J. Number Theory, 67 (1997), 229-245. [2] N. Blackburn, On prime-power groups in which the derived group has two generators, Proc. Cambridge Phil. Soc., 53 (1957), 19-27.
IMAGINARY QUADRATIC FIELDS 31 [3] , On prime power groups with two generators, Proc. Cambridge Phil. Soc., 54 (1958), 327-337. [4] G. Gras, Sur les ℓ-classes d’idéaux dans les extensions cycliques relatives de degré premier ℓ, Ann. Inst. Fourier, 23 (1973), 1-48. [5] P. Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc., 36 (1933), 29-95. [6] H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Teil II: Reziprozitätsgesetz, Physica Verlag, Würzburg, 1965. [7] D. Hilbert, Über die theorie des relativ-quadratischen Zahlkörpers, Math. Ann., 51 (1899), 1-127. [8] B. Huppert, Endliche Gruppen I, Springer Verlag, Heidelberg, 1967. [9] S. Lang, Cyclotomic Fields, I, II, Springer Verlag 1990. [10] F. Lemmermeyer, Kuroda’s class number formula, Acta Arith., 66(3) (1994), 245-260. [11] M. Moriya, Über die klassenzahl eines relativzyklischen zahlkörpers von primzahlgrad, Jap. J. Math., 10 (1933), 1-18. [12] L. Rédei and H. Reichardt, Die Anzahl der durch 4 teilbaren invarianten der klassengruppe eines beliebigen quadratischen zahlkörpers, J. Reine Angew. Math., 170 (1933), 69-74. [13] B. Schmithals, Konstruktion imaginärquadratischer körper mit unendlichem klassenkörperturm, Arch. Math., 34 (1980), 307-312. Received July 20, 1999 and revised April 5, 2000. Mathematics Department Unity College Unity, Maine 04988 E-mail address: benjamin@mint.net CSU San Marcos San Marcos, CA 92096 E-mail address: franzl@csosm.edu Department of Mathematics and Statistics University of Maine Research Institute of Mathematics, Orono, Maine 04469 E-mail address: snyder@gauss.umemat.maine.edu
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80 ROBERT F. BROWN AND HELGA SCHIRM
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82 GILLES CARRON à l’infini s’
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84 GILLES CARRON isométriques au d
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86 GILLES CARRON Comme D est ellipt
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88 GILLES CARRON Réciproquement si
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90 GILLES CARRON où H est la proje
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92 GILLES CARRON 2.a. Exemples repo
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94 GILLES CARRON alors si σ ∈ C
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96 GILLES CARRON Ainsi s’il exist
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98 GILLES CARRON Proposition 2.7. N
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100 GILLES CARRON l’espace des 1
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102 GILLES CARRON compact. De même
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104 GILLES CARRON 4. Application du
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106 GILLES CARRON Dans le cas où l
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BRAIDED OPERATOR COMMUTATORS 111 Re
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BRAIDED OPERATOR COMMUTATORS 113 Re
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BRAIDED OPERATOR COMMUTATORS 115 3.
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Then, for 1 < k ≤ m + 1, we have
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BRAIDED OPERATOR COMMUTATORS 119 Mo
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BRAIDED OPERATOR COMMUTATORS 121 It
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E-mail address: prosk@imath.kiev.ua
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124 DANIEL J. MADDEN times. Further
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126 DANIEL J. MADDEN Further, s t
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128 DANIEL J. MADDEN Now the whole
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130 DANIEL J. MADDEN and β = 1 −
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132 DANIEL J. MADDEN We can simplif
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134 DANIEL J. MADDEN surd, but we n
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136 DANIEL J. MADDEN and d = d(u, v
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138 DANIEL J. MADDEN (6l + 7) k +
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140 DANIEL J. MADDEN Then the conti
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142 DANIEL J. MADDEN Thus n + 1 1 =
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144 DANIEL J. MADDEN If we take b =
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146 DANIEL J. MADDEN b, 2b(2bn + 1)
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TANGLES, 2-HANDLE ADDITIONS AND DEH
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m n n n n TANGLES, 2-HANDLE ADDITIO
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SOME CHARACTERIZATION, UNIQUENESS A
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and SOME CHARACTERIZATION, UNIQUENE
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SYMPLECTIC SUBMANIFOLDS FROM SURFAC
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208 PAULO TIRAO Theorem. The affine
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210 PAULO TIRAO It follows from the
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212 PAULO TIRAO K ρ1 (0,−1,1) =
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214 PAULO TIRAO Theorem 2.7. Let ρ
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216 PAULO TIRAO Notation: By A = [
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218 PAULO TIRAO Now is not difficul
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220 PAULO TIRAO Regard H n (Z2 ⊕
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222 PAULO TIRAO is an isomorphism.
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224 PAULO TIRAO Lemma 4.2. Let Λ1
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226 PAULO TIRAO (c)1 B1 B2 B3 B1 0
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228 PAULO TIRAO ⎛ ⎜ B2 = ⎜
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230 PAULO TIRAO being those in whic
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232 PAULO TIRAO Hence, if ρ = m1χ
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(1.6) HÖLDER REGULARITY FOR ∂ 23
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HÖLDER REGULARITY FOR ∂ 239 can
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HÖLDER REGULARITY FOR ∂ 241 wher
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similarly, we can prove HÖLDER REG
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HÖLDER REGULARITY FOR ∂ 255 Refe
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