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14 JUNICHI ARAMAKI [5] H. Matsumoto, The short time asymptotics of the trace of the heat kernels for the magnetic Schrödinger operators, J. Math. Soc. Japan, 42(4) (1990), 677-689. [6] , Classical and non-classical eigenvalue asymptotics for the magnetic Schrödinger operators, J. Funct. Anal., 95 (1991), 460-482. [7] K. Odencrantz, The effect of a magnetic field on asymptotics of the trace of the heat kernel, J. Funct. Anal., 79 (1988), 398-422. [8] M. Reed and B. Simon, Method of Modern Mathematical Physics Vol. IV, Analysis of Operators, Academic Press, New York, 1977. [9] D. Robert, Comportement asymptotique des valeurs propres d’opérateurs du type Schrödinger a potentiel dégénéré, J. Math. Pures et Appl., 61 (1982), 275-300. [10] M. Schechter, Spectra of Partial Differential Operators, North-Holland, Amsterdam, 1971. [11] B. Simon, Non-classical eigenvalue asymptotics, J. Funct. Anal., 53 (1983), 84-98. [12] , Semiclassical analysis of low lying eigenvalues II, Tunneling, Ann. of Math., 120 (1984), 89-118. [13] , Functional integration and quantum physics, Academic Press, New York, San Francisco, London, 1979. Received May 17, 1999 and revised September 14, 1999. Department of Mathematical Sciences Tokyo Denki University Hatoyama-machi Hiki-gun Saitama 350-0394 Japan E-mail address: aramaki@r.dendai.ac.jp
PACIFIC JOURNAL OF MATHEMATICS Vol. 198, No. 1, 2001 IMAGINARY QUADRATIC FIELDS k WITH Cl2(k) (2, 2 m ) AND RANK Cl2(k 1 ) = 2 E. Benjamin, F. Lemmermeyer, and C. Snyder Let k be an imaginary quadratic number field and k 1 the Hilbert 2-class field of k. We give a characterization of those k with Cl2(k) (2, 2 m ) such that Cl2(k 1 ) has 2 generators. 1. Introduction. Let k be an algebraic number field with Cl2(k), the Sylow 2-subgroup of its ideal class group, Cl(k). Denote by k 1 the Hilbert 2-class field of k (in the wide sense). Also let k n (for n a nonnegative integer) be defined inductively as: k 0 = k and k n+1 = (k n ) 1 . Then k 0 ⊆ k 1 ⊆ k 2 ⊆ · · · ⊆ k n ⊆ · · · is called the 2-class field tower of k. If n is the minimal integer such that k n = k n+1 , then n is called the length of the tower. If no such n exists, then the tower is said to be of infinite length. At present there is no known decision procedure to determine whether or not the (2-)class field tower of a given field k is infinite. However, it is known by group theoretic results (see [2]) that if rank Cl2(k 1 ) ≤ 2, then the tower is finite, in fact of length at most 3. (Here the rank means minimal number of generators.) On the other hand, until now (see Table 1 and the penultimate paragraph of this introduction) all examples in the mathematical literature of imaginary quadratic fields with rank Cl2(k 1 ) ≥ 3 (let us mention in particular Schmithals [13]) have infinite 2-class field tower. Nevertheless, if we are interested in developing a decision procedure for determining if the 2-class field tower of a field is infinite, then a good starting point would be to find a procedure for sieving out those fields with rank Cl2(k 1 ) ≤ 2. We have already started this program for imaginary quadratic number fields k. In [1] we classified all imaginary quadratic fields whose 2-class field k 1 has cyclic 2-class group. In this paper we determine when Cl2(k 1 ) has rank 2 for imaginary quadratic fields k with Cl2(k) of type (2, 2 m ). (The notation (2, 2 m ) means the direct sum of a group of order 2 and a cyclic group of order 2 m .) The group theoretic results mentioned above also show that such fields have 2-class field tower of length 2. 15
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64 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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92 GILLES CARRON 2.a. Exemples repo
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98 GILLES CARRON Proposition 2.7. N
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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 115 3.
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Then, for 1 < k ≤ m + 1, we have
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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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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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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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