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Constructing hyperelliptic curves of genus 2 suitable for cryptography
 Math. Comp
, 2003
"... Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Cited by 29 (2 self)
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Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.
Ideal class groups of cyclotomic number fields
 II. Acta Arith
, 1998
"... Abstract. Following Hasse’s example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these results to CMfields by using class field theory. A ..."
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Cited by 10 (6 self)
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Abstract. Following Hasse’s example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these results to CMfields by using class field theory. Although we will only need some special cases, we have also decided to include a few results on Hasse’s unit index for CMfields as well, because it seems that our proofs are more direct than those in Hasse’s book [2]. 1. Notation Let K ⊂ L be number fields; we will use the following notation: • OK is the ring of integers of K; • EK is its group of units; • WK is the group of roots of unity contained in K; • wK is the order of WK; • Cl(K) is the ideal class group of K; • [a] is the ideal class generated by the ideal a; • K1 denotes the Hilbert class field of K, that is the maximal abelian extension of K that is unramified at all places; • jK→L denotes the transfer of ideal classes for number fields K ⊂ L, i.e. the homomorphism Cl(K) → Cl(L) induced by mapping an ideal a to aOL; • κL/K denotes the capitulation kernel ker jK→L; Now let K be a CMfield, i.e. a totally complex quadratic extension of a totally real number field; the following definitions are standard: • σ is complex conjugation; • K + denotes the maximal real subfield of K; this is the subfield fixed by σ. • Cl − (K) is the kernel of the map NK/K +: Cl(K) → Cl(K +) and is called the minus class group; • h − (K) is the order of Cl − (K), the minus class number; • Q(K) = (EK: WKEK +) ∈ {1, 2} is Hasse’s unit index. We will need a well known result from class field theory. Assume that K ⊂ L are CMfields; then ker(NL/K: Cl(L) → Cl(K)) has order (L ∩ K1: K). Since K/K + is ramified at the infinite places, the norm NK/K +: Cl(K) → Cl(K +) is onto. 2. Hasse’s unit index Hasse’s book [2] contains numerous theorems (Sätze 14 – 29) concerning the unit index Q(L) = (EL: WLEK), where K = L + is the maximal real subfield of a
Computation of relative class numbers of CMfields by using Hecke Lfunctions
 Math. Comp
"... Abstract. We develop an efficient technique for computing values at s =1 of Hecke Lfunctions. We apply this technique to the computation of relative class numbers of nonabelian CMfields N which are abelian extensions of some totally real subfield L. We note that the smaller the degree of L the mo ..."
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Cited by 4 (2 self)
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Abstract. We develop an efficient technique for computing values at s =1 of Hecke Lfunctions. We apply this technique to the computation of relative class numbers of nonabelian CMfields N which are abelian extensions of some totally real subfield L. We note that the smaller the degree of L the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing L = N + (the maximal totally real subfield of N) we can choose L real quadratic. We finally give examples of computations of relative class numbers of several dihedral CMfields of large degrees and of several quaternion octic CMfields with large discriminants. 1.
Computation of relative class numbers of CMfields
 Math. Comp
, 1997
"... Abstract. It was well known that it is easy to compute relative class numbers of abelian CMfields by using generalized Bernoulli numbers (see Theorem 4.17 in Introduction to cyclotomic fields by L. C. Washington, Grad. Texts in Math., vol. 83, SpringerVerlag, 1982). Here, we provide a technique fo ..."
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Cited by 3 (2 self)
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Abstract. It was well known that it is easy to compute relative class numbers of abelian CMfields by using generalized Bernoulli numbers (see Theorem 4.17 in Introduction to cyclotomic fields by L. C. Washington, Grad. Texts in Math., vol. 83, SpringerVerlag, 1982). Here, we provide a technique for computing the relative class number of any CMfield. 1. Statement of the results Proposition 1. Let n ≥ 1 be an integer and α>1be real. Set Pn(x) = ∑n−1