Results 1  10
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19
KMS states and complex multiplication
 the proceedings of the Abel Symposium
, 2005
"... The following problem in operator algebra has been open for several years. Problem 1.1. For some number field K (other than Q) exhibit an explicit quantum statistical mechanical system (A, σt) with the following properties: (1) The partition function Z(β) is the Dedekind zeta function of K. ..."
Abstract

Cited by 23 (6 self)
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The following problem in operator algebra has been open for several years. Problem 1.1. For some number field K (other than Q) exhibit an explicit quantum statistical mechanical system (A, σt) with the following properties: (1) The partition function Z(β) is the Dedekind zeta function of K.
Computing Hilbert Class Polynomials
"... Abstract. We present and analyze two algorithms for computing the Hilbert class polynomial HD. The first is a padic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CMcurves over fini ..."
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Cited by 11 (6 self)
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Abstract. We present and analyze two algorithms for computing the Hilbert class polynomial HD. The first is a padic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CMcurves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for computing HD, and we show that all methods have comparable run times. 1
Elliptic curves with a given number of points
 SYMPOSIUM VI, SPRINGER LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We present a nonarchimedean method to construct, given an integer N ≥ 1, a finite field Fq and an elliptic curve E/Fq such that E(Fq) has order N. ..."
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Cited by 11 (4 self)
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We present a nonarchimedean method to construct, given an integer N ≥ 1, a finite field Fq and an elliptic curve E/Fq such that E(Fq) has order N.
CONSTRUCTING ELLIPTIC CURVES OF PRIME ORDER
"... Abstract. We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time e O((log N) 3), and it is so fast that it may profitably be used to ta ..."
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Cited by 6 (3 self)
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Abstract. We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time e O((log N) 3), and it is so fast that it may profitably be used to tackle the related problem of finding elliptic curves with point groups of prime order of prescribed size. We also discuss the impact of the use of high level modular functions to reduce the run time by large constant factors and show that recent gonality bounds for modular curves imply limits on the time reduction that can be obtained. 1.
Efficient CMconstructions of elliptic curves over finite fields
 MATH. COMP.
, 2007
"... We present an algorithm that, on input of an integer N ≥ 1 together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) hasorderN. Although it is unproved that this can be done for all N, a heuristic analysis shows that the algorithm has an expect ..."
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Cited by 4 (2 self)
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We present an algorithm that, on input of an integer N ≥ 1 together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) hasorderN. Although it is unproved that this can be done for all N, a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2 ω(N) log N, whereω(N) isthe number of distinct prime factors of N. In the cryptographically relevant case where N is prime, an expected run time O((log N) 4+ε) can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order N =10 2004 and N = nextprime(10 2004)=10 2004 +4863.
PRINCIPAL MODULI AND CLASS FIELDS
, 2003
"... Abstract. We study the values taken by Γ0(n)modular functions at elliptic points of order 2 for the Fricke group Γ0(n) † that lie outside Γ0(n). In the case of a principal modulus (‘Hauptmodul’) for Γ0(n) or Γ0(n) † , we determine the class fields generated by these values. 1. ..."
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Abstract. We study the values taken by Γ0(n)modular functions at elliptic points of order 2 for the Fricke group Γ0(n) † that lie outside Γ0(n). In the case of a principal modulus (‘Hauptmodul’) for Γ0(n) or Γ0(n) † , we determine the class fields generated by these values. 1.