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Class invariants for quartic CM fields
, 2004
"... Abstract. One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. Such constructions were given in [DSG] and [Lau]. We provide explicit bounds on the primes appearing in the denominators of ..."
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Cited by 11 (7 self)
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Abstract. One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. Such constructions were given in [DSG] and [Lau]. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct Sunits in certain abelian extensions of a reflex field of K, where S is effectively determined by K, and to bound the primes appearing in the denominators of the Igusa class polynomials arising in the construction of genus 2 curves with CM, as conjectured in [Lau]. 1.
Genus 2 curves with complex multiplication
 International Mathematics Research Notices
"... While the main goal of this paper is to give a bound on the denominators of Igusa class polynomials of genus 2 curves, our motivation is twofold: on the one hand we are interested in applications to cryptography via the use of genus 2 curves with a prescribed number of points, and on the other han ..."
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Cited by 8 (4 self)
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While the main goal of this paper is to give a bound on the denominators of Igusa class polynomials of genus 2 curves, our motivation is twofold: on the one hand we are interested in applications to cryptography via the use of genus 2 curves with a prescribed number of points, and on the other hand, we are interested in construction of class invariants with a view towards
Computing Igusa class polynomials
, 2008
"... We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our alg ..."
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Cited by 6 (2 self)
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We give an algorithm that computes the genus two class polynomials of a primitive quartic CM field K, and we give a runtime bound and a proof of correctness of this algorithm. This is the first proof of correctness and the first runtime bound of any algorithm that computes these polynomials. Our algorithm uses complex analysis and runs in time e O( ∆ 7/2), where ∆ is the discriminant of K. 1
AN ARITHMETIC INTERSECTION FORMULA FOR DENOMINATORS OF IGUSA CLASS POLYNOMIALS
"... Abstract. In this paper we prove an explicit formula for the arithmetic intersection number (CM(K).G1)ℓ on the Siegel moduli space of abelian surfaces, generalizing the work of BruinierYang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which ..."
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Cited by 6 (2 self)
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Abstract. In this paper we prove an explicit formula for the arithmetic intersection number (CM(K).G1)ℓ on the Siegel moduli space of abelian surfaces, generalizing the work of BruinierYang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number (CM(K).G1)ℓ under strong assumptions on the ramification of the primitive quartic CM field K. Yang later proved this conjecture assuming that OK is freely generated by one element over the ring of integers of the real quadratic subfield. In this paper, we prove a formula for (CM(K).G1)ℓ for more general primitive quartic CM fields, and we use a different method of proof than Yang. We prove a tight bound on this intersection number which holds for all primitive quartic CM fields. As a consequence, we obtain a formula for a multiple of the denominators of the Igusa class polynomials for an arbitrary primitive quartic CM field. Our proof entails studying the Embedding Problem posed by Goren and Lauter and counting solutions using our previous article that generalized work of GrossZagier and Dorman to arbitrary discriminants. 1.