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Class Invariants for Quartic CM Fields

Eyal Z. Goren, Kristin E. Lauter (2007)

Annales de l’institut Fourier

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 by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct S -units in certain abelian extensions of a reflex field of K , where S is effectively determined by K , and to bound the primes appearing...

Complex Hyperbolic Surfaces of Abelian Type

Holzapfel, R. (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 11G15, 11G18, 14H52, 14J25, 32L07.We call a complex (quasiprojective) surface of hyperbolic type, iff – after removing finitely many points and/or curves – the universal cover is the complex two-dimensional unit ball. We characterize abelian surfaces which have a birational transform of hyperbolic type by the existence of a reduced divisor with only elliptic curve components and maximal singularity rate (equal to 4). We discover a Picard modular surface of...

Computing the cardinality of CM elliptic curves using torsion points

François Morain (2007)

Journal de Théorie des Nombres de Bordeaux

Let ℰ / ℚ ¯ be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field 𝕂 . The field of definition of ℰ is the ring class field Ω of the order. If the prime p splits completely in Ω , then we can reduce ℰ modulo one the factors of p and get a curve E defined over 𝔽 p . The trace of the Frobenius of E is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose...

Constructing elliptic curves over finite fields using double eta-quotients

Andreas Enge, Reinhard Schertz (2004)

Journal de Théorie des Nombres de Bordeaux

We examine a class of modular functions for Γ 0 ( N ) whose values generate ring class fields of imaginary quadratic orders. This fact leads to a new algorithm for constructing elliptic curves with complex multiplication. The difficulties arising when the genus of X 0 ( N ) is not zero are overcome by computing certain modular polynomials.Being a product of four η -functions, the proposed modular functions can be viewed as a natural generalisation of the functions examined by Weber and usually employed to construct...

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