Displaying similar documents to “Addendum and errata “Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields””

Good reduction of elliptic curves over imaginary quadratic fields

Masanari Kida (2001)

Journal de théorie des nombres de Bordeaux

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We prove that the j -invariant of an elliptic curve defined over an imaginary quadratic number field having good reduction everywhere satisfies certain Diophantine equations under some hypothesis on the arithmetic of the quadratic field. By solving the Diophantine equations explicitly in the rings of quadratic integers, we show the non-existence of such elliptic curve for certain imaginary quadratic fields. This extends the results due to Setzer and Stroeker.

On equations defining fake elliptic curves

Pilar Bayer, Jordi Guàrdia (2005)

Journal de Théorie des Nombres de Bordeaux

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Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as . We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explicit knowledge of the normalized period matrices and on the use of theta functions with characteristics. As in the...

Fundamental domains for Shimura curves

David R. Kohel, Helena A. Verrill (2003)

Journal de théorie des nombres de Bordeaux

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We describe a process for defining and computing a fundamental domain in the upper half plane of a Shimura curve X 0 D ( N ) associated with an order in a quaternion algebra A / 𝐐 . A fundamental domain for X 0 D ( N ) realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves X 0 6 ( 1 ) , X 0 15 ( 1 ) , and X 0 35 ( 1 ) . The first example is a classical example of a triangle group and the second is a corrected version of that appearing in the book of Vignéras [13], due...

Constructing elliptic curves over finite fields using double eta-quotients

Andreas Enge, Reinhard Schertz (2004)

Journal de Théorie des Nombres de Bordeaux

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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...

Equivalences between elliptic curves and real quadratic congruence function fields

Andreas Stein (1997)

Journal de théorie des nombres de Bordeaux

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In 1994, the well-known Diffie-Hellman key exchange protocol was for the first time implemented in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a so-called infrastructure. More recently, the scheme was extended to real quadratic congruence function fields, whose set of reduced principal ideals has a similar infrastructure. As always,...