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On equations defining fake elliptic curves

Pilar Bayer, Jordi Guàrdia (2005)

Journal de Théorie des Nombres de Bordeaux

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 fake elliptic curves. 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...

On Equations y² = xⁿ+k in a Finite Field

A. Schinzel, M. Skałba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Solutions of the equations y² = xⁿ+k (n = 3,4) in a finite field are given almost explicitly in terms of k.

On Exceptions in the Brauer-Kuroda Relations

Jerzy Browkin, Juliusz Brzeziński, Kejian Xu (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

Let F be a Galois extension of a number field k with the Galois group G. The Brauer-Kuroda theorem gives an expression of the Dedekind zeta function of the field F as a product of zeta functions of some of its subfields containing k, provided the group G is not exceptional. In this paper, we investigate the exceptional groups. In particular, we determine all nilpotent exceptional groups, and give a sufficient condition for a group to be exceptional. We give many examples of nonnilpotent solvable...

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