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On computing quaternion quotient graphs for function fields

Gebhard Böckle, Ralf Butenuth (2012)

Journal de Théorie des Nombres de Bordeaux

Let Λ be a maximal 𝔽 q [ T ] -order in a division quaternion algebra over 𝔽 q ( T ) which is split at the place . The present article gives an algorithm to compute a fundamental domain for the action of the group of units Λ * on the Bruhat-Tits tree 𝒯 associated to PGL 2 ( 𝔽 q ( ( 1 / T ) ) ) . This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group Λ * in terms of generators and relations. Moreover we determine an upper bound...

On congruent primes and class numbers of imaginary quadratic fields

Nils Bruin, Brett Hemenway (2013)

Acta Arithmetica

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes p such that 16 divides the class number of the imaginary quadratic field ℚ(√-p)....

On elementary equivalence, isomorphism and isogeny

Pete L. Clark (2006)

Journal de Théorie des Nombres de Bordeaux

Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...

On Elkies subgroups of -torsion points in elliptic curves defined over a finite field

Reynald Lercier, Thomas Sirvent (2008)

Journal de Théorie des Nombres de Bordeaux

As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic p , there exists an algorithm that computes, for an Elkies prime, -torsion points in an extension of degree - 1 at cost O ˜ ( max ( , log q ) 2 ) bit operations in the favorable case where p / 2 .We combine in this work a fast algorithm for computing isogenies due to Bostan, Morain, Salvy and Schost with the p -adic approach followed by Joux and Lercier to get an algorithm valid without any limitation...

On elliptic curves and random matrix theory

Mark Watkins (2008)

Journal de Théorie des Nombres de Bordeaux

Rubinstein has produced a substantial amount of data about the even parity quadratic twists of various elliptic curves, and compared the results to predictions from random matrix theory. We use the method of Heegner points to obtain a comparable (yet smaller) amount of data for the case of odd parity. We again see that at least one of the principal predictions of random matrix theory is well-evidenced by the data.

On elliptic Galois representations and genus-zero modular units

Julio Fernández, Joan-C. Lario (2007)

Journal de Théorie des Nombres de Bordeaux

Given an odd prime   p   and a representation ϱ   of the absolute Galois group of a number field k onto PGL 2 ( 𝔽 p ) with cyclotomic determinant, the moduli space of elliptic curves defined over k with p -torsion giving rise to ϱ consists of two twists of the modular curve X ( p ) . We make here explicit the only genus-zero cases p = 3 and p = 5 , which are also the only symmetric cases: PGL 2 ( 𝔽 p ) 𝒮 n for n = 4 or n = 5 , respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which...

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 extensions of mixed motives.

Christopher Deninger (1997)

Collectanea Mathematica

In this article we give an introduction to mixed motives and sketch a couple of ways to construct examples.

On families of 9-congruent elliptic curves

Tom Fisher (2015)

Acta Arithmetica

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose 9-torsion subgroups are isomorphic as Galois modules.

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