Julián Aguirre | University of the Basque Country, Euskal Herriko Unibertsitatea (original) (raw)
Supervisors: Ronald Coiffman
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Papers by Julián Aguirre
We construct a family of Diophantine triples {c 1 (t), c 2 (t), c 3 (t)} such that the elliptic c... more We construct a family of Diophantine triples {c 1 (t), c 2 (t), c 3 (t)} such that the elliptic curve over Q(t) induced by this triple, i.e.:
Mathematics of Computation, 2003
We develop an algorithm for bounding the rank of elliptic curves in the family y 2 = x 3 −B x, al... more We develop an algorithm for bounding the rank of elliptic curves in the family y 2 = x 3 −B x, all of them with torsion group Z/(2 Z) and modular invariant j = 1728. We use it to look for curves of high rank in this family and present four such curves of rank 13 and 22 of rank 12.
Periodica Mathematica Hungarica, 2014
There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number... more There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over some quadratic field with this torsion group and with rank ≥ 2.
Glasnik Matematicki, 2013
Given two integers m and n consider N = m 4 +n 4 and the elliptic curve y 2 = x 3 − N x The rank ... more Given two integers m and n consider N = m 4 +n 4 and the elliptic curve y 2 = x 3 − N x The rank of this family over Q(m, n) is at least 2.
Mathematics of Computation, 2005
For all totally positive algebraic numbers α except a finite number of explicit exceptions, the f... more For all totally positive algebraic numbers α except a finite number of explicit exceptions, the following inequality holds:
There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number... more There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over some quadratic field with this torsion group and with rank >= 2.
We construct a family of Diophantine triples {c 1 (t), c 2 (t), c 3 (t)} such that the elliptic c... more We construct a family of Diophantine triples {c 1 (t), c 2 (t), c 3 (t)} such that the elliptic curve over Q(t) induced by this triple, i.e.:
Mathematics of Computation, 2003
We develop an algorithm for bounding the rank of elliptic curves in the family y 2 = x 3 −B x, al... more We develop an algorithm for bounding the rank of elliptic curves in the family y 2 = x 3 −B x, all of them with torsion group Z/(2 Z) and modular invariant j = 1728. We use it to look for curves of high rank in this family and present four such curves of rank 13 and 22 of rank 12.
Periodica Mathematica Hungarica, 2014
There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number... more There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over some quadratic field with this torsion group and with rank ≥ 2.
Glasnik Matematicki, 2013
Given two integers m and n consider N = m 4 +n 4 and the elliptic curve y 2 = x 3 − N x The rank ... more Given two integers m and n consider N = m 4 +n 4 and the elliptic curve y 2 = x 3 − N x The rank of this family over Q(m, n) is at least 2.
Mathematics of Computation, 2005
For all totally positive algebraic numbers α except a finite number of explicit exceptions, the f... more For all totally positive algebraic numbers α except a finite number of explicit exceptions, the following inequality holds:
There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number... more There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over some quadratic field with this torsion group and with rank >= 2.