arXiv subjects
Publications and source records attributed to Juan Carlos Peral.
In this note we present the main details of the construction of an elliptic curve over $\mathbb{Q}(u)$ with torsion $\mathbb{Z}/4\mathbb{Z}$ and rank 6. Previously only rank 5 examples for such curves were known.
In this paper, we present details of seven elliptic curves over $\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/ 8\mathbb{Z}$ and five curves over $\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/ 2\mathbb{Z} \times \mathbb{Z}/ 6\mathbb{Z}$. We also exhibit some particular examples of curves with high rank over $\mathbb{Q}$ by specialization of the parameter. We present several sets of infinitely many elliptic curves in both torsion groups and rank at least $3$ parametrized by elliptic curves having positive rank. In some of these sets we have performed calculations about the distribution of the root number. This has relation with recent heuristics concerning the rank bound for elliptic curves by Park, Poonen, Voight and Wood.
A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y^2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over Q with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank >= 7, which are both the current records for that kind of curves.
Given a Diophantine triple $\{c_1(t),c_2(t),c_3(t)\}$, the elliptic curve over Q(t) induced by this triple, i.e. $y^2=(c_1(t) x+1) (c_2(t) x+1) (c_3(t) x+1)$, can have as torsion group one of the non-cyclic groups in Mazur's theorem, i.e. Z/2Z x Z/2Z, Z/2Z x Z/4Z, Z/2Z x Z/6Z or Z/2Z x Z/8Z. In this paper we present results concerning the rank over Q(t) of these curves improving in some of the cases the previously known results.
We show the existence of families of elliptic curves over Q whose generic rank is at least 2 for the torsion groups Z/8Z and Z/2Z x Z/6Z. Also in both cases we prove the existence of infinitely many elliptic curves, which are parameterized by the points of an elliptic curve with positive rank, with such torsion group and rank at least 3. These results represent an improvement of previous results by Campbell, Kulesz, Lecacheux, Dujella and Rabarison where families with rank at least 1 were constructed in both cases.
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.
The elliptic curve y^2= x^3-Nx where N=m^4+n^4 has rank at least 2 over Q(m,n). When N can be written in two different ways as sum of two fourth powers, then we prove that the rank is at least 4.