1,314 research outputs found
On Mordell-Weil groups of elliptic curves induced by Diophantine triples
We study the possible structure of the groups of rational points on elliptic
curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals
such that the product of any two of them is one less than a square.Comment: 17 pages; to appear in Glasnik Matematicki 42 (2007
Descent Via Isogeny on Elliptic Curves with Large Rational Torsion Subgroups.
We outline PARI programs which assist with various algorithms related to descent via isogeny on elliptic curves. We describe, in this context, variations of standard inequalities which aid the computation of members of the Tate-Shafarevich group. We apply these techniques to several examples: in one case we use descent via 9-isogeny to determine the rank of an elliptic curve; in another case we find nontrivial members of the 9-part of the Tate-Shafarevich group, and in a further case, nontrivial members of the 13-part of the Tate-Shafarevich group
Calabi-Yau threefolds with large h^{2, 1}
We carry out a systematic analysis of Calabi-Yau threefolds that are
elliptically fibered with section ("EFS") and have a large Hodge number h^{2,
1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of
moduli space associated with different topologies connected through transitions
that can be understood in terms of singular Weierstrass models. We determine
the complete set of such threefolds that have h^{2, 1} >= 350 by tuning
coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set
of Hodge numbers includes those of all known Calabi-Yau threefolds with h^{2,
1} >= 350, as well as three apparently new Calabi-Yau threefolds. We speculate
that there are no other Calabi-Yau threefolds (elliptically fibered or not)
with Hodge numbers that exceed this bound. We summarize the theoretical and
practical obstacles to a complete enumeration of all possible EFS Calabi-Yau
threefolds and fourfolds, including those with small Hodge numbers, using this
approach.Comment: 44 pages, 5 tables, 5 figures; v2: minor corrections; v3: minor
corrections, moved figure; v4: typo in Table 2 correcte
Elliptic curves with torsion group
We exhibit several families of elliptic curves with torsion group isomorphic
to and generic rank at least . Families of this kind have been
constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo.
We mention the details of some of them and we add other examples developed more
recently by Dujella and Peral, and MacLeod.
Then we apply an algorithm of Gusi\'c and Tadi\'c and we find the exact rank
over \Q(t) to be 3 and we also determine free generators of the Mordell-Weil
group for each family. By suitable specializations, we obtain the known and new
examples of curves over \Q with torsion and rank , which is the
current record
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