Mikić, Miljen (2014) MordellWeil groups and isogenies of the families of elliptic curves. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Abstract
In this work it is proved that the torsion group of elliptic curves induced by $D(4)$triples can be either $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$. Therefore, as a special case it is proved that the torsion group of elliptic curves induced by Diophantine triples can be one of these two, as well. Families of elliptic curves induced by Diophantine triples of the form $\{k − 1, k + 1, c_{l}(k)\}$ are examined and the possible form of the torsion group and rank (i.e. MordellWeil group) of these curves is determined for the large number of values of $k$ and $l$. Finally, by studying modular curves $X_0(n)$ the number of elliptic curves with cyclic isogeny of degree $n$ over various quartic fields is found.
Item Type:  Thesis (Doctoral thesis) 

Supervisor:  Dujella, Andrej and Najman, Filip 
Date:  2014 
Number of Pages:  83 
Subjects:  NATURAL SCIENCES > Mathematics > Algebra 
Divisions:  Faculty of Science > Department of Mathematics 
Depositing User:  Iva Prah 
Date Deposited:  27 Apr 2015 12:19 
Last Modified:  05 Jun 2015 11:32 
URI:  http://digre.pmf.unizg.hr/id/eprint/3883 
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