SEMINARIO DE TEORIA DE NUMEROS

"Mordell-Weil groups of elliptic curves over number fields"

Filip Najman (Universidad de Zagreb / Sveučilište u Zagrebu)

DÍA: Martes, 15 de Marzo de 2016.

HORA: 12:00 h.

LUGAR: 17-520.

RESUMEN:

The Mordell-Weil group E(K) of K-rational points of an elliptic curve E
over a number field K, is a finitely generated abelian group and hence
isomorphic to the direct product of its torsion subgroup and Z^r, where r
is the rank of E/K.

In this talk we will consider the question of what this group can be over
number fields of certain type, e.g. over all number fields of degree d or
over a fixed number field. After surveying known results, both old and new,
about torsion groups, we will show that prescribing the torsion over number
fields (as opposed to over the rationals!) can force various properties on
the elliptic curve. For instance, all elliptic curves with points of order 13 or 18 over
quadratic fields have to have even rank and elliptic curves with points of
order 16 over quadratic fields are base changes of elliptic curves defined
over the rationals. We show that these properties arise from the geometry of the
corresponding modular curves.