**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. **