Cecília Salgado

Large Mordell-Weil rank jumps and the Hilbert Property

We discuss recent progress on the variation of the Mordell-Weil rank in families of elliptic curves. Recently, in collaboration with D. Loughran, we showed that if the underlying surface admits a conic bundle structure then the subset of fibres for which the Mordell-Weil rank is strictly larger than the generic rank is not thin, as a subset of the base of the fibration. We also obtained larger rank jumps (+2) under extra hypotheses on the fibration. In this talk, we discuss even larger rank jumps (+3) on non-thin subsets for elliptic fibrations with a non-reduced fibre. The new results are based on joint work with R. Dias (UFRJ).