### Density of rational points on elliptic surfaces

Heilbronn Number Theory Seminar

26th April 2017, 4:00 pm – 5:00 pm

Howard House, 4th Floor Seminar Room

Let X be an algebraic surface. We are interested in the set of rational points X(ℚ). Is it non-empty? Is it infinite? Is it Zariski-dense? We are concerned with elliptic surfaces, i.e. 1-parameter families of elliptic curves. The density of rational points is not well known in general. When the surface is isotrivial, one shows the density in most cases when it is also rational. Moreover, by studying the variation of the root number of the fibers, one predicts the density on non-isotrivial elliptic surfaces conditionally to some conjectures (parity conjecture, squarefree conjecture, Chowla's conjecture). The last two conjectures impose a restriction on the degree of the factors of the discriminant. We manage to avoid the squarefree conjecture in certain cases, and thus show unconditionally the variation of the root number, without imposing a bound for the degree of the irreducible factors.

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