Number Theory Seminar: Gabriel Dill (Université Neuchâtel)

Titel: Points of small height and where to find them - a group-theoretic criterion

Abstract: The height of an algebraic number is a measure for its arithmetic complexity. While numbers of height zero are classified by Kronecker's theorem (they are precisely 0 and the roots of unity), many questions remain open about numbers of small but non-zero height. The famous conjecture of Lehmer predicts an essentially best possible lower bound for the height of such numbers. A related question is whether, given an algebraic extension of the rationals, it contains numbers of arbitrarily small height.

Rémond formulated a generalization of Lehmer’s conjecture, which yields a characterization of points of small height on abelian varieties or algebraic tori that are defined over some infinite algebraic extensions of the rational numbers, generated by the division points of certain finitely generated subgroups. For instance, the field generated over the rationals by all roots of 2 contains some obvious points of very small height (0, small fractional powers of 2 multiplied by roots of unity) and the conjecture implies that it contains no others.

If the finitely generated subgroup is of positive rank, then even weaker versions of Rémond's conjecture (analogues of Dobrowolski’s theorem) are wide open. Recently, Pottmeyer identified a necessary group-theoretic condition for Rémond’s conjecture to hold and showed that it is satisfied for the multiplicative group. In joint work in progress with Sara Checcoli, we show that Pottmeyer’s condition is also satisfied for arbitrary almost split semiabelian varieties, using results from Kummer theory.

Spiegelgasse 5, Seminarraum 05.002

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