The workshop will take place on December 17th on Zoom from 15:00-18:20 (UK time).
15:00-15:50, Philipp Habegger (University of Basel)
Title: Some cases of the Schinzel-Zassenhaus Conjecture in Arithmetic Dynamics
Abstract: This is joint work with Harry Schmidt. We prove a variant of the Schinzel-Zassenhaus Conjecture for a class of polynomials in the setting of arithmetic dynamics. The class contains $T^2-1$. For this polynomial we conclude a lower bound for the Call-Silverman height of a wandering point that decays like the inverse of the square of the field degree. Our method is based on a recent breakthrough by Dimitrov who proved the Schinzel-Zassenhaus Conjecture.
16:15-17:05, Sara Checcoli (Université Grenoble Alpes)
Title: On small height and local degrees, (joint work with A. Fehm)
Abstract: A field of algebraic numbers has the Northcott property (N) if it contains only finitely many elements of bounded absolute logarithmic Weil height. While for number fields property (N) follows immediately by Northcott's theorem, to establish its validity for an infinite extension of the rationals is, in general, a difficult problem.
This property was introduced in 2001 by Bombieri and Zannier, who raised the question of whether it holds for fields with uniformly bounded local degrees. They also remarked that, for a (possibly infinite) Galois extension of the rationals whose local degrees are bounded at (at least) one prime, property (N) is implied by the divergence of a certain sum, but suggested that this phenomenon might occur only for number fields. In 2011 Widmer gave a criterion for an infinite extension of the rationals to have property (N) under some condition on the growth of the discriminants of certain finite sub-extensions of the field.
In this talk I will present several results obtained in this context with A. Fehm. In particular, we show the existence of infinite Galois extensions of the rationals for which the sum considered by Bombieri and Zannier is divergent and to which Widmer's criterion does not apply and we also show the existence of fields without property (N) and having (non-uniformly) bounded local degrees at all primes. This last result is a corollary of a theorem of Fili on totally $p$-adic numbers of small height, of which I will present an effective version.
17:30-18:20, Jeff Vaaler (University of Texas at Austin)
Title: Schinzel's determinant inequality and a conjecture of Rodriguez Villegas