Université de Versailles-St Quentin, Laboratoire de Mathématiques, 45 avenue des États-Unis 78035 Versailles cedex, France
e-mail: divizio[at]math.cnrs.fr          Office: bâtiment Fermat, office 3305

Organisateurs : Alin Bostan et Lucia Di Vizio

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### Programme

11h00-12h00Sara Checcoli (Institut Fourier, UGA, Grenoble)

Resumé : 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 decide property (N) 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 subextensions 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 $$S$$-adic numbers of small height, of which I will present an effective version.

14h30-15h30Veronika Pillwein (RISC, Linz, Austria)

Resumé : Kapteyn series are series expansions in terms of the Bessel function of the first kind. The first researcher to investigate such series in a systematic way was Willem Kapteyn (not to be confused with his brother Jacobus Cornelius Kapteyn). The topic of this talk is the explicit representation for a particular Kapteyn series of the second kind in terms of a family of polynomials. The appearing sums and sequences involve Stirling numbers and are just outside the class of holonomic functions. Still, an extension of the holonomic systems approach due to Chyzak, Kauers, and Salvy, allows to use symbolic computation to find a recurrence for the coefficients of this family of polynomials.

Resumé : We will discuss some new developments arising from a solution of the unbounded denominators conjecture in the theory of noncongruence and vector-valued modular forms. One form of our result, confirming in particular certain new cases of the algebraicity conjectures of Grothendieck and Christol, is the complete determination of those integer coefficients formal power series that fulfill a linear ODE without singularities outside of 0, 1/16 and infinity, and whose local monodromy around 0 is semisimple. We also raise a few related open questions, notably what can be said whenever one lifts the constraint on semisimple local monodromy. This is a joint work with Frank Calegari and Yunqing Tang.

Bon ton du Séminaire différentiel en ligne : merci de vous connecter en utilisant vos nom et prénom réels et complets. À défaut de voir le visage des personnes qui suivent l'exposé, nous pensons que l'orateur doit au moins savoir qui sont les collègues connectés.

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