Humboldt Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

Forschungsseminar "Arithmetische Geometrie"

Wintersemester 2019/20

Das Forschungsseminar findet dienstags in der Zeit von 13:15 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 3.006 (Haus 3, Erdgeschoss), statt.

22.10.2019 kein Seminar
29.10.2019 Yuri Bilu (U Bordeaux)
Diversity in Parametric Families of Number Fields
Let $X$ be a projective curve over $\mathbb{Q}$ of genus $g$ and $t$ a non-constant $\mathbb{Q}$-rational function of degree $m>1$. For every integer $n$ pick a point $P_n$ on $X$ such that $t(P_n)=n$. Hilbert's Irreducibility Theorem (HIT) tells us that, for infinitely many $n$, the field $\mathbb{Q}(P_n)$ is of degree $m$ over $\mathbb{Q}$. Moreover, this holds true for overwhelmingly many $n$, in the following sense: among the number fields $\mathbb{Q}(P_1), ... ,\mathbb{Q}(P_n)$ there is only $o(n)$ fields of degree less than $m$. However, HIT fails to answer the following question: how many distinct field are there among $\mathbb{Q}(P_1), ... ,\mathbb{Q}(P_n)$? A 1994 result of Dvornicich and Zannier implies that, for large $n$, among those fields there are at least $cn/\log n$ distinct, with $c=c(m,g)>0$. Conjecturally, there should be a positive proportion (that is, $cn$) of distinct fields. This conjecture is proved in many special cases in the work of Zannier and his collaborators, but, in general case, getting rid of the log term seems to be a very hard problem. We make a little step towards proving this conjecture. While we cannot remove the log term altogether, we can replace it by log n raised to a power strictly smaller than 1. To be precise, we prove that for large $n$ there are at least $n/(\log n)^{1-e}$ distinct fields, where $e=e(m,g)>0$. A joint work with Florian Luca.
05.11.2019 Workshop on Brill-Noether theory
[see here for details]
03.12.2019 Martin Gallauer (U Oxford)
Title: tba
Abstract: tba
10.12.2019 Anil Aryasomayajula (IISER Tirupati, India)
Title: tba
Abstract: tba
14.01.2020 Dragos Fratila (U Strasbourg)
Title: tba
Abstract: tba
11.02.2020 Adrian Langer (U Warsaw)
Title: tba
Abstract: tba


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