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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Wintersemester 2007/08
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