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.

 29.10.2019 Yuri Bilu (U Bordeaux) Diversity in Parametric Families of Number Fields Abstract: 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 $n\in \mathbb{N}$, pick $P_n\in X$ with $t(P_n)=n$. Hilbert's Irreducibility Theorem (HIT) says that for infinitely many $n$ the field $\mathbb{Q}(P_n)$ is of degree $m$ over $\mathbb{Q}$. Moreover, this holds for overwhelmingly many $n$: Among the number fields $\mathbb{Q}(P_1), \dots ,\mathbb{Q}(P_n)$ there is only $o(n)$ fields of degree less than $m$. However, HIT does not say how many distinct field there are among $\mathbb{Q}(P_1), ... ,\mathbb{Q}(P_n)$. A 1994 result of Dvornicich and Zannier implies that for large $n$, there are at least $cn/\log n$ distinct among those fields, 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, getting rid of the log term seems very hard. 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] 12.11.2019 Ziyang Gao (IMJ-PRG Paris) Functional transcendence on the universal abelian variety Abstract: The main topic of this talk is the mixed Ax-Schanuel theorem for the universal abelian variety. I will explain the statement and sketch its proof. We will start from the analogous statement for abelian varieties over the field of complex numbers. 19.11.2019 Marco Flores (HU Berlin) Cohomological dimension of projective schemes in pro-p towers Abstract: If $X$ is a variety of dimension $d$ over an algebraically closed field, its étale cohomology groups with coefficients in any constructible sheaf vanish in degree greater than $2d$. Moreover, if $X$ is affine they already vanish in degree greater than $d$ by Artin's vanishing theorem. The last is not true for projective varieties. However, Scholze showed that if $X$ is a complex projective variety of dimension $d$ and $p$ is a prime number, there is a specific tower of $p$-power degree covers of $X$ such that the direct limit of étale cohomology groups with $\mathbb{F}_p$-coefficients does vanish in degree greater than $d$. In this talk we present a new proof of this result, by Hélène Esnault, which moreover works over any algebraically closed field of characteristic different from $p$. 26.11.2019 Thorsten Heidersdorf (U Bonn) Tensor categories in positive characteristic Abstract: Some of the most important tensor categories over a field come from representations of algebraic groups. A celebrated theorem of Deligne asserts that every tensor category of subexponential growth is the representation category of an algebraic supergroup scheme. The theorem is no longer true in characteristic $p > 0$. Counterexamples arise from representations of the cyclic group $\mathbb{Z}/p\mathbb{Z}$. Ostrik proposed a conjectural extension, but recently Benson-Etingof constructed an infinite chain of counterexamples in characteristic 2. These categories are closely related to representations of algebraic groups and the question if monoidal categories admit abelian envelopes. I will give an overview about these results and discuss some recent developments. 03.12.2019 Martin Gallauer (U Oxford) Simplicity of Tannakian categories, and applications Abstract: In this talk, I want to discuss two classification problems in algebraic geometry: (1) Given a variety, classify its constructible sheaves (in the derived sense). (2) Given a field, classify its motives (in the derived sense). One aspect which connects the two problems is the appearance of Tannakian categories (in the derived sense). I will draw attention to the fact that these are "simple", and explain how this allows us to make progress on the two classification problems. 10.12.2019 Anil Aryasomayajula (IISER Tirupati, India) On the holomorphic version of a conjecture by Sarnak Abstract: In 1995, Iwaniec and Sarnak computed estimates of Hecke Eigen Maass forms associated to co-compact arithmetic subgroups of SL(2,R), and Sarnak went on to make a conjecture on the growth of Hecke Eigen Maass forms. Adapting the arguments of Iwaniec and Sarnak to the setting of Hecke Eigen cusp forms, we discuss a holomorphic version of the conjecture of Sarnak. 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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