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

## Forschungsseminar "Arithmetische Geometrie"

Sommersemester 2020

Das Forschungsseminar findet im SS 2020 wegen der Corona-Pandemie vorerst nicht statt!!! The research seminar won't take place until further notice due to coronavirus!!!

 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 $\mathrm{SL}(2,\mathbb{R})$, 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) The Jordan-Chevalley decomposition for semistable $G$-bundles on elliptic curves Abstract: For projective curves of arithmetic genus one, there are three possibilities: nodal curve, cuspidal curve, elliptic curve. For $G$ a reductive group one can consider semistable $G$-bundles on such a curve and their moduli stack. In the nodal case one recovers the adjoint quotient $G/G$ and in the cuspidal case the Lie algebra version of the adjoint quotient. The elliptic curve case could be considered as a further "exponentiation" of the group $G$. We will review the Jordan-Chevalley decomposition in the Lie algebra and Lie group case and then proceed to explain how one can formulate such a decomposition for $G$-bundles over an elliptic curve. Then we'll see that the Jordan-Chevalley decomposition can be also expressed in terms of the existence of a certain stratification of the moduli stack of semistable $G$-bundles. In the nodal or cuspidal case the strata are described in terms of nilpotent cones of certain Levi or pseudo-Levi subgroups. I will explain a similar description in the elliptic case. Finally, I hope to explain how such a stratification can be used to study certain automorphic sheaves on $\mathrm{Bun}_G(E)$ for an elliptic curve in analogy to the study of character sheaves for groups or Lie algebras. This is joint work with Sam Gunningham and Penghui Li. 21.01.2020 Antareep Mandal (HU Berlin) Sup-norm bound for the average over an orthonormal basis of Siegel cusp forms Abstract: This talk updates our progress towards obtaining a sup-norm bound for the average over an orthonormal basis of Siegel cusp forms by relating it to the discrete spectrum of the heat kernel of the Siegel-Maass Laplacian. We begin by generalizing the relation between the cusp forms and the Maass forms in higher dimensions. Then we obtain the spherical functions for the Laplace-Beltrami operator in Siegel upper half-space by reduction to the complex case and make a weight correction to make them correspond to the Siegel-Maass Laplacian. Then we use these to construct the heat kernel, whose analysis leads to the weight dependence of the sup-norm bound. 28.01.2020 Ezra Waxman (TU Dresden) Random Models for the Distribution of Primes with a Prescribed Primitive Root Abstract: The Hardy-Littlewood Conjecture (also known as the k-tuple conjecture) is a vast generalization of the twin prime conjecture. In particular, it gives an asymptotic count for the number of integer pairs $(n,n+d)$ such that both $n$ and $n+d$ are prime. In this talk, I will discuss preliminary results concerning the application of these heuristic models to the distribution of primes with a prescribed primitive root. Specifically, for integers $\mathbb{g} \geq 2$ and $d \in 2\mathbb{N}$, I will present a conjecture for the number of prime pairs $(p,p+d)$ such that $g$ is a primitive root modulo both $p$ and $p+d$. Time permitting, I will also introduce preliminary findings concerning the distribution of primes with a prescribed primitive root, across short intervals. Joint work with Magdaléna Tinková & Mikuláš Zindulka. 11.02.2020 Adrian Langer (U Warsaw) On smooth projective $\mathscr{D}$-affine varieties Abstract: A $\mathscr{D}$-affine variety is a variety on which left $\mathscr{D}$-modules behave as on an affine variety. Unlike in the case of $\mathscr{O}$-modules there exist smooth projective varieties that are $\mathscr{D}$-affine (e.g., rational homogeneous varieties in characteristic zero). I will survey known results on smooth projective $\mathscr{D}$-affine varieties. In particular, I will classify $\mathscr{D}$-affine smooth projective surfaces. In positive characteristic, a basic tool that I use is a new generalization of Miyaoka's generic semipositivity theorem.

Archiv:

Wintersemester 2007/08
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