Humboldt Universität zu Berlin
Mathem.Naturwissenschaftliche Fakultät
Institut für Mathematik
Sommersemester 2019
Das Forschungsseminar findet dienstags in der Zeit von 13:15  15:00 Uhr in der Rudower Chaussee 25, 12489 BerlinAdlershof, Raum 3.006 (Haus 3, Erdgeschoss), statt.
23.04.2019  Giulio Bresciani (FU Berlin) 
Title: Essential dimension and profinite group schemes
Abstract: The essential dimension of an algebraic group is a measure of the complexity of its functor of torsors, i.e. $H^1(,G)$. Classically, essential dimension has only been studied for group schemes of finite type. We study the case of profinite group schemes, and prove two very general criteria that show that essential dimension is almost always infinite for profinite group schemes: It does not provide much information about them. We thus propose a new, natural refinement of essential dimension, the fce dimension. The fce dimension coincides with essential dimension for group schemes of finite type but has a better behaviour otherwise. Over any field, we compute the fce dimension of the Tate module of a torus. Over fields finitely generated over $\mathbb{Q}$, we compute the fce dimension of $\mathbb{Z}_p$ and of the Tate module of an abelian variety. 

30.04.2019  Workshop DModules and the RiemannHilbert Correspondence (29. April  03. Mai, BerlinBrandenburgische Akademie der Wissenschaften) 
07.05.2019  Johan Commelin (U Freiburg) 
Title: On the cohomology of smooth projective surfaces with $p_g = q = 2$ and maximal Albanese dimension
Abstract: In this talk I will report on a joint project with Matteo Penegini (Genova). The second cohomology of a surface S as mentioned in the title splits up as a sum of two pieces. One piece comes from the Albanese variety. The other piece looks like the cohomology of a K3 surface, which we call a K3 partner X of S. If the surface S is a productquotient, then we can geometrically construct the K3 partner X and an algebraic correspondence that relates the cohomology of S and X. Finally, we prove the Tate and MumfordTate conjectures for all surfaces S that lie in the same connected component of the Gieseker moduli space as a productquotient surface. 

28.05.2019  Ezra Waxman (U Prague) 
Title: Hecke Characters and the LFunction Ratios Conjecture
Abstract: A Gaussian prime is a prime element in the ring of Gaussian integers $\mathbb{Z}[i]$. As the Gaussian integers lie on the plane, interesting questions about their geometric properties can be asked which have no classical analogue among the ordinary primes. Hecke proved that the Gaussian primes are equidistributed across sectors of the complex plane by making use of Hecke characters and their associated Lfunctions. In this talk I will present several applications obtained upon applying the Lfunctions Ratios Conjecture to this family of Lfunctions. In particular, I will present a refined conjecture for the variance of Gaussian primes across sectors, and a conjecture for the one level density across this family. 

04.06.2019  Quentin Guignard (ENS Paris) 
Title: Geometric $\ell$adic local factors
Abstract: I will explain how to give a cohomological definition of epsilon factors for $\ell$adic sheaves over a henselian trait of positive equicharacteristic distinct from $\ell$. The resulting formula is reminiscent of the cohomological construction by Katz of the $\ell$adic Swan representation, and involves GabberKatz extensions as well. These local factors provide a product formula for the determinant of the cohomology of an $\ell$adic sheaf on a curve over a field of positive characteristic distinct from $\ell$. When the base field is finite, this specializes to the classical theory of Dwork, Langlands, Deligne, and Laumon. 

11.06.2019  13:15 14:15 Patrick Graf (U Bayreuth) 
Title: Reflexive differential forms in positive characteristic
Abstract: Given a differential form on the smooth locus of a normal variety defined over a field of positive characteristic, we discuss under what conditions it extends to a resolution of singularities (possibly with logarithmic poles). Our main result works for log canonical surface pairs over a perfect field of characteristic at least seven. We also give a number of examples showing that our results are sharp in the surface case, and that they fail in higher dimensions. If time permits, we will give applications to the study of the LipmanZariski conjecture. 

14:30 15:30 Antonio Rojas León (U Sevilla)  
Title: Local systems with sporadic monodromy groups
Abstract: Let $X$ be a curve over a separably closed field $k$ of characteristic $p$ and $\mathcal F$ an $\ell$adic local system on $X$, where $\ell\neq p$. If we view this local system as a representation of $\pi_1(X)$, its Zariski closure is the global monodromy group $G$ of $\mathcal F$. When $X$ and $\mathcal F$ are defined over a finite field, this group determines (after a normalization) the distribution of the Frobenius traces of $\mathcal F$ on the points of $X$ with values over sufficiently large finite extensions of the base field. In general, one expects this group to be as large as allowed for by the geometric properties of $\mathcal F$. In particular, only in exceptional cases it will be finite. Abhkanyar's conjecture determines which finite groups can appear as monodromy groups of such local systems. In particular, if $G$ is simple finite, it can be the monodromy group of a local system on the affine line in characteristic $p$ if and only if $p$ divides the order of $G$. In this talk we will give some naturally constructed examples of local systems on the affine line and the punctured affine line on small characteristic whose monodromy groups are sporadic finite: the Conway groups $Co_1$, $Co_2$, $Co_3$, the Suzuki group $6.Suz$ or the McLaughlin group $McL$. This is joint work with Nicholas M. Katz (Princeton) and Pham H. Tiep (Rutgers). 

18.06.2019  Giulio Orecchia (U Rennes) 
Title: A tropical / monodromy criterion for the existence of Néron models
Abstract: Néron models are central objects to the study of degenerations of abelian varieties over Dedekind schemes. However, over bases of higher dimension, they do not always exist. In this talk I will introduce a criterion, called toric additivity, for an abelian family with semistable reduction to admit a Néron model. It can be expressed in terms of monodromy action on the ladic Tate module, and, in the case of jacobians of curves, in terms of finiteness of the tropical jacobian. 

25.06.2019  Ana Maria Botero (U Regensburg) 
Title: The convex set algebra and the bChow ring of toric varieties
Abstract: We extend McMullen's polytope algebra to the so called convex set algebra. We show that the convex set algebra embeds in the projective limit of the Chow cohomology rings of all smooth toric compactifications of a given torus, with image generated by the classes of all nef toric bdivisors. It follows that the convexset algebra can be viewed as a universal ring for intersection theory of nef toric $b$cocycles on the toric RiemannZariski space. We further discuss some applications of this viewpoint towards a combinatorial interpretation of nonarchimidean Arakelov theory of toric varieties over discretely valued fields in the sense of BlochGilletSoulé. 

02.07.2019  Yuri Bilu (U Bordeaux) 
Title: Singular units do not exist
Abstract: It is classically known that a singular modulus (a jinvariant of a CM elliptic curve) is an algebraic integer. Habegger (2015) proved that at most finitely many singular moduli are units, answering a question of Masser (2011). However, his argument, being noneffective, did not imply any bound for the size of these "singular units". I will report on a recent work with Philipp Habegger and Lars Kühne, where we prove that singular units do not exist at all. First, we bound the discriminant of any singular unit by $10^{15}$. Next, we rule out the remaining singular units using computerassisted arguments. 

09.07.2019  Grégoire Menet (U Grenoble) 
Title: Integral cohomology of quotients via toric geometry
Abstract: I will provide some new methods, based on toric blowups, to determine the integral cohomology of complex manifolds quotiented by automorphisms groups of prime order. Indeed, quotient singularities can locally be interpreted as toric varieties, and the framework of toric geometry is well adapted to deal with integral cohomology. The original motivation to study this problem was the computation of an important invariant in the context of hyperhähler geometry: the BeauvilleBogomolov form. 
Wintersemester 2007/08
