Forschungsseminar Algebraische Geometrie

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

Forschungsseminar "Arithmetische Geometrie"

Sommersemester 2019

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.

23.04.2019	Giulio Bresciani (FU Berlin)
	Title: Essential dimension and pro-finite 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 pro-finite group schemes, and prove two very general criteria that show that essential dimension is almost always infinite for pro-finite 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 D-Modules and the Riemann-Hilbert Correspondence (29. April - 03. Mai, Berlin-Brandenburgische 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 product-quotient, 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 Mumford-Tate conjectures for all surfaces S that lie in the same connected component of the Gieseker moduli space as a product-quotient surface.

28.05.2019	Ezra Waxman (U Prague)
	Title: Hecke Characters and the L-Function 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 L-functions. In this talk I will present several applications obtained upon applying the L-functions Ratios Conjecture to this family of L-functions. 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 Gabber-Katz 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 Lipman-Zariski 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 l-adic 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 b-Chow 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 b-divisors. It follows that the convex-set algebra can be viewed as a universal ring for intersection theory of nef toric $b$-cocycles on the toric Riemann-Zariski space. We further discuss some applications of this viewpoint towards a combinatorial interpretation of non-archimidean Arakelov theory of toric varieties over discretely valued fields in the sense of Bloch-Gillet-Soulé.

02.07.2019	Yuri Bilu (U Bordeaux)
	Title: Singular units do not exist Abstract: It is classically known that a singular modulus (a j-invariant 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 non-effective, 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 computer-assisted 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 blow-ups, 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 Beauville--Bogomolov form.