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.

09.04.2019 kein Seminar
16.04.2019 kein Seminar
23.04.2019 Speaker: 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 Speaker: 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 Speaker: 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.
02.07.2019 Speaker: 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 Speaker: Grégoire Menet (U Grenoble)
Title: tba
Abstract: tba


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