## Humboldt Arithmetic Geometry Seminar

Humboldt University Berlin, Winter 2022/23

Time: Tuesdays 13:15 - 14:45

Venue: Room 3.006, Rudower Chaussee 25, 12489 Berlin

Title: Hecke orbits on Shimura varieties
Abstract: I will talk about the proof of the Hecke orbit conjecture, proposed by Chai and Oort. I will mainly focus on two of the tools that are used. The first main ingredient is a new local result on the monodromy groups of F-isocrystals, which enhances Crew's parabolicity conjecture. Another one is the Cartier-Witt stack constructed by Bhatt-Lurie. This is a joint work with Pol van Hoften.
01.11.2022 Constantin Podelski (HU Berlin)
Title: The degree of the Gauss map for certain Prym varieties and the Schottky problem
Abstract: One approach to the Schottky problem is to study the stratification of the moduli space $$\mathcal{A}_g$$ of principally polarized abelian varieties by the dimension of the singular locus of the theta divisor: Andreotti and Mayer have shown that the locus of Jacobians is an irreducible component of the stratum $$\mathcal{N}_{g-4} \subset \mathcal{A}_g$$. In a similar spirit one can stratify $$\mathcal{A}_g$$ by the degree of the Gauss map: Recently, Codogni and Krämer have shown that the locus of Jacobians is also an irreducible component of the corresponding Gauss stratum. Motivated by this, we study the Gauss map on various other irreducible components of $$\mathcal{N}_{g-4}$$ that parametrize certain Prym varieties, and we show that for each $$g \ge 4$$ one of these components has the same Gauss degree as Jacobians.
08.11.2022 Riccardo Pengo (MPI Bonn)
Title: On the Northcott property for special values of L-functions
Abstract: According to Northcott's theorem, each set of algebraic numbers whose height and degree are bounded is finite. Analogous finiteness properties are also satisfied by many other heights, as for instance the Faltings height. Given the many (expected and proven) links between heights and special values of L-functions (with the BSD conjecture as the most remarkable example), it is natural to ask whether the special values of an L-functions satisfy a Northcott property. In this talk, based on a joint work with Fabien Pazuki, and on another joint work in progress with Jerson Caro and Fabien Pazuki, we will show how this Northcott property is often satisfied at the left of the critical strip, and not satisfied on the right. We will also overview the links between these Northcott properties and those of the motivic heights defined by Kato, and also some effective aspects of our work, which aim at giving some explicit bounds for the cardinality of the finite sets that we come across.
15.11.2022 Annette Huber-Klawitter (Freiburg)
Title: Period Numbers
Abstract: Period numbers are complex numbers like $$\pi, \log(2)$$ or $$\zeta(5)$$. They can be described as values of integrals. As apparent from the examples, such numbers appear in many places in mathematics and are of great interest in transcendence theory. More recently, a variant involving also the exponential function has come into focus. I am going to explain the definition and its conceptual interpretation.
22.11.2022 Siddarth Mathur (Orsay)
Title: Searching for the impossible Azumaya algebra
Abstract: In two 1968 seminars, Grothendieck used the framework of étale cohomology to extend the definition of the Brauer group to all schemes. Over a field, the objects admit a well-known algebro-geometric description: They are represented by $$\mathbb{P}^n$$-bundles (equivalently: Azumaya Algebras). Despite the utility and success of Grothendieck's definition, an important foundational aspect remains open: Is every cohomological Brauer class over a scheme represented by a $$\mathbb{P}^n$$-bundle? It is not even known if smooth proper threefolds over the complex numbers have enough Azumaya algebras! In this talk, I will outline a strategy to construct a Brauer class that cannot be represented by an Azumaya algebra. Although the candidate is algebraic, the method will leave the category of schemes and use formal-analytic line bundles to create Brauer classes. I will then explain a strange criterion for the existence of a corresponding Azumaya Algebra. At the end, I will reveal the unexpected conclusion of the experiment.
29.11.2022 Laurentiu Maxim (Wisconsin)
Title: On the topology of aspherical complex projective manifolds and related questions
Abstract: I will report on recent progress on various open problems involving aspherical complex projective manifolds, including the Singer-Hopf conjecture and the Bobadilla-Kollar conjecture. The first part will be very informal, stating the main problems, historical developments, and some recent results. In the second part, I will introduce the main technical tools and sketch proofs of some of the recent results.

Note: This is the first of a series of two talks. The second talk will be in the algebraic geometry seminar on Wednesday, see here.
06.12.2022 Roberto Gualdi (Regensburg)
Title: How to guess the height of the solutions of a system of polynomial equations
Abstract: A beautiful result due to Bernstein and Kushnirenko allows to predict the number of solutions of a system of Laurent polynomial equations from the combinatorial properties of the defining Laurent polynomials. In a joint work with Martín Sombra (ICREA and Universitat de Barcelona), we give intuitions for an arithmetic version of such a theorem. In particular, in the easy case of the planar curve x + y + 1 = 0, we show how to guess the height of its intersection with a twist of itself by a torsion point. The talk will involve the Arakelov geometry of toric varieties, special values of the Riemann zeta function and, unexpectedly, the most famous detective of the world literature.
20.12.2022 Moritz Kerz (Regensburg)
Title:
Abstract:
31.01.2023 Martín Sombra (ICREA and Universitat de Barcelona)
Title: The zero set of the independence polynomial of a graph
Abstract: In statistical mechanics, the independence polynomial of a graph $$G$$ arises as the partition function of the hard-core lattice gas model on $$G$$. The distribution of the zeros of these polynomials when $$G \to +\infty$$ is relevant for the study of this model and, in particular, for the determination of its phase transitions. In this talk, I will review the known results on the location of these zeros, with emphasis on the case of rooted regular trees of fixed degree and varying depth $$k \ge 0$$. In an on-going work with Juan Rivera-Letelier (Rochester, USA) we show that for these graphs, the zero sets of their independence polynomials converge as $$k \to \infty$$ to the bifurcation measure of a certain family of dynamical systems on the Riemann sphere. In turn, this allows to show that the pressure function of this model has a unique phase transition, and that it is of infinite order.

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