# Arithmetic and Algebraic Geometry seminar

## University of Amsterdam

### Practical information

The seminar takes place every week on Wednesday, from 11:00 to 12:00, starting September 13th. The lectures will take place in room F3.20 at the Nikhef. The seminar is organized by Arno Kret (ArnoKret at gmail dot com) and Mingmin Shen (M.Shen at uva dot nl).

### Upcoming lectures

Click on the name of the speakers below to see the abstract.

### Past events

#### June 13, 2018. Ramla Abdellatif: Iwahori-Hecke algebras and masures for split Kac-Moody groups

Abstract: Let $$F$$ be a non-Archimedean local field and $$G$$ be the group of $$F$$-rational points of a connected reductive group defined over $$F$$. The study of (complex smooth) representations of $$G$$ imply various tools coming from different nature. These include in particular induction functors, Hecke algebras (seen as convolution algebras or as intertwinning algebras) and Bruhat-Tits buildings.

When seeing Kac-Moody groups as a natural generalization of reductive groups, one can wonder how far the setting developed for reductive groups can be extended to the Kac-Moody case. Thanks to Rousseau, Gaussent-Rousseau and Bardy-Panse-Gaussent Rousseau, there is a suitable generalization of Bruhat-Tits buildings (called masures) as well as handful definitions of spherical and Iwahori-Hecke algebras. Nevertheless, these algebras are not really fully satisfying as they do not, for instance, satisfy the analogue of Bernstein's theorem in this setting. Another frustrating lack was that there was so far no natural construction attaching a Hecke algebra to a suitable analogue of open compact subgroups.

In this talk, we discuss some results, obtained in collaboration with Auguste Hebert, addressing these questions for split Kac-Moody groups. In particular, we explain why the Iwahori-Hecke algebra as defined by Rousseau and his collaborators is not the right generalization of the usual Iwahori-Hecke algebra, as its center is too small', then we define a suitable generalization (using a sort of completion process) that satisfies a Bernstein-like theorem. If enough time is left, we will also explain how to attach a suitable Hecke algebra to each type 0 spherical facet of the masure that gives back the well-known Hecke algebras in the reductive case.

#### May 30, 2018. Stefan Schreieder: Stably irrational hypersurfaces of small slopes

Abstract: We show that a very general complex projective hypersurface of dimension $$N$$ and degree at least $${\rm log}_2 (N+2)$$ is not stably rational. The same statement holds over any uncountable field of characteristic $$p \gg N$$.

#### May 16, 2018. Junliang Shen: Special subvarieties in holomorphic symplectic varieties

Abstract: By the result of Bogomolov and Mumford, every projective K3 surface contains a rational curve. Holomorphic symplectic varieties are higher dimensional analogs of K3 surfaces. We will discuss a conjecture of Voisin about algebraically coisotropic subvarieties of holomorphic symplectic varieties, which can be viewed as a higher dimensional generalization of the Bogomolov-Mumford theorem. We show that Voisin's conjecture holds when the holomorphic symplectic variety arises as a moduli space of sheaves on a K3 surface. Finally, we discuss the structure of rational curves in holomophic symplectic varieties. Based on joint work with Georg Oberdieck, Qizheng Yin, and Xiaolei Zhao.

#### May 9, 2018. Gregorio Baldi: An open image theorem for points on Shimura varieties defined over function fields

Abstract: We will recall how to attach $$\ell$$-adic Galois representations to points of arbitrary Shimura varieties $${\rm Sh}_K(G,X)$$ defined over finitely generated fields of characteristic zero and why one inclusion of the Mumford-Tate conjecture holds in this setting. Tools from the theory of Shimura varieties can be used to study the image of such representations. In the case of function fields, when such points are not contained in a smaller Shimura subvariety and are without isotrivial component, this approach is especially fruitful. We show indeed that, for $$\ell$$ large enough, the image of the $$\ell$$-adic representation contains the $${\mathbb Z}_\ell$$-points coming from the simply connected cover of the derived subgroup of $$G$$.

#### April 25, 2018. Will Sawin: The cohomology of the moduli spaces of rational curves on hypersurfaces

Abstract: The Grothendieck-Lefschetz fixed point formula relates the cohomology of varieties over finite fields to their number of points. In many cases, we can use our understanding of the cohomology to prove bounds for the number of points. For the moduli spaces of rational curves on low-degree hypersurfaces, we have a good understanding of the point counts from analytic number theory, and we would like to translate that into cohomological information. This is not possible directly because the spaces can fail to be smooth and always fail to be proper. However, in joint work with Tim Browning, we adapted the classical analytic number theory techniques into a geometric argument to compute the high-degree cohomology groups.

#### April 18, 2018. Chenyang Xu: Volume and stability of singularities

Abstract: One guiding principle for the class of kawamata log terminal (klt) singularities is that it is the local analogue of Fano varieties. In this talk, I will discuss our work (joint with Chi Li) on establishing an algebraic stability theory, which is the analogue to the K-stability of Fano varieties, for a klt singularity. This is achieved by using Chi Li’s definition of normalised volumes on the ’non-archimedean link'. The conjectural picture can be considered as a purely local construction which algebrizes the metric tangent cone in complex geometry. As an application, we solve Donaldson-Sun’s conjecture.

#### April 11, 2018. Jean Fasel: Borel-Moore homology and Chow-Witt groups of singular varieties

Abstract: (joint work with Frédéric Déglise). In this talk, we present MW-motivic cohomology and its associated Borel-Moore homology. We will perform a few computations of these (co-)homology groups, and recover Chow-Witt groups of singular schemes as a special case. This shows that the latter are obtained via an explicit complex, covariantly functorial for proper morphisms (with an explicit push-forward morphism) and covariantly functorial for étale morphisms.

#### March 21, 2018. Maxim Mornev: Regulator theory for elliptic shtukas.

Abstract: Let $$X$$ be a smooth projective curve over a finite field $$\mathbb{F}_{\!q}$$. To an $$\ell$$-adic sheaf $$\mathcal{F}$$ on $$X$$ one can associate its $$L$$-function $$L(\mathcal{F},T)$$. Kato [1] discovered that the value $$L^*(\mathcal{F},1)$$ has a cohomological interpretation: multiplication by $$L^*(\mathcal{F},1)$$ on the one-dimensional $$\mathbb{Q}_\ell$$-vector space $$\det\nolimits_{\mathbb{Q}_\ell} \mathrm{R}\Gamma(X, \mathcal{F})$$ is a composition of certain natural operations on cohomology.

Roughly speaking, a shtuka on $$X$$ is the correct analog of an $$\ell$$-adic sheaf in the situation when $$\mathbb{Q}_\ell$$ is replaced by a finite extension of $$\mathbb{F}_{\!q}(\!(z)\!)$$. Shtukas also have a cohomology theory and $$L$$-functions so it is tempting to try Kato's approach for them. This line of research was successfully pursued by V.~Lafforgue [2]. His theory is quite involved and differs considerably from the one of Kato in the $$\ell$$-adic setting.

In this talk I would like to discuss one more such theory. It is somewhat parallel to Lafforgue's but applies to a different class of shtukas, the elliptic shtukas. One can construct a natural map on cohomology of elliptic shtukas called the regulator. I will explain how the regulator can be used to give a simple formula for the values of shtuka-theoretic $$L$$-functions at $$T = 1$$.

[1] K. Kato. Lectures on the approach to Iwasawa theory for Hasse-Weil $$L$$-functions via $$B_{\textrm{dR}}$$. I. Arithmetic algebraic geometry (Trento, 1991), 50--163, Lecture Notes in Math. 1553, Springer, Berlin (1993)

[2] V. Lafforgue. Valeurs speciales des fonctions $$L$$ en caracteristique $$p$$. J. Number Theory 129 (2009), no. 10, 2600--2634.

#### March 14, 2018. Sofia Tirabassi: Derived categories of canonical covers in positive characteristic.

Abstract: We show that abelian varieties and K3 surfaces arising as canonical covers of bielliptic and Enriques surfaces do not admit any non-trivial Fourier-Mukai partners This is a joint work with K. Honigs and L. Lombardi

#### March 7, 2018. Celine Maistret: Parity of ranks of abelian surfaces.

Abstract: Let $$K$$ be a number field and $$A/K$$ an abelian surface. By the Mordell-Weil theorem, the group of $$K$$-rational points on $$A$$ is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the $$L$$-series determines the parity of the rank of $$A/K$$. Under suitable local constraints and finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces. We also prove analogous unconditional results for Selmer groups.

#### March 1, 2017. Xuanyu Pan: Automorphism and Cohomology of Varieties.

Abstract: I will give a survey of my works and joint works on the following fundamental problem "Does the automorphism of variety act faithfully on cohomology?" The answer is positive for cubic fourfolds and its Fano variety of lines, complete Intersections and cyclic coverings with a few exceptions. There are some applications to arithmetic and geometry. For instance, Javanpeykar and Loughran relate this question to the Lang- Vojta conjecture and the Shafarevich conjecture, and the symmetry of cubic fourfolds and their middle Picard numbers. The program also involves a proof of (crystalline) variational Hodge conjecture for some graph cycles.

#### February 22, 2017. Wushi Goldring: Geometry engendered by G-Zips: Shimura varieties and beyond.

Abstract: Moonen, Pink, Wedhorn and Ziegler initiated a theory of $$G$$-Zips, which is modelled on the de Rham cohomology of varieties in characteristic $$p>0$$ "with $$G$$-structure", where $$G$$ is a connected reductive $${\mathbb F}_p$$-group. Building on their work, when $$X$$ is a good reduction special fiber of a Hodge-type Shimura variety, it has been shown that there exists a smooth, surjective morphism from X to a quotient stack $$G{\mathrm{-Zip}}^{\mu}$$. When $$X$$ is of PEL type, the fibers of this morphism recover the Ekedahl-Oort stratification defined earlier in terms of flags by Moonen. It is commonly believed that much of the geometry of $$X$$ lies beyond the structure of $$\zeta$$. I will report on a project, initiated jointly with J.-S. Koskivirta and developed further in joint work with Koskivirta, B. Stroh and Y. Brunebarbe, which contests this common view in two stages: The first consists in showing that fundamental geometric properties of $$X$$ are explained purely by means of (and its generalizations). The second is that, while these geometric properties may appear to be special to Shimura varieties, the $$G$$-Zip viewpoint shows that they hold much more generally, for geometry engendered by $$G$$-Zips, i.e. any scheme $$Z$$ equipped with a morphism to $${\mathrm {GZip}}^{\mu}$$ satisfying some general scheme-theoretic properties. To illustrate our program concretely, I will describe results and conjectures regarding two basic geometric questions about $$X, Z$$: (i) Which automorphic vector bundles on $$X, Z$$ admit global sections? (ii) Which of these bundles are ample?

#### February 8, 2017. Peter Bruin: On elliptic curves with prescribed torsion over number fields.

Abstract: This talk is about Mordell--Weil groups, and in particular torsion subgroups, of elliptic curves over number fields of given degree. I will explain some results showing that prescribing a suitable torsion structure on an elliptic curve over a number field can force the curve to have unexpected properties. For example, if $$E$$ is an elliptic curve over a cubic number field $$K$$ such that $$E(K)$$ has full 2-torsion and a point of order 7, then $$E$$ arises by base change from an elliptic curve over $$\mathbf{Q}$$.

#### January 31, 2017. Mini-workshop Arithmetic and Moduli of K3 surfaces.

Lectures by Martin Orr (Imperial College, London), Tony Várilly-Alvarado (Rice U, Houston), Alexei Skorobogatov (Imperial College, London) and Bianca Viray (U of Washington, Seattle). See https://staff.fnwi.uva.nl/l.d.j.taelman/k3day.html

#### December 7, 2016. Bhargav Bhatt: Comparing crystalline and etale cohomology.

Abstract: For smooth proper varieties over $$p$$-adic fields with good reduction, Grothendieck's ideas on the mysterious functor predicted a comparison isomorphism relating the rational crystalline cohomology of the special fiber with the rational $$p$$-adic etale cohomology of the generic fibre. A precise version of this conjecture was formulated by Fontaine in the early days of $$p$$-adic Hodge theory, and has since been established by various mathematicians. In my talk, I will review this story, and then explain our current understanding of the torsion aspects of this comparison: the torsion on the crystalline side provides an upper bound for the torsion on the etale side, and the inequality can be strict. This talk is based on recent joint work with Matthew Morrow and Peter Scholze.

#### November 16, 2016. Ekaterina Amerik: Characteristic foliation on a smooth divisor in a holomorphic symplectic manifold.

Abstract: Let $$D$$ be a smooth hypersurface in a holomorphic symplectic manifold. The kernel of the restriction of the symplectic form to $$D$$ defines a foliation in curves called the characterisic foliation. Hwang and Vehweg proved in 2008 that if $$D$$ is of general type this foliation cannot be algebraic unless in the trivial case when $$X$$ is a surface and D is a curve. I shall explain a refinement of this result, joint with F. Campana: the characteristic foliation is algebraic if and only if $$D$$ is uniruled or a finite covering of $$X$$ is a product with a symplectic surface and $$D$$ comes from a curve on that surface. I shall also explain a recent joint work with L. Guseva, concerning the particular case of an irreducible holomorphic symplectic fourfold: we show that if Zariski closure of a general leaf is a surface, then $$X$$ is a lagrangian fibration and $$D$$ is the inverse image of a curve on its base.

See here.

#### October 26, 2016. Jean-Stefan Koskivirta: Generalized Hasse invariants and some applications.

Abstract: This talk is a report on a paper with Wushi Goldring. If $$A$$ is an abelian variety over a scheme $$S$$ of characteristic $$p$$, the isomorphism class of the $$p$$-torsion gives rise to a stratification on $$S$$. When it is nonempty, the ordinary stratum is open and the classical Hasse invariant is a section of the $$p-1$$ power of the Hodge bundle which vanishes exactly on its complement. In this talk, we will explain a group-theoretical construction of generalized Hasse invariants based on the stack of $$G$$-zips introduced by Pink, Wedhorn, Ziegler Moonen. When $$S$$ is the good reduction special fiber of a Shimura variety of Hodge-type, we show that the Ekedahl-Oort stratification is principally pure. We apply Hasse invariants to attach Galois representations to certain automorphic representations whose archimedean part is a limit of discrete series, and to study systems of Hecke-eigenvalues that appear in coherent cohomology.

#### October 19, 2016. Javier Fresan: Exponential motives.

Abstract: Exponential periods form a class of complex numbers containing the special values of the gamma and the Bessel functions, the Euler constant and other interesting numbers which are not expected to be periods in the usual sense of algebraic geometry. However, one can interpret them as coefficients of the comparison isomorphism between two cohomology theories associated to pairs consisting of an algebraic variety and a regular function: the de Rham cohomology of a connection with irregular singularities and the so-called “rapid decay” cohomology. Following ideas of Kontsevich and Nori, I will explain how this point of view allows one to construct a Tannakian category of exponential motives and to produce Galois groups which conjecturally govern all algebraic relations among these numbers. I will give many examples and show how some classical results from transcendence theory can be reinterpreted in this way. This is a joint work with Peter Jossen.