# 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

#### November 8, 2017. Jeanine van Order: Bounds for Mordell-Weil ranks via bounds for Fourier coefficients of metaplectic forms.

Abstract: I will explain how to use bounds for Fourier coefficients of metaplectic forms to derive bounds for Mordell-Weil ranks of elliptic curves twisted by certain Artin characters. This makes it possible (for instance) to derive bounds for Mordell-Weil ranks of elliptic curves over ring class extensions of real quadratic fields which had not been accessible previously.

#### November 1, 2017. Jazmin Matz: Effective Plancherel equidistribution.

Abstract: Families of automorphic forms have seen a lot of interest over recent years, and the distribution of their local components is one of the most obvious properties to study. In this talk I want to explain current work on this distribution in families of spherical automorphic representations of split reductive groups. (Joint work with T. Finis.)

#### October 24, 2017. Raphaël Beuzart-Plessis: Recent progress on the Gan-Gross-Prasad and Ichino-Ikeda conjectures.

Abstract: The global Gan-Gross-Prasad conjectures relate the non-vanishing of central values of certain automorphic $$L$$-functions to the non-vanishing of certain explicit integrals of automorphic forms that are called 'automorphic periods'. They have been subsequently refined by Ichino-Ikeda and N.Harris into precise conjectural identities relating these two kind of invariants. These predictions also have local counterparts which concern certain branching laws in the representation theory of real or p-adic groups. This talk aims to give an introduction to this circle of ideas and to review recent results on the subject. If time permits, I will also describe the general stucture of the proof.

#### October 17, 2017. Rebecca Bellovin: Wach modules for formal families of Galois representations.

Abstract: If $$T$$ is a $${\mathbb Z}_p$$-lattice in a crystalline representation of the absolute Galois group of $${\mathbb Q}_p$$, then we can associate to it a certain finite free $${\mathbb Z}_p[[\pi]]$$ module $$N(T)$$ by work of Wach and Berger. I will explain how to extend this to lattices in certain families of crystalline representations, and I will discuss some properties of this construction. If time permits, I will discuss applications to the theory of epsilon-isomorphisms. This is joint work with Otmar Venjakob.

#### October 11, 2017. Renji Lyu: Two isogenous K3 surfaces are twisted derived equivalent.

Abstract: We will continue the discussion of Huybrechts' paper. In this talk, I first introduce some necessary generalization of notations in the twisted setting, e.g. twisted Chern characters. In the end, I will show that Huybrechts use some lattice theory and twisted derived Torelli theorem to obtain the twisted derived equivalence between two isogenous K3 surfaces.

#### October 11, 2017. Maarten Derickx: $$A$$-gonalities of curves and the existence of infinitely many points of degree $$d$$.

Abstract: The goal of this talk is to study when a curve $$C$$ over $$\mathbb Q$$ contains infinitely many points whose field of definition is of degree $$d$$ over $$\mathbb Q$$. Faltings famous Theorem on subvarieties of abelian varieties with infinitely many rational points implies the following: If a curve has infinitely many points of degree $$d$$, then either there exists a function of degree $$d$$ on $$C$$ or a certain subvariety of the jacobian called $$W_d^0$$ contains a translate of a positive rank abelian variety.

The degree of the smallest degree function on $$C$$ is called the gonality and it has the following two nice properties. The gonality can only decrease when reducing modulo primes $$p$$, and over finite fields it is computable in theory and often also in practice using linear algebra. However the question about containment of a translate of a positive rank abelian variety is more difficult to answer, especially in practice. To solve this problem I introduce a concept called the $$A$$-gonality for any abelian variety $$A$$ that is a quotient of $${\rm Jac}(C)$$, the Jacobian of $$C$$. Taking $$A = {\rm Jac}(C)$$ recovers the standard definition of gonality. The $$A$$-gonality can, like the standard definition of gonality, only decrease reducing modulo $$p$$ and is computable in terms of linear algebra as well. An application to degree $$9$$ points on the modular curve $$X_1(37)$$ will be given.

#### September 27, 2017. Renjie Lyun: Two isogenous K3 surfaces are twisted derived equivalent.

Abstract: Two complex projective K3 surfaces $$S$$ and $$S'$$ are isogenous if there exists a Hodge isometry $$H^2(S, \mathcal{Q}) \cong_\varphi H^2(S', \mathcal{Q})$$. Safarevic conjectures that the associated Hodge class $$[\varphi]\in H^{2,2}(S\times S', \mathcal{Q})$$is algebraic. This question was first solved by Mukai for K3 surfaces with Picard rank $$\rho\geq 11$$, and later Buskin completed the answer by differential geometry methods. Recently, Huybrechts gave a purely algebraic approach, which combines Mukai's and Buskin's ideas, to show that the twisted derived categories of isogenous K3 surfaces are derived equivalence in suitable sense. In this talk, we will briefly review Mukai's original work, and then explain Huybrechts' proof.

#### September 27, 2017. Wessel Bindt: the Chow motives of twisted derived equivalent K3 surfaces.

Abstract: As a lemma to his theorem on the Chow motives of isogenous K3 surfaces (see Renjie's talk), Huybrechts proves that, under certain conditions, K3 surfaces with equivalent derived categories of twisted coherent sheaves have isomorphic Chow motives. In this talk we give an introduction to Chow motives, as well as an introduction to Fourier-Mukai functors in the twisted setting. As an application, we prove Huybrechts' lemma.

#### September 20, 2017. Richard Griffon: A Brauer-Siegel theorem for Fermat surfaces over finite fields.

Abstract: The classical Brauer-Siegel theorem gives asymptotic upper and lower bounds on the product of the class-number times the regulator of units of a number field in terms of its discriminant. In this talk, I will describe an analogous result in a more geometric context. Namely, for a Fermat surface $$F$$ over a finite field, we consider the product of the order of its Brauer group (which is known to be finite) by the Gram determinant of a basis of its Néron-Severi group for the intersection form, and we describe the growth of this product in terms of the geometric genus of $$F$$ when the latter grows to infinity. As in the classical setting, the proof of the asymptotic estimate is rather analytic: it relies on obtaining asymptotic bounds on the size of the residue’’ of the zeta function of $$F$$ at its pole at $$s=1$$.

#### September 13, 2017. Christopher Lazda: A Néron-Ogg-Shafarevich criterion for K3 surfaces.

Abstact: The naive analogue of the Néron-Ogg-Shafarevich criterion is false for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields $$K$$, with unramified etale cohomology groups, but which do not admit good reduction over $$K$$. Assuming potential semi-stable reduction, we show how to correct this by proving that a K3 surface has good reduction if and only if $$H^2_{{\mathrm{ét}}}(X, {\mathbb Q}_\ell)$$ is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain “canonical reduction” of $$X$$. This is joint work with B. Chiarellotto and C. Liedtke.

#### September 5, 2017. Toshiyuki Katsura: Classification of Enriques surfaces with finite automorphism group in characteristic $$2$$.

Abstract: In characteristic $$0$$, S. Kondo classified Enriques surfaces with finite automorphism group into seven types. In this talk, we consider Enriques surfaces with finite automorphism group in characteristic $$2$$, and give the complete classification of them. We have 3 types for singular Enriques surfaces, 5 types for supersingular Enriques surfaces and 8 types for classical Enriques surfaces. We also determine the structure of automorphism groups. This is a joint work with S. Kondo and G. Martin.

#### June 20, 2017. Florian Sprung: Elliptic curves and the Birch and Swinnerton-Dyer conjecture.

Abstract: The Birch and Swinnerton-Dyer conjecture, one of the millenium problems, is a bridge between algebraic invariants of an elliptic curve and its (complex analytic) $$L$$-function. In the case of low ranks, we prove this conjecture up to the finitely many bad primes and the prime $$2$$, by proving the Iwasawa main conjecture in full generality. The ideas in the proof and formulation also lead us to new and mysterious phenomena. This talk assumes no specialized background in number theory.

#### June 15, 2017. Lance Gurney: Perfectoid algebras and tilting.

Abstract: I will give a self-contained lecture on (integral) perfectoid algebras, relative to any discrete valuation ring $$O$$ with finite residue field $$k$$, and the theory of tilting which states that the categories of perfectoid algebras over certain pairs $$A, A'$$ of perfectoid algebras are equivalent. I will also emphasise how this ties in with the philosophy that such an equivalence arises via a k-linear isomorphism $$A \to A'$$, where we view $$A$$ and $$A'$$ as $$k$$-algebras via the non-sense homomorphism $$k \to O$$. In particular, I will explain how in the case where $$O$$ is equal characteristic this non-sense statement is literally true.

#### May 31, 2017. Davide Lombardo: Reductions of points of infinite order on algebraic groups.

Abstract: Over 50 years ago, Hasse proved that the set of prime numbers dividing at least one integer of the form $$2n + 1$$ has natural density $$17/24$$. One can interpret this result as a statement about the rational point (of infinite order) $$2 \in {\mathbb G}_m({\mathbb Q})$$, and this point of view leads to the following general question: fix an algebraic group $$A$$ over a number field $$K$$, a point $$\alpha \in A(K)$$ of infinite order, and a prime $$\ell$$. For “how many” places $${\mathfrak p}$$ of $$K$$ does $$\ell$$ divide the order of $$(\alpha {\mathrm{ mod }} {\mathfrak p}$$? I will describe a general framework to tackle this and similar questions, and provide an answer when $$A$$ is the product of an abelian variety and a torus. If time permits I will also discuss an unexpected consequence of the result: the density of such places $${\mathfrak p}$$ is a rational number whose denominator is almost independent of the underlying algebraic group and of the choice of $$\alpha$$. This is joint work with Antonella Perucca (Universitat Regensburg).

#### May 17, 2017. Erick Knight: Patching and the p-adic Jacquet-Langlands correspondence.

Abstract: I will explain how to use the Taylor-Wiles-Kisin patching method to study the p-adic Jacquet-Langlands correspondence. I will show that the two constructions of the p-adic Jacquet-Langlands correspondence due to myself and Scholze agree, and also determine the locally algebraic vectors inside the representations of the quaternion algebra. This is joint work with Przemyslaw Chojecki.

#### May 10, 2017. Stefano Morra: Generalized Serre Conjectures, local-global compatibility and the p-adic local Langlands program.

Abstract: We discuss about the generalization of the weight part of Serre conjecture for $$\mathrm{GL}_n$$ and how these conjectures are related to the mod $$p$$ and $$p$$-adic local Langlands program. Let $$F/{\mathrm Q}$$ be a number field where $$p$$ is unramified and $$r \colon {\mathrm{Gal}}(\overline F/F) \to {\mathrm GL}_n({{\mathbb F}}_p)$$ be a continuous, totally odd Galois representation. When $$n = 2$$ and $$F = {\mathbb Q}$$, J-P. Serre conjectured that $$r$$ should indeed be modular, the minimal weights of the modular forms being predicted by the local behavior of r at the decomposition group at $$p$$. Since then, the progress in understanding the cohomology of arithmetic manifolds showed that the strong form of Serre’s modularity conjecture is indeed a description of the $${\mathrm{GL}}_n({\mathbb F}_p$$-action on Hecke isotypical parts in the cohomology of Shimura varieties with principal level at $$p$$, in terms of the inertial behavior of $$r$$ at places above $$p$$. This can be interpreted as an avatar of an hypothetical p-adic local Langlands correspondence, and its local-global compatibility in the cohomology of Shimura varieties. In this talk we will discuss about recent progress on the weight part of Serre’s modularity conjectures, and generalizations, for $$U(n)$$ arithmetic manifolds, using modularity lifting techniques, a deep understanding of deformation spaces beyond the Barsotti-Tate case, and combinatorial methods in modular representation theory. This is joint work with Dan Le, Viet-Bao Le Hung et Brandon Levin.

#### May 3, 2017. Christian Johansson: Mod $$p$$ cohomology of the Lubin--Tate tower for induced representations.

Abstract: Let $$F/{\mathbb Q}_p$$ be a finite extension. The Local Langlands correspondence was first constructed by Harris--Taylor by studying the $$\ell$$-adic cohomology of the Lubin--Tate tower for $$\ell$$ not equal to $$p$$. One could hope to construct the mod p Langlands correspondence by studying the cohomology when $$\ell = p$$, but this seems difficult. Scholze has proposed an alternative, which is a functor from admissible mod p representations of $${\mathrm GL}(n,F)$$ to etale sheaves on $${\mathbb P}^{n-1}$$, and the cohomology groups of these sheaves have good finiteness properties. So far, little progress has been made on their computation. I will discuss work of Judith Ludwig in the case $$n=2$$, $$F={\mathbb Q}_p$$, which shows that the cohomology groups vanish above degree $$n-1$$ for principal series representations. If there is time at the end, I will discuss work in progress between Ludwig and myself to generalize this to arbitrary $$n$$ and $$F$$.

#### April 26, 2017. John Welliaveetil: An application of model theory to non-Archimedean geometry.

Abstract: In 2010, Hrushovski and Loeser used techniques from Model theory to study the homotopy types of of the Berkovich analytification of algebraic varieties. They showed that the Berkovich analytic varieties involved admitted strong deformation retractions onto finite simplicial complexes embedded in the varieties. In this talk, we present an application of this striking result where in we study the extent to which the Berkovich analytification of an algebraic morphism deviates from being a topological fibration.

#### April 19, 2017. Jaclyn Lang: Images of Galois representations associated to Hida families.

Abstract: We explain a sense in which Galois representations associated to non-CM Hida families have large images. This is analogous to results of Ribet and Momose for Galois representations associated to classical modular forms. In particular, we show how extra twists of the Hida family decreases the size of the image.

#### April 12, 2017. Gabriel Dospinescu: $$p$$-adic étale cohomology and coverings of the Drinfeld upper half-plane.

Abstract: The $$\ell$$-adic (with $$\ell \neq p$$) cohomology of the coverings of the Drinfeld upper half-plane is known to realize the local Langlands and Jacquet-Langlands correspondences for $$\mathrm{GL}_2(F)$$ ($$F$$ a $$p$$-adic field), for a large class of representations. We will describe a $$p$$-adic analogue of this picture, in which the p-adic local Langlands correspondence for $$\mathrm {GL}_2({\mathbb Q}_p)$$ naturally appears.

Abstract: Let $$X\subset \mathbb P^{n+1}_{\mathbb C}$$ be a smooth cubic hypersurface. In view of rationality questions for cubic $$5$$-folds, we present some (sketch) of results on birational invariants of those hypersurfaces, related to $$2$$-cycles. The method we use is to reduce to problems on \$14-cycles on the variety of lines $$F(X)$$ of the cubic. We present also bounds for the group of $$1$$-cycles of $$F(X)$$ to be generated by lines.

#### 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.