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).

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

*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.

*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 \(p \gg N\).

*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.

*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 \(G\).

*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.

*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.

*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.

*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.

*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

*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.

*Abstract*: Zagier and Bengoechea introduced weight \(2k\) cusp forms $f_{k,D} for \(k>1\) associated to quadratic forms of discriminant \(D\). We investigate analogues of these functions for negative discriminants in weight \(2\) and higher level. Their Fourier coefficients are given by traces of singular moduli of Niebur-Poincaré series, such as the modular \(j\)-function. This allows us to compute their regularized inner products, which in the higher weight case have been related double traces over CM-values of higher Green's functions.

*Abstract*: I will discuss joint work with Kentaro Mitsui, in which we study the index of a smooth, proper variety over a strictly local field of residual characteristic p > 0 under the assumption that it admits a log smooth model over the ring of integers. We prove that the index is prime to p under a rather mild additional hypothesis (non-vanishing of the l-adic Euler characteristic). We also show that - perhaps surprisingly - this hypothesis is really necessary, since the result fails for curves of genus 1, and we fully classify torsors under elliptic curves with logarithmic good reduction.

*Abstract*: The quotients groups of the filtration of the Witt group of quadratic nonsingular forms over a given field are known to be isomorphic to either the Milnor K-groups \(k_n F\) (``the Milnor conjecture" proven by Voevodsky) or to the Kato-Milne cohomology groups \(H_2^n(F)\), depending on the characteristic of the base-field. Both groups are generated by symbols which correspond to n-fold Pfister forms. We say that \(k_n F\) (or \(H_2^n(F)\)) is triple linked if every three symbols share a common factor in \(k_{n-1} F\) (\(H_2^{n-1}(F)\)). We prove that if \(k_n(F)\) (\(H_2^n(F)\)) is triple linked then \(k_{n+1} F=0\) (\(H_2^{n+1}(F)=0\)).The statement in characteristic not 2 had been proven earlier by Karim Becher, but the new proof applies also to characteristic 2. This talk is based on a joint work with Andrew Dolphin and David Leep.

*Abstract*: Given an abelian scheme over a smooth curve over a number field, we can associate two height function: the fiberwise defined Neron-Tate height and a height function on the base curve. For any irreducible subvariety X of this abelian scheme, we prove that the Neron-Tate height of any point in an explicit Zariski open subset of X can be uniformly bounded from below by the height of its projection to the base curve. We use this height inequality to prove the Geometric Bogomolov Conjecture over characteristic 0. This is joint work with Philipp Habegger.

*Abstract*: For every vector bundle on a smooth projective surface, there is an associated tautological bundle on the Hilbert scheme of points on the surface. We study the behavior of these tautological bundles and their exterior powers under the McKay correspondence and use this to deduce formulas for homological invariants of these bundles.

*Abstract*: The generalized Franchetta conjecture as formulated by O’Grady is about algebraic cycles on the universal K3 surface. It is natural to consider a similar conjecture for algebraic cycles on universal families of hyperkaehler varieties. This has close ties to Beauville’s conjectural ``splitting property’’, and the Beauville-Voisin conjecture (stating that the Chow ring of a hyperkaehler variety has a certain subring injecting into cohomology). I will attempt to give an overview of these conjectures, and present some cases where they can be proven. This is joint work with Lie Fu and Charles Vial.

*Abstract*: The Mumford-Tate conjecture (for abelian varieties) roughly states that for an abelian variety over a finitely generated subfield of \({\mathbb C}\) the Hodge structure on its Betti cohomology and the Galois representation on its \(l\)-adic étale cohomology 'carry the same information'. The conjecture is far from proven, with the first unsolved case appearing in dimension 4 already. While the Mumford-Tate conjecture is usually stated in terms of an equality of algebraic groups over \({\mathbb Q}_l\), it is actually equivalent to an analogous statement of group schemes over \({\mathbb Z}_l\). In this talk, I will show how this integral approach may be helpful in proving cases of the Mumford-Tate conjecture.

*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.

*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.)

*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.

*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.

*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.

*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.

*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.

*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.

*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 \(s=1\).

*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.

*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.

*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.

*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.

*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).

*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.

*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.

*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 \(F\).

*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.

*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.

*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.

*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.

*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?

*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 \(\mathbf{Q}\).

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

*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.

*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.

*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.

*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.