Aravind Asok: Vector bundles and A1-homotopy theory
This will be a survey talk about recent applications of A1-homotopy theory to the study of vector bundles on algebraic varieties. In particular, we will discuss how techniques from obstruction theory can be used to study questions such as: when does an algebraic vector bundle split off a trivial rank 1 summand? Time permitting, we will show how these techniques can be used to produce existence results for vector bundles of low rank (e.g., rank 2) on smooth affine varieties of large dimension.
Asher Auel: Brill-Noether special cubic fourfolds
Probably the first class of smooth cubic fourfolds known to be rational are those containing two skew planes. Such cubic fourfolds also turn out to be distinguished by the property of having an associated K3 surface of degree 14 that is Brill-Noether special in the sense of Lazarsfeld and Mukai. More generally, I will discuss how the geometry of cubic fourfolds is reflected in the Brill-Noether properties of special divisors on curves in their associated K3 surfaces. One application is to the rationality problem for cubic four folds.
Joseph Ayoub: Conjectures on motives and algebraic cycles
I'll start with a quick review of Voevodsky motives. I'll then describe a few conjectures about motives and explain their consequences on algebraic cycles. Special emphasis will be placed on the interplay between abstract properties of motives and concrete consequences on algebraic cycles.
Rebecca Bellovin: Local \varepsilon-isomorphisms in families
Given a representation of Gal_{Q_p} with coefficients in a p-adically complete local ring R, Fukaya and Kato have conjectured the existence of a canonical trivialization of the determinant of a certain cohomology complex. When R=Z_p and the representation is a lattice in a de Rham representation, this trivialization should be related to the \varepsilon-factor of the corresponding Weil--Deligne representation. Such a trivialization has been constructed for certain crystalline Galois representations, by the work of a number of authors. I will explain how to extend these trivializations to certain families of crystalline Galois representations. This is joint work with Otmar Venjakob.
Laurent Berger: Iterated extensions and relative Lubin-Tate groups
An important construction in p-adic Hodge theory is the 'field of norms' corresponding to an infinite extension K_infty/K. For the cyclotomic extension, it is possible to lift the field of norms to characteristic zero, and we can ask for which other extensions K_infty/K this is possible. The goal of this talk is to explain this question and discuss some partial answers. This involves p-adic dynamical systems, Coleman power series and relative Lubin-Tate groups.
Nicolas Bergeron: Special cycles in ball quotients and moduli spaces of quasi-polarized K3 surfaces
Abstract : In a joint work with J. Millson and C. Moeglin we verify the Hodge conjecture for n-ball quotients in every degree away from the neighborhood ]n/3,2n/3[ of the middle degree. More recently with the further help of Z. Li we have proved the Noether-Lefschetz conjecture of Maulik and Pandharipande on moduli spaces of quasi-polarized K3 surfaces. I will review these results and explain how they are related. Both proofs make heavy use of special cycles. Eventually the method is purely automorphic, making use of the recent "endoscopic classification of automorphic representations" by J. Arthur, and can be extended beyond algebraic geometry e.g. to real hyperbolic manifolds.
Bhargav Bhatt: Perfect algebraic geometry
I will describe some surprisingly nice features of algebraic geometry in the world of perfect schemes of characteristic p (i.e, schemes where Frobenius is an isomorphism). As an application, we will see why the p-adic analog of the affine Grassmannian is ind-projective (which improves on recent work of X. Zhu). This is based on joint work with P. Scholze.
Ana Maria Botero: On the integrability of b-divisors on toric varieties
We introduce toric b-divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions, toric b-divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric b-divisor corresponds to the number of lattice points in this convex set and we give a Hilbert-Samuel type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. We further investigate the question of extending these results to arbitrary toroidal varieties. Examples in which such b-divisors naturally appear are invariant metrics on line bundles over toroidal compactifications of mixed Shimura varieties. Indeed, the singularity type which the metric acquires along the boundary can be encoded using toroidal b-divisors.
Patrick Brosnan: Nilpotent orbits in Hodge Theory
Nilpotent orbits are linear algebraic objects which approximate degenerations of Hodge structure. They are very important for understanding the limits and asymptotics of variations of mixed Hodge structures. However, although nilpotent orbits are rather simple to define, it is not clear how to classify them, and it is somewhat challenging to write down several variable nilpotent orbits with given properties. I will discuss joint work with Colleen Robles and Gregory Pearlstein whose aim is to give an algorithm for classifying several variable nilpotent orbits.
Tim Browning: Counting failures of weak approximation
Given a family of varieties over a number field, we give conditions under which a positive proportion of the varieties in the family are everywhere locally soluble, together with conditions which guarantee that there is a Brauer-Manin obstruction to weak approximation on 100% of these fibres. This is joint work with Martin Bright and Dan Loughran.
Anna Cadoret: Specialization of adelic representations of étale fundamental groups of schemes
I will discuss the issue of specializing adelic representations of the étale fundamental group of schemes, especially those attached to Abelian schemes and Shimura varieties. In particular, I will give applications to integral and adelic variants of the Mumford-Tate conjecture.
Bryden Cais: On F-crystalline representations
For a p-adic field K, the theory of Kisin modules provides a powerful classification of Galois stable lattices in Q_p-valued representations of G_K. Throughout Kisin's theory, the non-Galois "Kummer" extension K_{\infty}/K obtained by adjoining to K a compatible system of p-power roots of a uniformizer plays a central role. We describe a generalization of Kisin's theory allowing arbitrary (p-adic) coefficient fields and more general Frobenius lifts in which the role of K_{infty} is replaced by a general iterated extension. As an application, we describe a class of infinite and totally wildly ramified extensions L of K for which restriction of G_K-representations to G_L-representations is fully-faithful on F-crystalline representations. This is joint work with TONG LIU.
Pierre Colmez: Locally analytic representations de ${\rm GL}_2({\bf Q}_p)$ and coverings of Drinfeld's upper half plane.
Dospinescu and Lebras recently proved a conjecture of Breuil and Strauch describing the de Rham complex of coverings of Drinfeld's upper half plane in terms of the $p$-adic local Langlands correspondence for ${\rm GL}_2({\bf Q}_p)$. I will survey some of the ingredients that come into the proof, and related results about locally analytic representations of ${\rm GL}_2({\bf Q}_p)$.
Charlotte Chan: p-adic Deligne-Lusztig constructions and the local Langlands correspondence
The representation theory of SL2(Fq) can be studied by studying the geometry of the Drinfeld curve. This is a special case of Deligne-Lusztig theory, which gives a beautiful geometric construction of the irreducible representations of finite reductive groups. I will discuss recent progress in studying Lusztig's conjectural construction of a p-adic analogue of this story. More precisely, the homology of the p-adic Deligne-Lusztig (ind-)scheme X associated to a division algebra gives rise to supercuspidal representations of arbitrary depth and furthermore gives a geometric realization of the local Langlands and Jacquet-Langlands correspondences. We also give a description of X as an "infinite level" mirabolic analogue of affine Deligne-Lusztig varieties for GLn.
This talk is based on arXiv:1406.6122 and arXiv:1505.07185.
Ishai Dan-Cohen: Towards Chabauty-Kim loci for the polylogarithmic quotient over an arbitrary number field
Let $K$ be a number field and let $S$ be an open subscheme of $\mathrm{Spec} \; \mathcal{O}_K$. Minhyong Kim has developed a method for bounding the set of $S$-valued points on a hyperbolic curve $X$ over $S$; his method opens a new avenue in the quest for an \textit{effective Mordell}. But although Kim's approach has lead to the construction of explicit bounds in special cases, the problem of realizing the potential effectivity of his methods remains a difficult and beautiful open problem. In the case of the thrice punctured line, this problem may be approached via the methods of mixed Tate motives. Using these methods we are able to describe an algorithm; its output upon halting is provably the set of integral points, while its halting depends on conjectures. This ongoing project builds on joint work with Stefan Wewers, and relies on another ongoing joint project with Andr\'e Chatzistamatiou.
Johan de Jong: Local Picard groups
I will explain joint work with Bhargav Bhatt on Lefschetz type theorems for local Picard groups.
Carl Erickson: Singularities along the Eisenstein locus of the ordinary eigencurve
Hida theory says that any ordinary eigenform lies in a family of ordinary eigenforms with p-adically varying coefficients and weight. Sometimes, these families collide. We will discuss joint work with Preston Wake in which we investigate the collisions between the Eisenstein family and cuspidal families, showing that if we assume a mild condition on class groups, the collision is a plane singularity. We also determine when it is a simple normal crossing and draw consequences in Iwasawa theory, namely, new cases of Sharifi's conjecture.
Hélène Esnault: Some Fundamental Groups in Arithmetic Geometry
Content: 1) Deligne’s conjectures: ℓ-adic theory
2) Deligne’s conjectures: crystalline theory
3) Maltsev-Grothendieck theorem; Gieseker conjecture; de Jong conjecture
4) Relative 0-cycles
1) We review Deligne’s Weil II ℓ-adic conjecture 1.2.10, its analogous finiteness statement in Hodge theory (Deligne ’87), on what (Lefschetz type) theorems the solution the Weill II conjecture relies, for the existence of weights (how to go from Lafforgue’s theorem in dimension 1 to higher dimension), of ℓ'-companions (solved by Drinfeld) and for finiteness (solved by Deligne).
2) We review Deligne’s Weil II p-adic conjecture 1.2.10, mention Abe’s recent analogue of Lafforgue theorem, and explain what Lefschetz theorems are missing to go further, even to show the existence of weights.
3) We present a ‘conservativity’ program, the origins of which are rooted in Maltsev-Grothendieck theorem: in general one may ask what corresponds, over a perfect field of characteristic p>0, to complex local systems. We discuss the category of crystals we treat, mention Gieseker conjecture and its positive solution for crystals on the infinitesimal site (Esnault-Mehta), de Jong conjecture and its partial positive solution for isocrystals coming from crystals in the crystalline site (Esnault-Shiho).
4) We pose the problem of a motivic version of the Grothendieck ℓ-adic base change theorem and give a partial answer for relative 0-cycles (Kerz-Esnault-Wittenberg).
Laurent Fargues: From local class field theory to the curve and vice versa
I will speak about results contained in my article "G-torseurs en théorie de Hodge p-adique" linked to local class field theory. I will in particular explain the computation of the Brauer group of the curve and why its fundamental class is the one from local class field theory.
Tom Fisher: On families of n-congruent elliptic curves
Elliptic curves E and E' are said to be n-congruent if their n-torsion subgroups are isomorphic as Galois modules. The elliptic curves n-congruent to a given elliptic curve are parametrised by (the non-cuspidal points of) certain twists of the modular curve X(n). I will discuss methods for computing equations for these curves, and also for the surfaces that parametrise pairs of n-congruent elliptic curves.
Richard Hain: Mixed motives associated to classical modular forms
This talk will survey recent work whose goal is to understand mixed motives associated to classical modular forms and their relation to the study of mixed Tate motives. The talk will survey work of Francis Brown on ``mixed modular values'' and joint work with Makoto Matsumoto on ``universal mixed elliptic motives''.
Roger Heath-Brown: Rational Points on Intersections of Quadrics
This talk will survey what is known about the Hasse principle and weak approximation for intersections of quadrics in higher dimension. This will lead in to a progress report about ongoing work on intersections of three quadrics.
Eugen Hellmann: Degenerations of trianguline representations
Trianguline representations are a class of representations of the Galois group of a local p-adic field defined by Colmez. By definition the (phi,Gamma)-module over the Robba ring associated to a trianguline representation admits a complete flag stable under phi and Gamma. These representations naturally vary in rigid analytic families and one can define some kind of universal family of trianguline representations. We describe some results about the local geometry of this space at points where the complete (phi,Gamma)-stable flag degenerates. The study of the geometry of the space of trianguline representations is motivated by questions about p-adic automorphic forms of finite slope. In this talk we will focus on the Galois theoretic aspects. This is joint work with C. Breuil and B. Schraen.
David Holmes: A Néron model of the universal Jacobian
Néron models for 1-parameter families of abelian varieties were defined and constructed by André Néron in the 1960’s, and provide perhaps the best possible model for the degenerating family. If we naively extend Néron’s original definition to degenerating families of abelian varieties over higher-dimensional bases then we will see that such objects fail to exist, even if we allow blowups or alterations of the base space of the family. In the case of the Jacobian of the universal curve we will describe the minimal base-change required in order that a Néron model exist, giving a possible answer to the shape of the `best possible degeneration’. It time allows, we will relate our constructions to moduli of enriched structures on semistable curves.
Yuichiro Hoshi: Classical p-adic Teichmuller theory in characteristic three
A nilpotent admissible indigenous bundle, as well as a nilpotent ordinary indigenous bundle, is a certain projective line bundle equipped with a connection over a hyperbolic curve in positive characteristic and plays a central role in the positive characteristic portion of the classical p-adic Teichmuller theory established by Shinichi Mochizuki. In this talk, I will discuss a characterization, via the Cartier operators, of the supersingular divisor (i.e., the divisor determined by the Hasse invariant) of a nilpotent admissible/ordinary indigenous bundle in characteristic three. I will also explain that this characterization yields respective negative, partial positive answers to two basic questions in p-adic Teichmuller theory.
Sean Howe: p-adic Modular Forms and the Hodge-Tate Period Map
We explain how the Hodge-Tate period map and the pro-etale site can be used to provide a clean construction of classical spaces of overconvergent modular forms of arbitrary p-adic weight, starting from infinite level and using reduction of structure group on the flag variety. We also explain how this interpretation leads us to new and mysterious spaces of overconvergent modular forms supported in neighborhoods of the locus of supersingular curves whose p-divisible groups have extra endomorphisms. In the talk we will focus on the the modular curve, but the techniques generalize to Shimura varietes of Hodge Type.
Eric Katz: Uniform bounds on rational and torsion points on curves
I will discuss the recent proof with Joseph Rabinoff and David Zureick-Brown that there is a uniform bound for the number of rational points on genus g curves of Mordell-Weill rank at most g-3, extending a result of Stoll on hyperelliptic curves. Our work also gives unconditional bounds on the number of rational torsion points and bounds on the number of geometric torsion points on curves with very degenerate reduction type. I will outline the Chabauty-Coleman method for bounding the number of rational points on a curve of low Mordell-Weil rank and discuss the challenges to making the bound uniform. These challenges involve p-adic integration and Newton polygon estimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.
Kiran Kedlaya: (phi, Gamma)-modules on analytic, adic, and perfectoid spaces
Until recently, p-adic Hodge theory tended to break down into two related but very distinct subfields: the study of algebraic varieties and comparison isomorphisms, and the study of p-adic Galois representations and their associated structures. In this talk, I will try to indicate how these two subfields are coming together, by explaining how the study of p-adic Galois representations is really the case "over a point" of a more general theory of p-adic etale local systems. In particular, the linear algebraic data used to describe Galois representations, such as (phi, Gamma)-modules, admit a natural generalization described in the language of perfectoid spaces, and the comparison isomorphism for analytic spaces described in Scholze's plenary lectures amounts to a statement about higher direct images in the category of geometric (phi, Gamma)-modules. Based on ongoing joint work with Ruochuan Liu (on which Liu will also speak in this session).
Moritz Kerz: K-theory of non-Archimedean algebras and spaces
In joint work with S. Saito and G. Tamme we generalize Karoubi-Villamayor K-theory and homotopy K-theory to non-Archimedean analytic algebras and spaces. This analytic homotopy K-theory of spaces satisfies the analog of the Eilenberg-Steenrod axioms except the dimension axiom and it can be related to the continuous K-theory of formal models.
Lars Kindler: Ramification theory for D-modules in positive characteristic
On a smooth variety in positive characteristic, a vector bundle carrying an action of the sheaf of differential operators is called a stratified bundle. The notion of regular singularity for such objects is fairly well understood. For stratified bundles on a curve, I will sketch the beginning of a higher ramification theory of stratified bundles, analogous to higher ramification.
Mark Kisin: Integral models of Shimura varieties
Shimura varieties may be viewed as moduli spaces, for certain Hodge structures, with example being given by families of abelian varieties equipped with a collection of Hodge cycles. Despite their apparently transcendental definition, these varieties have a rich arithmetic theory.
I will discuss the construction of good integral models for these varieties, which involves p-adic Hodge theory. Their mod p points admit an explicit description, which is in some sense analogous to the description of their complex points, via the complex uniformization.
Bruno Klingher: An André-Oort conjecture for variations of Hodge structures
Given a smooth family X of complex algebraic varieties (or more generally an admissible variation of Hodge structures) over a smooth quasi-projective base S, can we describe the locus in S of fibers "with complex multiplication"? In this talk I will describe a conjecture generalizing the classical Andre-Oort conjecture (corresponding essentially to the case where X is a family of Abelian varieties) and partial results towards it.
Daniel Krashen: Field patching and higher dimensional local-global principles
Abstract: Local-global principles have played a prominent role in understanding the interplay between the structure of algebraic objects and the arithmetic of fields. The problem of understanding the behavior of quadratic forms and central simple algebras over finitely generated fields remains a central issue with connections to a many other problems in mathematics. In this talk I will describe some progress and open problems in the technique of field patching, and the development of "higher dimensional'' local-global principles to approach these and related algebraic structures.
Ching-Jui Lai: Surfaces with maximal canonical degree
It was shown by A. Beauville that if the canonical map φ of a complex smooth projective surface M is generically finite, then deg(φ) ≤ 36. A surface with maximal canonical degree 36 was only recently obtained by S.-K. Yeung, arising from a very special fake projective plane. In this talk, I will explain a joint work with S.-K. Yeung where we construct many more examples of surfaces with deg(φ)=36 arising from normal coverings of all fake projective planes. Canonical degree in dimension three will also be discussed.
Max Lieblich: Twisted sheaves, ten years later
At the last Summer Institute in Algebraic Geometry, twisted sheaves had just found their way into arithmetic algebraic geometry. I will discuss how they have blossomed since then, touching on their usefulness in abstract algebra, local-to-global principles, geometric lifting problems, non-commutative algebraic geometry, and the geometry of rational curves on moduli spaces.
Ruochuan Liu: Finiteness of cohomology of relative (phi,Gamma)-modules.
We will show finiteness of the higher direct images of geometric (phi, Gamma)-modules for proper smooth morphisms of smooth rigid analytic varieties. As a consequence, for proper smooth rigid analytic varieties over finite extensions of Qp, the proetale cohomology groups of Qp-local systems are all finite dimensional Qp-vector spaces. This is joint work with Kedlaya.
Melanie Matchett Wood: Heuristics for boundedness of ranks of elliptic curves
The set of rational points on an elliptic curve E over Q has the structure of an abelian group, and in 1922 Mordell proved that this group is finitely generated. We present heuristics that suggest that there is a uniform upper bound on its rank as E varies over all elliptic curves over Q. The heuristics model ranks and Tate-Shafarevich groups of elliptic curves by coranks and cokernels of skew-symmetric matrices with integer coefficients. This is joint work with Jennifer Park, Bjorn Poonen, and John Voight.
Swarnava Mukhopadhyay: Strange Duality of Conformal Blocks and Nef divisors on \bar-M_{0,n}
Conformal blocks are vector bundles on moduli space of curves with marked points that arise naturally in rational conformal field theory. They also give rise to a very interesting family of nef divisors and hence relate to questions on nef cone of moduli space of genus zero curves with n-marked points. Strange duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this talk, we discuss relations among conformal blocks divisors that arise from strange duality.
Andrew Niles: The Picard groups of the stacks Y_0(2) and Y_0(3)
We compute the Picard group of the stack of elliptic curves equipped with a cyclic subgroup of order 2, and of the stack of elliptic curves equipped with a cyclic subgroup of order 3, over any base scheme on which 6 is invertible. This generalizes a result of Fulton-Olsson, who computed the Picard group of the stack of elliptic curves (with no level structure) over a wide variety of base schemes.
Wieslawa Niziol: Syntomic complexes and $p$-adic nearby cycles.
For a semistable scheme over a mixed characteristic local ring I will present a proof of a comparison isomorphism, up to some universal constants, between truncated sheaves of $p$-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. I will also explain how to combine this local comparison isomorphism with the theory of finite dimensional Banach Spaces and finitness of \'etale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction. This is a joint work with Pierre Colmez.
Andrew Obus: A Generalization of the Oort Conjecture
We generalize the Oort conjecture on lifting curves with cyclic groups of automorphisms from characteristic p to characteristic 0 to the case where the automorphism group has a cyclic p-Sylow subgroup. We reduce the conjecture to a pure characteristic p statement and prove it in many cases. In particular, we show that D_9 is a so-called "local Oort group."
Emmanuel Peyre: The upgraded version of Batyrev-Manin program
From the point of view of Batyrev-Manin program about rational points of bounded height, accumulating subspaces hinder the equidistribution of points. Slopes à la Bost enable one to select points the distribution of which ought to be more uniform. This leads to a new version of empiric formulae which is compatible with the example of Batyrev and Tschinkel.
Marta Pieropan: Generalized Cox rings over non closed fields
The relations between Cox rings and universal torsors for varieties over algebraically closed fields of characteristic 0 are well known. See, for example, the book "Cox rings" by Arzhantsev, Derenthal, Hausen and Laface, 2015. Universal torsors are certain torsors under quasi-tori that have been introduced by Colliot-Thélène and Sansuc in the 1970s for varieties over arbitrary fields. In joint work with U. Derenthal, we introduce a notion of generalized Cox rings associated to torsors under quasi-tori that extends the usual notion of Picard graded Cox rings. We define the generalized Cox rings from an axiomatic point of view, we discuss their classification and their existence over nonclosed fields, and we present an arithmetic application.
Alena Pirutka: On stable rationality
A variety X is stably rational if a product of X and some projective space is rational. There exists examples of stably rational non rational complex varieties. In this talk we will discuss recent series of examples of varieties, which are not stably rational, in particular some quartic double solids (C. Voisin), quartic threefolds (a joint work with Colliot-Thélène), some hypersurfaces (Totaro) and others.
Jonathan Pottharst: On the parity conjecture in p-adic analytic families
Work of Nekovar shows how the validity of the parity conjecture for a p-adic Galois representation of a number field is often constant as the representation varies an in irreducible p-adic analytic family, subject to certain symplectic self-duality and "Panchishkin" conditions. We explain joint work with Xiao that pushes these techniques towards generality.
Mohamed Saidi: On the Grothendieck anabelian section conjecture over finitely generated fields
In my talk I will present new results on the Grothendieck anabelian section conjecture for hyperbolic curves, as well as its birational version, over finitely generated fields.
Shuji Saito: Motives with modulus
Recently several attempts have been made to introduce theory of motivic cohomology with modulus, which motivate a quest for motives with modulus. In this talk I report a work in progress with B. Kahn and T. Yamazaki on an attempt to extend Voevodsky's theory of motives to motives with modulus. Full Abstract
Takeshi Saito: The characteristic cycle and the singular support of an étale sheaf
We define the characteristic cycle of an étale sheaf on a smooth variety of arbitrary dimension in positive characteristic using the singular support, constructed by Beilinson recently. The characteristic cycle satisfies a Milnor formula for vanishing cycles and an index formula for the Euler-Poincaré characteristic.
Will Sawin: Applications of Algebraic Geometry to Analytic Number Theory
At least since Weil proved the Riemann hypothesis for algebraic curves using the Hodge index theorem there have been applications of algebraic geometry to analytic number theory. Recently some new ways have been found to use etale cohomology to solve difficult problems from analytic number theory, including improving recent results of Zhang and Maynard on bounded tuples of prime numbers. I will explain a few of them.
Tomer Schlank: Stable obstruction to degree one zero cycles.
The Brauer-Manin obstruction is probably the best known obstruction to the existence of points on an algebraic variety. The BM obstruction can also be used to obstruct the existence of zero cycles (Galois invariant formal sums of geometric points). For rational points stronger obstruction exists. In 99' Skorobogatov defined the more refined etale-Brauer-Manin obstruction, which is a finer obstruction to the existence of such points. However, this obstruction cannot be applied to zero cycles . The theory of e'tale homotopy gives us a way to understand this fact. The difference between BM and e'tale-BM lies in the difference between homotopy and homology, and it is homology's abelian nature that allows to extend the obstruction. This suggest that an intermediate obstruction should exist in stable e'tale homotopy. We'll present such an obstruction and explain how to compute it. This is a work in progress with V. Stojanoska.
Peter Scholze: p-adic Hodge theory and q-de Rham cohomology
In the first two talks, I will explain the proof of the Hodge-Tate decomposition for the cohomology of proper smooth rigid spaces, giving the necessary background on perfectoid spaces. In the final talk, I will explain the emerging formalism of q-de Rham cohomology, which is a cohomology theory that knows about (integral) p-adic Hodge theory for all p at once. For schemes over R (an etale Z-algebra), it has coefficients R[[q-1]], and is a q-deformation of de Rham cohomology based on stipulating that the (q-)derivative of X^n is not n X^{n-1}, but (q^{n-1}+...+q+1) X^{n-1}.
Stefan Schreieder: The construction problem for Hodge numbers
What are the possible Hodge numbers of a smooth complex projective variety? We construct enough varieties to show that many of the Hodge numbers can take all possible values satisfying the constraints given by Hodge theory. For example, the k-th cohomology group of a smooth complex projective variety can take arbitrary Hodge numbers if k is odd; the same result holds for even k as long as the given middle Hodge number is larger than some quadratic bound in k.
Romyar Sharifi: Modular symbols and arithmetic
I will describe an explicit conjectural relationship between modular symbols in the homology of modular curves modulo Eisenstein ideals and products of cyclotomic units in the cohomology of cyclotomic integer rings. I intend to survey results and work-in-progress on this conjecture and its refinements, along with the development of higher-dimensional analogues, in particular over function fields. Much of this is joint work with Takako Fukaya and Kazuya Kato..
Sug Woo Shin: From Langlands-Rapoport conjecture to cohomology of Shimura varieties
Kisin proved the Langlands-Rapoport conjecture for Shimura varieties of abelian type (under a very mild condition). The conjecture describes the special fibers of Shimura varieties at primes of good reduction with natural group actions. So it should in principle lead to the description of the cohomology with such actions, which is one of the central problems in the Langlands program. I will explain how this is done based on work of Langlands-Rapoport, Kottiwtz and Milne as well as my work in progress with Mark Kisin and Yihang Zhu.
Alexei Skorobogatov: Variation of the Selmer group of quadratic twists and the Hasse principle for Kummer varieties
This is a joint work with Yonatan Harpaz. We show, developing some ideas of Swinnerton-Dyer, Rubin and Mazur, that the finiteness of the relevant Tate-Shafarevich groups implies the Hasse principle for certain Kummer surfaces and their higher-dimensional analogues.
Padmavathi Srinivasan: Conductors and discriminants for a class of hyperelliptic curves
Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the naive minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.
Peter Stiller: Aspects of Algebraic Geometry in Computer Vision
This talk will focus on a number of problems in computer vision where techniques from algebraic geometry can be applied to good effect. This will include the use of invariants and the geometry of Grassmannians in object recognition and 3D reconstruction from multiple images, calculation of essential matrices using Grobner basis techniques (and problems with this method), techniques for navigation, and feature extraction from point clouds using GPCA, to name a few.
Junecue Suh: New vanishing theorems for mixed Hodge modules and applications
We present two new vanishing theorems of Kawamata-Viehweg type, one for general mixed Hodge modules and the other for polarisable variations of Hodge structures (the latter yielding sharper results when applicable). The focus will be placed on applications.
Roberto Svaldi: Hyperbolicity for log pairs
A classical result in birational geometry, Mori's Cone Theorem, implies that if the canonical bundle of a variety X is not nef then X contains rational curves. This is the starting point of the so-called Minimal Model Program. In particular, hyperbolic varieties are positive from the point of view of birational geometry. Very much in the same vein, one could ask what happens for a quasi projective variety, Y. Using resolution of singularity, then one is lead to consider pairs (X, D) of a variety and a divisor, such that Y=X\setminus D. I will show how to obtain a theorem analogous to Mori's Cone Theorem in this context. Instead of rational complete curves, algebraic copies of the complex plane will make their appearance. I will also discuss an ampleness criterion for hyperbolic pairs.
Tamás Szamuely: Variations on a theme by Ribet
Some 35 years ago, Ken Ribet proved that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. In joint work with Damian Roessler, we show that Ribet's theorem is an instance of a general cohomological statement about smooth projective varieties over K. We also present a largely conjectural generalization to torsion cycles of higher codimension, as well as an analogue in positive characteristic.
Akia Tamagawa: Specialization of l-adic representations of arithmetic fundamental groups and applications to arithmetic of abelian varieties
In this talk, I will survey some recent developments concerning specialization of l-adic representations of arithmetic fundamental groups and their applications to arithmetic of abelian varieties. Among other things, I will discuss certain uniform open image theorems and their applications to uniform boundedness of l-primary torsion of abelian varieties, as well as to the modular tower conjecture in inverse Galois theory. (This first part is a joint work with Anna Cadoret and related to her talk.) I will also discuss certain specialization results for first cohomology groups and their applications to arithmetic of abelian varieties over finitely generated fields. In particular, I will introduce new notions of "discrete Selmer groups" and "discrete Shafarevich-Tate groups", which are finitely generated abelian groups. (This second part is a joint work with Mohamed Saidi and related to his talk.);
Yunqing Tang: Algebraic solutions of differential equations over the projective line minus three points
The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for almost all p, has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on the projective line minus three points. In this talk, I will focus on this case and introduce a p-adic convergence condition, which would hold if the p-curvature is defined and vanishes. Using the algebraicity criteria established by Andr\'e, Bost, and Chambert-Loir, I will prove a variant of this conjecture for the projective line minus three points, which asserts that if the equation satisfies the above convergence condition for all p, then its monodromy is trivial.
Sho Tanimoto: Towards a refinement of Manin's conjecture
Manin's conjecture predicts the generic distribution of rational points on Fano varieties, however, the strongest form of Manin's conjecture admits many counterexamples. Every failure of Manin's conjecture can be explained via compatibility of geometric invariants for the underlying variety and its subvarieties or its finite extensions. In this presentation, we will discuss our recent analysis of the geometry of invariants appearing in Manin's conjecture and introduce the notion of balanced line bundles which explains failures of Manin's conjecture. This is joint work with Brian Lehmann and Yuri Tschinkel. Manin's conjecture predicts the generic distribution of rational points on Fano varieties, however, the strongest form of Manin's conjecture admits many counterexamples. Every failure of Manin's conjecture can be explained via compatibility of geometric invariants for the underlying variety and its subvarieties or its finite extensions. In this presentation, we will discuss our recent analysis of the geometry of invariants appearing in Manin's conjecture and introduce the notion of balanced line bundles which explains failures of Manin's conjecture. This is joint work with Brian Lehmann and Yuri Tschinkel.
Nicola Tarasca: Loci of curves with subcanonical points in low genus
The locus of curves of genus 3 with a marked subcanonical point has two components: the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. In this talk, I will show how to compute the classes of the closures of these codimension-two loci in the moduli space of stable curves of genus 3 with a marked point. Similarly, I will present the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. Finally, I will discuss the geometric consequences of these computations. This is joint work with Dawei Chen.
Jean-Baptiste Teyssier: Nearby slopes. Applications and open problems
The goal of this talk is to introduce the audience to an enhancement in arbitrary dimension of the analogies between wild ramification for l-adic sheaves and irregularity for complex D-modules.
We will namely introduce a new notion of slope making sense in both contexts, with an emphasis on boundedness (still conjectural for l-adic sheaves, but now a theorem for complex D-modules). If time permits, an application of boundedness to periods of connections will be given.
Zhiyu Tian: Fundamental group of Fano varieties
I will talk about finiteness of the fundamental group of the smooth locus of some Fano 3-folds. This is work-in-progress with Chenyang Xu.
Adam Topaz: On Mod-ℓ Birational Anabelian Geometry
This talk will discuss some recent progress in birational anabelian geometry for function fields of higher-dimensional varieties, using “mod-ℓ” abelian-by-central Galois groups. We will also discuss how this progress leads to new results concerning nilpotent mod-ℓ quotients of étale fundamental groups.
Yuri Tschinkel: Almost abelian anabelian geometry
I will discuss some geometric constructions arising in the study of Galois groups of function fields of algebraic varieties over algebraically closed ground fields (joint with F. Bogomolov).
Takeshi Tsuji: On p-adic etale cohomology of perverse sheaves
I will talk about a generalization of the comparison theorem of the cohomologies of crystalline p-adic etale sheaves and filtered F-isocrystals by G. Faltings, to p-adic perverse sheaves for the stratification associated to a simple normal crossing divisor. A comparison map is constructed via descriptions of both cohomologies in terms of "nearby cycles" along each stratum; we use certain fibered topos whose fibers are topos associated to strata with some log structures.
Douglas Ulmer: Ranks of abelian varieties over function fields
We know a lot about ranks of abelian varieties in towers of rational function fields over finite fields (unbounded ranks are ubiquitous) and a little about about the corresponding question over complex function fields (generically there are no points). I'll explain why this is so and speculate about what might be true over other function fields.
Anthony Várilly-Alvarado: Kodaira Dimension of certain orthogonal modular varieties
In 2007 Gritsenko, Hulek, and Sankaran showed that the coarse moduli space K_d of K3 surfaces of degree d is of general type as soon as d is at least 124. They did this by constructing enough pluricanonical forms out of a single special modular form for a group associated to the space K_d, by leveraging work of Borcherds. In this talk I will review the circle of ideas behind their proof, and explain how one can use these ideas to show that certain moduli spaces parametrizing cubic fourfolds that contain more surfaces than one might expect are also of general type (joint with Sho Tanimoto). I will explain connections of this result to the arithmetic of K3 surfaces over number fields.
Kirsten Wickelgren: Splitting varieties for triple Massey products in Galois cohomology
The Brauer-Severi variety a x^2 + b y^2 = z^2 has a rational point if and only if the cup product of cohomology classes associated to a and b vanish. The cup product is the order-2 Massey product. Higher Massey products give further structure to Galois cohomology, and more generally, they measure information carried in a differential graded algebra which can be lost on passing to the associated cohomology ring. We show that the variety X defined by b x^2 = (y_1^2 - a y_2^2 + c y_3^2 - ac y_4^2)^2 - c(2 y_1 y_3 - 2 a y_2 y_4)^2 with x invertible is a splitting variety for the triple Massey product , and give examples of rings over which X has no points following Jochen Gärtner. We then show that this variety satisfies the Hasse principle. It follows that all triple Massey products over global fields vanish when they are defined. More generally, one can show this vanishing over any field of characteristic different from 2; Ján Mináč and Nguyễn Duy Tân, and independently Suresh Venapally, found an explicit rational point on X(a,b,c). Mináč and Tân have other nice results in this direction. The method could produce splitting varieties for higher order Massey products. This is joint work with Michael Hopkins.
Olivier Wittenberg: On the fibration method for zero-cycles and rational points
I will discuss recent progress (joint with Y. Harpaz) on the fibration method for proving the existence of zero-cycles of a given degree and rational points on varieties defined over number fields. The results are unconditional for zero-cycles, while in the context of rational points they depend on a conjecture on locally split values of polynomials which a recent work of Matthiesen establishes in the case of linear polynomials over the rationals.
Jesse Wolfson: Topology and arithmetic of resultants
Topological relationships between algebraic varieties should have algebraic explanations. In this talk, I will report on ongoing joint work with Benson Farb guided by this principle. We consider the interplay of point counts, singular and \'etale cohomology, and eigenvalues of the Frobenius for complements of resultants, including spaces of rational maps and moduli of marked, degree $d$ rational curves in $\Pb^n$. We deduce as special cases algebro-geometric and arithmetic analogues of topological computations of Segal, Cohen--Cohen--Mann--Milgram, Vassiliev and others.
Shou-Wu Zhang: Faltings heights and Zariski density of CM abelian varieties
The moduli of CM abelian varieties are the simplest objects in the category of Shimura varieties, and have been intensively studied related to Hilbert 's12th problem and the BSD conjecture. In this talk, I will discuss some recent progress on two different kinds of problems:
1) Colmez' conjecture on Faltings' heights of CM abelian varieties in terms of Artin L-functions, and
2) André-Oort's conjecture on Zariski density of CM points on Shimura varieties.
David Zureick-Brown: The canonical ring of a stacky curve
We generalize to stacks the classical theorem of Petri -- we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. This is joint work with John Voight.