Arithmetic and Algebraic Geometry Seminar

Lenny Taelman: Moduli of K3 surfaces

Arno Kret: K3 surfaces and orthogonal Shimura varieties

Arno Kret: The spin representation and the Kuga-Satake construction

Giovanni Rosso: Eigenvarieties for non-cuspidal Siegel modular forms (joint with Riccardo Brasca)

Since the seminal work of Serre and Swinnerton-Dyer, people have been interested in congruences between eigenforms. After recalling some foundational results of Hida and Coleman on p-adic family of modular forms, we shall explain how Andreatta, Iovita and Pilloni extended their construction to families of cuspidal Siegel forms and how their work can be generalised to non-cuspidal forms.

Ariyan Javanpeykar: Finiteness results for smooth hypersurfaces over finitely generated domains over Z

The Lang-Vojta conjecture states that a smooth quasi-projective variety is Brody hyperbolic if and only if it is arithmetically hyperbolic. In this talk we will first explain the statement of the Lang-Vojta conjecture and then, assuming the Lang-Vojta conjecture, prove that the set of smooth hypersurfaces over a finitely generated domain over Z of fixed dimension and fixed degree is finite. We will explain that this conditional finiteness result is a consequence of the fact that the complex algebraic stack of smooth hypersurfaces admits a representable immersive period map, under suitable assumptions on the degree and dimension. Finally, we will prove that, for all finitely generated domains A over Z, the set of smooth sextic surfaces in P^3_A is finite. This is joint work with Daniel Loughran.

Lance Gurney: Witt vectors, Frobenius lifts and canonical lifts in families

The standard theory of canonical lifts states that if A is an ordinary abelian variety over a perfect field k of characteristic p then there is a unique lift, the "canonical lift", of A to the Witt vectors of k with the property that the Frobenius endomorphism of A lifts as well. I will explain how this result can be reinterpreted in terms of a universal property satisfied by the Witt vectors and a certain structure possessed by the moduli space of ordinary abelian varieties (the key phrase here is "Frobenius lift"), and how this reinterpretation leads naturally to canonical lifts for families of ordinary abelian varieties over any scheme on which p is nilpotent.

Lenny Taelman: Exterior powers, polynomial functors, and universal exact sequences

The Grothendieck group of vector bundles on a manifold or variety is not just a group, but has extra structure: it comes with a product (induced by tensor product) and with operations \lambda^n (induced by n-th exterior power). These operations satisfy a number of axioms, and make the Grothendieck group into what is called a "lambda ring". In this talk, I'll compute the Grothendieck group of finite degree polynomial functors over Z, following arguments of Serre and Friedlander and Suslin. This provides a collection of `universal' short exact sequences that can be used to show that many other Grothendieck groups are lambda rings. In particular, it can be used to put a \lambda-ring structure on higher algebraic K-theory. This is based on joint work with Tom Harris and Bernhard Köck. No prior knowledge of lambda-rings or polynomial functors will be assumed.

This week there will be no seminar, due to the BeNeLux Congress of Mathematics which will be in Amsterdam on March 22 and 23.

Emre Sertoz: Enumerative geometry of theta characteristics

The geometry of theta characteristics in the form of contact hyperplanes to the canonical curve has been a classical topic of study. The study of the 28 bitangents to a quadric plane curve has been especially popular but for higher genera classical methods are far less revealing. Cornalba (1989) compactified the moduli space of theta characteristics, allowing the study of a single theta characteristic via degenerate stable curves. Following Cornalba, we compactify the moduli of pairs of theta characteristics, thereby initiating the study of the respective positions of 2 contact hyperplanes via degeneration. In particular, we calculate the class of the divisor of curves admitting two contact hyperplanes sharing a point of tangency on the curve. We end with a description of the limit stable curves in this divisor and what properties they imply about a general point on the divisor.

Netan Dogra: The Chabauty-Kim method for rational points on curves

Given a curve C of genus bigger than one, it is open problem to find a practical method for determining the set of rational points of C. In the Chabauty-Kim method, one approaches this problem by associating to a rational point a family of nonabelian Galois representations arising from the etale fundamental group of X at x, and instead trying to determine when such nonabelian Galois representations locally come from points on X. In this talk I will explain some recent developments in the theory. This is joint work with Jennifer Balakrishnan.

Riccardo Brasca: Hida theory over some Shimura varieties without ordinary locus.

The notion of ordinary modular form is meaningless if the ordinary locus of the relevant Shimura variety is empty, so it seems impossible to generalize Hida theory to this situation. We can nevertheless replace the ordinary locus by the so called mu-ordinary locus introduced by Wedhorn, that is always not empty. In this talk we will explain how to generalize Hida theory over the mu-ordinary locus in some cases.

Room F1.15 of the ILLC, first floor Science Park 107, at 16:30 (as opposed to 16:00 ) .

Giuseppe Ancona: The Chow ring of a commutative group scheme.

A classical result of Beauville shows that the action of the multiplication by n on the Chow ring of an abelian variety is semisimple with a finite number of explicit eigenvalues. Beauville's method is based on a Fourier transform using the dual abelian variety. We will generalize this result to commutative group schemes (semiabelian varieties, Néron models of abelian varieties, mixed Shimura varieties,...). The Fourier transform cannot be generalized to this context, we will show that Voevodsky's motives are a useful tool for this question. This is a joint work with Annette Huber and Simon Pepin Lehalleur.

There was no seminar today.