# Algebra and Number Theory Seminar Fall 2016

**September 23, 2016**

Kiyokazu Nagatomo, Osaka University

Please click for a full description of the presentation.*"Characterization of minimal models (an introduction)."*Kiyokazu Nagatomo, Osaka University

Please click for a full description of the presentation.

**September 30, 2016
** The theory of vertex operator algebras (VOAs) has developed quite rapidly since its inception not too long ago. Its importance reaches many areas of mathematics and physics. We will discuss some of the main motivations behind the theory, give the d nition of a VOA and the notion of its modules, and visit a few concrete examples. This short discussion is meant as a very basic introduction to vertex operator algebras, paving the way for some upcoming talks on the subject.

*"A Friendly Introduction to Vertex Operator Algebras"*Danquynh Nguyen, University of California, Santa Cruz

**October 7, 2016**

**NO SEMINAR**

**This talk will give an introduction to vertex operator algebras over an arbitrary field. In particular, we discuss in detail the vertex operator algebra associated to the highest weight modules for the Virasoro algebra with central charge 1/2, such as the rationality and classification of the irreducible modules.**

**October 14, 2016**

Li Ren, Sichuan University, Chengdu, China

*"Modular vertex operator algebras"*Li Ren, Sichuan University, Chengdu, China

**October 21, 2016**

**"Descending cohomology geometrically"****Jeff Achter, Colorado State University
**

Motivated by a question of Mazur, we consider the problem of modelingvthe cohomology of an arbitrary smooth projective variety by that of an abelian variety. Our main strategy is to understand the extent to which the group of algebraically trivial cycles on a variety is represented by the points of an abelian variety. In particular, we extend Murre's theory of algebraic representatives to varieties over arbitrary perfect fields; and show that for a variety over a subfield K of the complex numbers, the image of the (complex) Abel-Jacobi map in the (transcendentally constructed) intermediate Jacobian admits a distinguished model over K. (This is joint work with Sebastian Casalaina-Martin and Charles Vial)

**October 28, 2016
**

**One day mini-conference on October 28 to celebrate Geoff Mason's retirement.**

1-2pm: Terry Gannon (University of Alberta)

Title:

Abstract:Because of applications primarily to geometry and physics, there has been increasing attention paid to modular forms with say rational Fourier coefficients but denominators which get arbitrarily large. According to an old conjecture of Atkin and Swinnerton-Dyer, modular forms of noncongruence subgroups of the modular group must have this property (unless they're also modular forms for a congruence subgroup, of course). Despite work by many people, there has been relatively little progress towards the general conjecture, which perhaps is an indication that a new idea is needed. In my talk I'll explain a new approach. This work is evolving from conversations with Geoff Mason and Cam Franc.

1-2pm: Terry Gannon (University of Alberta)

Title:

*"Modular forms and p-curvature"*Abstract:

**2-3pm: Cameron Franc (University of Saskatchewan)
Title: "Structural results on modules of modular forms"**

**Abstract:**Marks and Mason initiated the general study of structural properties of modules of vector valued modular forms associated with complex finite dimensional representations of SL2(Z). In particular, they proved that all such modules are free over the ring of scalar modular forms of level one. Recently, geometric results, such as the splitting principle for vector bundles, have allowed us to gain a more conceptual understanding of their work. In this talk we will review some of our recent joint work with Candelori and Mason on this topic. In the latter half of the talk, we will outline joint work with Candelori on a conjecture of Mason on the projectivity of modules of modular forms on the subgroup of SL2(Z) of index two. We show that, up to taking direct sums and shifting gradings, there are really only two modules that arise: the free module, and another module that can be described in terms of theta series. This reduces the projective module conjecture in this case to the study of a single module.

**The study of Frobenius-Schur indicators has provided new insights on the arithmetic properties of spherical fusion categories. In particular, the congruence subgroup theorem of modular categories and their conjectural congruence properties by Coste and Gannon were established via the generalized Frobenius-Schur indicators. These new results allude to a new approach on the classification modular categories of small rank or dimension from the arithmetic of their associated representations of SL(2,Z). In this talk, we will discussion this approach and its consequence of the on the classification modular categories.**

3-3:30: Tea break

3:30-4:30pm: Richard Ng (Louisiana State University)

Title:

Abstract:

3-3:30: Tea break

3:30-4:30pm: Richard Ng (Louisiana State University)

Title:

*"On the classification of modular categories"*Abstract:

All the talks will be in Room 4130. There will not be an algebra and number theory seminar on 10/28/16.

Organizers: Chongying Dong & Robert Boltje

All the talks will be in Room 4130. There will not be an algebra and number theory seminar on 10/28/16.

Organizers: Chongying Dong & Robert Boltje

**November 4, 2016**

*"A Stark conjecture and definition of a p-adic L-function"*

Joe Ferrara, University of California, Santa Cruz

In many instances there are p-adic L-functions associated to ray class characters and Stark type conjectures for the values of these p-adic L-functions at certian points. In this talk, the definition of a p-adic L-function of a ray class character will be given in a case when a p-adic L-function has not yet been defined. A conjecture in the spirit of the Stark conjectures will then be stated about the value of the p-adic L-function at s=1. The definition uses the theory of p-adic families of modular forms, so modular forms and Hida families of p-adic modular forms will be introduced. Also, as motivation, the classical rank one abelian Stark conjecture will be given, and the cases when p-adic L-functions of ray class characters are defined will be discussed.

**November 11, 2016
**

**NO SEMINAR-VETERAN'S DAY**

**November 18, 2016
**

***Please note the change in time***

Brian Conrad, Stanford University

Tamagawa numbers are canonical (finite) volumes attached to smooth connected affine groups G over global fields k; they arise in mass formulas and local-global formulas for adelic integrals. A conjecture of Weil (proved long ago for number fields, and recently by Lurie and Gaitsgory for function fields) asserts that the Tamagawa number of a simply connected semisimple group is equal to 1; for special orthogonal groups this expresses the Siegel Mass Formula. Sansuc pushed this further (using a lot of class field theory) to give a formula for the Tamagawa number of any connected reductive G in terms of two finite arithmetic invariants: its Picard group and degree-1 Tate-Shafarevich group.

*"Sansuc’s formula and Tate global duality (d’après Rosengarten)."*Brian Conrad, Stanford University

Over number fields it is elementary to remove the reductivity hypothesis from Sansuc’s formula, but over function fields that is a much harder problem; e.g., the Picard group can be infinite. Work in progress by my PhD student Zev Rosengarten is likely to completely solve this problem. He has formulated an alternative version, proved it is always finite, and established the formula in many new cases. We will discuss some aspects of this result, including one of its key ingredients: a generalization of Tate local and global duality to the case of coefficients in any positive-dimensional (possibly non-smooth) commutative affine algebraic k-group scheme and its (typically non-representable) GL1-dual.

**November 25, 2016
**

**NO SEMINAR-THANKSGIVING HOLIDAY**

**December 2, 2016
Jessica Fintzen, University of Michigan
**

*"On the Moy-Prasad filtration and supercuspidal*

**representations"**Reeder and Yu gave recently a new construction of certain supercuspidal representations of p-adic reductive groups (called epipelagic representations). Their construction relies on the existence of stable vectors in the first Moy-Prasad filtration quotient under the action of a reductive quotient. We will explain these ingredients and present a theorem about the existence of such stable vectors for all primes p. This builds on a result of Reeder and Yu about the existence of stable vectors for large primes and generalizes the paper of the speaker and Romano, which treats the case of an absolutely simple split reductive group.

In addition, we will present a general set-up that allows us to compare the Moy-Prasad filtration representations for different primes p. This provides a tool to transfer results about the Moy-Prasad filtration from one prime to arbitrary primes and also yields new descriptions of the Moy-Prasad filtration representations.

**December 9, 2016
**

**NO SEMINAR**

** **