# Algebra and Number Theory Seminar Winter 2016

Fridays at 12:00 noon

McHenry Library Room 1240

For more information please contact Professor Samit Dasgupta or call the Mathematics Department at 831-459-2969

**January 8, 2016 -** McHenry ROOM 1240

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*Cycles in the de Rham cohomology of abelian varieties over number fields.*

**Yunqing Tang, Harvard**

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus and Blasius, and Ogus predicted that all such cycles are Hodge. In this talk, I will first introduce Ogus' conjecture as a crystalline analogue of Mumford--Tate conjecture and explain how a theorem of Bost on algebraic foliation is related. After this, I will discuss my proof of Ogus' conjecture for some families of abelian varieties under the assumption that the cycles lie in the Betti cohomology with real coefficients.

**January 15, 2016** - McHenry ROOM 1240

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** Orbifold theory studies the action of a finite group G on a vertex operator algebra V. A vertex operator algebra is called rational if the module category is semisimple. The orbifold theory conjecture says: If V is rational then 1) V^G is rational, 2) every irreducible V^G-module occurs in an irreducible g-twisted V-module. I will report the recent progress on proving this conjecture.

*On orbifold theory.*Chongying Dong, University of California, Santa Cruz

** January 22, 2016-** McHenry ROOM 1240

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*Formulas for Iwasawa Invariants of Imaginary Quadratic Number Fields.*

**Jordan Schettler, San Jose State University**

Given a prime p and number field F, one can associate a nonnegative integer lambda (an Iwasawa invariant) which measures the growth of p-parts of class numbers in certain extensions of F, i.e., the layers in the cyclotomic Z_p-extension of F. We will discuss formulas for lambda when F is imaginary quadratic. In the p = 2 case, there is a classically-known and simple formula; however, by finding an alternate proof based on an analogy with Riemann surfaces, a generalization involving Fermat primes is revealed. In the p > 2 case, one can use entirely different methods to determine a formula involving the continued fraction data of a primitive element, and again the formula simplifies nicely when Fermat primes are involved.

**January 29, 2016**- McHenry ROOM 1240 from 1:00-2:00 pm

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** We discuss some recent research on modularity properties of intertwining operators in rational conformal field theory, mostly in the setting of the Virasoro minimal models. After presenting some general theorems we will focus on low-dimension representations of the modular group, including a description of the spaces of 1-point functions associated to modules of minimal models.

*Intertwining operators for rational VOAs and vector-valued modular forms.*Christopher Marks, California State University, Chico

**February 5, 2016**- McHenry ROOM 1240 from 1:00-2:00 pm

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**Polynomials representing primes**

**James Maynard,** **University of Oxford **

It is a famous conjecture that any one variable polynomial satisfying some simple conditions should take infinitely many prime values. Unfortunately, this isn't known in any case except for linear polynomials - the sparsity of values of higher degree polynomials causes substantial difficulties. If we look at polynomials in multiple variables, then there are a few polynomials known to represent infinitely many primes whilst still taking on `few' values; Friedlander-Iwaniec showed X2+Y4 is prime infinitely often, and Heath-Brown showed the same for X3+2Y3. We will describe recent work which gives a family of multivariate sparse polynomials all of which take infinitely many prime values.

**February 12, 2016**- McHenry ROOM 1240 from 1:00-2:00 pm

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**Introduction to the Davenport-Heilbronn Theorems**

**Shawn Tsosie, University of California, Santa Cruz
** In anticipation of Ila Varma's upcoming talk, I will introduce the theorems of Davenport-Heilbronn and provide a sketch of their proofs. The theorems of Davenport-Heilbronn give an asymptotic formula for the number of cubic fields whose discriminants are bounded and the number of 3-torsion elements in the class groups of quadratic fields whose discriminants are bounded. This talk is aimed at graduate students.

**Monday, February 22, 2016**- McHenry ROOM 4130 from 2:00-3:00 pm

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**The mean number of 3-torsion elements in ray class groups of quadratic fields****Ila Varma, Harvard University**

In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn’s theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over Q. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn’s asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields. If time permits, I will discuss the analogous theorems computing the mean size of 2-torsion elements in ray class groups of cubic fields ordered by discriminant, generalizing Bhargava.

**February 26, 2016
**

*Completed cohomology versus overconvergent cohomology*

**David Hansen, Columbia University**

Emerton's completed cohomology and Ash-Stevens's overconvergent cohomology are two approaches to a theory of p-adic automorphic forms and eigenvarieties on a general reductive group, each with their own unique structures and strengths. A priori, there is no direct relationship between the two theories. I'll explain such a relationship, namely a spectral sequence which computes overconvergent cohomology in terms of certain locally analytic Ext groups of completed cohomology. I'll also discuss some applications of this spectral sequence (and related ideas) to p-adic L-functions and eigenvarieties. This is joint work in progress with Christian Johansson.

**March 4, 2016-McHenry ROOM 1240
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**

**Künneth conjecture with torsion coefficients****Junecue Suh, University of California, Santa Cruz**

Grothendieck's Künneth conjecture predicts that the Künneth projectors (the idempotents giving the grading on the total cohomology) on the _rational_ cohomology of a projective smooth variety are algebraic, i.e., induced by the cohomology class of an algebraic cycle.

A. Venkatesh asked, in the context of torsion automorphic forms: Can the analogous statement hold true with _torsion_ coefficients? We first explain two reasons that make the question interesting, and then provide answers (a definite no in general, but a subtle yes in particular cases of interest).

**March 11, 2016-McHenry ROOM 1240
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** It is known that the blocks of the group algebra are in bijection with the blocks of the cohomological Mackey algebra. In this talk we look at equivalences (Morita or derived) between blocks of cohomological Mackey algebras. We will explain the relations between these equivalences and the equivalences between the corresponding blocks of group algebras.

*Baptiste Rognerud, UNAM, Morelia*

**Equivalences between blocks of cohomological algebras**