# Algebra and Number Theory Seminar Fall 2015

Fridays at 12:00 noon

McHenry Library Room 4130

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

**Friday, October 2, 2015 (ROOM 1240) * Please note the room change***

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*Introduction to Bogomolov's birational anabelian geometry."***Joe Ferrara, University of California, Santa Cruz**

This talk is aimed as an introduction for graduate students to some of the topics Adam Topaz will speak about the following week when he speaks on mod-$ell$ anabelian geometry. It will begin with an example which motivates Grothendieck's anabelian geometry. After that, the Isom conjecture of Grothendieck's anabelian geometry will be introduced. The rest of the talk will be spent explaining the connection between the Isom conjecture and Bogomolov's birational anabelian geometry program. It is my goal to make the talk as accessible as possible. Many examples will be given.

**Friday, October 9, 2015 (ROOM 1240) * Please note the room change***

**"On mod-l anabelian geometry."****Adam Topaz, University of California, Berkeley**

In the early 90's, Bogomolov introduced a program whose ultimate goal is to reconstruct higher-dimensional geometric function fields from their pro-

*l*abelian-by-central Galois groups. This program has since been carried through in several important cases, but the general case remains open. In this talk, we will discuss some recent progress towards a mod-

*l*abelian-by-central variant of Bogomolov's program, as well as some of the inherent problems that arise. If time permits, we will discuss how this work leads to new results about mod-

*l*abelian-by-central quotients of geometric fundamental groups.

**Friday, October 16, 2015**

Alex Beloi, University of California, Santa Cruz

In many areas of math, specifically number theory, zeta functions have been a key area of interest. They have been shown to often encode many important features of the objects they are associated to. I'll be talking about a method developed originally by Takuro Shintani for equating algebraic zeta functions (associated to number fields) to certain analytic (Shintani) zeta functions defined over a lattice. The latter zeta functions were well studied by Shintani. Many of their properties are well documented, specifically Shintani showed explicit formulas for their values at non-positive integers. Equating these to algebraic zeta functions allows us to also give formulas for the values of algebraic zeta functions and their relation to conjectures of Stark.

**"Shintani's method for computing zeta values."**Alex Beloi, University of California, Santa Cruz

**Friday, October 23, 2015**

"** Ordinary pseudorepresentations, modular forms, and Iwasawa theory."
Preston Wake, University of California, Los Angeles
** Pseudorepresentations appear naturally when we talk about modular forms that are congruent to Eisenstein series. I'll talk about the difficulties that arise when defining "ordinary pseudorepresentation", and how to resolve these difficulties. I'll explain how the deformation theory of pseudorepresentations is related to cyclotomic Iwasawa theory and the geometry of the ordinary eigencurve. This is joint work with Carl Wang Erickson.

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**Friday, October 30, 2015
** Modular symbols are a fundamental tool for studying the L-functions of cuspforms. Unfortunately, classical modular symbols are ill-suited for studying the L-functions of Eisenstein series-- the corresponding modular symbols are, in some sense, trivial. We will discuss two ways of incorporating non-trivial Eisenstein symbols into the framework of modular symbols: the partial modular symbols of Dasgupta and the modular symbols with rational poles of Stevens. We will state a precise comparison and discuss arithmetic applications.

*Ander Steele, University of California, Santa Cruz*

**"Eisenstein modular symbols and L-values."**

**Friday, November 6, 2015
** Modular symbols are a fundamental tool for studying the $L$-functions of cuspforms. Unfortunately, classical modular symbols are ill-suited for studying the $L$-functions of Eisenstein series---the corresponding modular symbols are, in some sense, trivial.

**"Eisenstein modular symbols and L-values, part II."****Ander Steele, University of California, Santa Cruz**

In part two, we give details of Dasgupta's construction of Eisenstein partial modular symbols, and introduce the Shintani modular symbol. We will state a precise comparison theorem and give applications of these constructions.

**Friday, November 13, 2015
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*Generalizing Tensor Products of Bimodules over Group Algebras."*

**Robert Boltje, University of California, Santa Cruz.**

This talk reports on joint work with Philipp Perepelitsky. We study a construction that was initially introduced by Serge Bouc in 2010 and generalizes the usual tensor product of bimodules over group algebras. This elementary construction turns out to have beautiful functorial properties and to be a key tool for proving long desired results in modular representation theory and block theory. Most of the time will be spent with the introduction to the generalized tensor product and its properties, which happens on an elementary level. Block theoretic applications will be given at the end.

**Friday, November 20, 2015
** In 1980, Gross stated a conjecture relating the leading term of the p-adic L-function of a ray class character of a totally real field at s=0 to a p-adic regulator of p-units in the field cut out by the character. In previous joint work with Darmon and Pollack, we proved this conjecture in the rank one case under certain assumptions; these assumptions were later removed by Ventullo. In this talk, we describe work in progress with Ventullo and Kakde on the higher rank case. In particular, we present a proof in the rank two setting under a certain assumption. As a corollary of our result, we obtain an unconditional proof of the conjecture when the ground field is real quadratic.

*"On the higher rank Gross-Stark conjecture."*Samit Dasgupta, University of California, Santa Cruz

**Friday, November 27, 2015 * NO SEMINAR* THANKSGIVING HOLIDAY**

**Friday, December 4, 2015
**I will discuss how Gross's formalism of algebraic modular forms allows one to prove the algebricity of certain period integral functionals that arise in the formulas of Ichino and Ichino-Ikeda for special values of L-functions when the relevant group is compact-mod-center at infinity. In my Bay Area Number Theory Day talk, I will consider the problem of p-adically interpolating such functionals. This is joint work with Marco Seveso.

*"Algebraic modular forms and algebricity of period integrals."*Matt Greenberg, University of Calgary

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