# Algebra and Number Theory Seminar Winter 2015

Friddays from 12:00-1:00pm

McHenry Library room 1257

Friday, January 9, 2015

"The Eisenstein Cocycle and Gross's Tower of Fields Conjecture"

Samit Dasgupta, University of California, Santa Cruz

Let $F$ be a totally real number field, and let $L/F$ be a field extension with $G=\Gal(L/F)$ finite abelian.  For a complex number $s$, let $\Theta(s)$ denote the unique element of $\C[G]$ such that $\chi(\Theta) = L_S(\chi, s)$ for all 1-dimensional characters $\chi$ of $G$.  Here $S$ is a finite set of primes of $F$ and $L_S$ denotes the $L$-function with Euler factors at elements of $S$ removed.  If $r$ denotes the number of places in $S$ that split completely in $L$, it is clear that $\Theta(s)$ vanishes to order at least $r$ at $s=0$, since the removed Euler factors at splitting places vanish at $s=0$.

In 1988, Gross  conjectured  an integral refinement of this order of vanishing statement.   Siegel and Klingen proved that $\Theta(0) \in \Q[G]$, and later Deligne--Ribet and Cassou-Nogu\`es proved that $\Theta^*(0) \in \Z[G]$, where $\Theta^*$ is the product of $\Theta$ and a simple smoothing factor. Let $K$ be a field extension of $F$ contained in $L$, and let $I$ denote the kernel of the natural projection from $\Z[G]$ to $\Z[\Gal(K/F)]$.  Let $r$ denote the number of places in $S$ that split completely in $K$.   Gross conjectured that $\Theta^*$ lies in $I^r$ (with a certain simple exception).  We describe a proof of this conjecture using the Eisenstein cocycle.

Friday, January 16, 2015

"Introduction to the three-point algebra"

Elizabeth Jurisich, College of Charleston

I will introduce the definition of the three point algebra and introduce two field representations for this algebra. The three-point algebra is perhaps the simplest nontrivial example of a Krichever– Novikov algebra beyond an affine Kac–Moody algebra. We provide a natural free field realization in terms of a beta- gamma system and the oscillator algebra of the three-point affine Lie algebra when g = sl(2, C).

Friday, January 23, 2015

"The Arithmetic Lefschetz Principle"

In complex algebraic geometry there is a general technique called the "Lefschetz Principle" which roughly says that for certain appropriately-formulated algebro-geometric problems over a general field of characteristic 0, it is sufficient to solve the problem over the complex numbers (where one has access to analytic and topological methods).  But this idea has relevance far beyond its traditional setting, and in fact underlies a wide array of powerful methods for relating problems in characteristic 0 (even over the complex field) or over the integers to problems over (many) finite fields, where new tools become available (point-counting, Frobenius, etc.).  This is widely used by experts but not well-explained in the literature outside of EGA.  In this expository seminar talk we will illustrate the basic ideas with a few interesting examples, not assuming prior familiarity with the usual Lefschetz Principle (or EGA).

Friday, January 30, 2015

"The p-parity conjecture for elliptic curves with a p-isogeny"

Kestutis Cesnavicius, UC Berkeley

For an elliptic curve E over a number field K, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number should match the parity of the Mordell-Weil rank. Its weaker but more approachable version is the p-parity conjecture for a fixed prime p: the global root number should match the parity of the Z_p-corank of the p-infinity Selmer group. After surveying what is known on the p-parity conjecture, we will discuss its proof in the case when E has a K-rational p-isogeny.

Friday, February 6, 2015

"Searching for sugar man: a CM-free definition of Heegner points"

Victor Rotger, Universitat Politecnica Catalunya (Barcelona)

I will describe my recent work with Darmon and Lauder from a point of view completely different to the one presented in the paper. The lecture will be accessible to non-experts with a basic background on modular curves and L-functions.  I will define what a Heegner point on an elliptic curve is and will introduce several equivalent formulations of these points. The ideal goal which will serve as motivation for the whole lecture is the search of a definition of Heegner point which should generalize in a natural way to other scenarios where the conjecture of Birch and Swinnerton-Dyer predicts the existence of a non-torsion canonical point.

Friday, February 13, 2015

"Orthogonal Units in The Trivial Source Ring of a Finite Group"

Rob Carman, University of California, Santa Cruz

I will begin this talk by defining and discussing a few basic rings used in studying the representations of a finite group: the Burnside, character, and Brauer character rings. We then move onto p-permutation modules aka trivial source modules and define the trivial source ring. Each of these four rings has a duality operation, and so we can ask which units have inverses coming from their dual elements. This question can be answered for the first three rings, but not in general for the trivial source ring. After discussing some motivation for finding these orthogonal units in the trivial source ring, I will describe some results towards answering this question.

February 20, 2015

"Certain families of Krichever-Novikov algebras and their automorphism groups"

Ben Cox, College of Charleston

We describe recent joint work with Xiangqian Guo, Rencai Lu and  Kaiming Zhao on an explicit description of the universal central extensions of the algebra of derivations whose coordinate ring is the ring of rational functions on the Riemann sphere with a finite number of fixed poles and also the ring of functions on certain superelliptic curves. In the description of the universal central extensions in the superelliptic case, associated Legendre polynomials naturally appear. Moreover, we give a description of the automorphism group of these algebras of derivations. In the Riemann sphere case, the automorphism group has to be one of the families of finite groups C_n, D_n, A_4,S_4 and A_5 given by F. Klein.

Friday, February 27, 2015

"Geometric theta series"

Luis Garcia, Imperial College (London)

We will review the construction by Kudla and Millson of theta series that give Poincare duals for special cycles on locally symmetric spaces. Then we will speculate on extensions of this theory related to regulators and higher K-theory classes.

Friday, March 6, 2015

"The conjectures of McKay, Alperin, Dade, and Broué"

Robert Boltje, University of California, Santa Cruz

The conjectures of McKay, Alperin, Dade and Broué, formulate several "local-global" principles in the representation theory of finite groups. Roughly speaking, they state that, for a prime $p$, much of the $p$-arithmetical information on the representation theory of a finite group $G$ is determined by the same $p$-arithmetical information for the $p$-local subgroups of $G$. These conjectures span statements from mere numerical equalities to more structural statements as for instance derived equivalences. The talk is intended to give an introduction to these conjectures.

Friday, March 13, 2015

"Vanishing theorems in algebraic geometry"

Junecue Suh, University of California, Santa Cruz

We'll start with an introduction to vanishing theorems for the general audience.  After explaining the notions involved with examples, we'll state (and perhaps sketch the proof of, as time permits) our recent theorem of Kawamata-Viehweg type.