# Algebra and Number Theory Seminar Spring 2012

Tuesday - 2:30p.m.-3:30p.m. in
McHenry Building - Room 4130
For information http://people.ucsc.edu/~cfranc/seminars/ucscANT.php
Cameron Franc - cfranc@ucsc.edu - (831) 459-4180

### Rob Laber, UCSC Graduate Student

In this talk, I will briefly recall the definitions of modular forms and cusp forms, and I will show how a modular form gives rise to a q-expansion. I will introduce the Eisenstein series and present their q-expansions, and we will use these to compute the discriminant Δ, which is the lowest weight nonzero cusp form. The space of modular forms carries the structure of a Z-graded algebra, and I will present some results which describe the structure of this algebra, including an isomorphism with a certain weighted polynomial algebra. I will conclude with a brief introduction to the Jacobi group and Jacobi forms. This talk should be accessible to all graduate students.

### Matthew Krauel, UCSC Graduate Student

We will introduce vertex operator algebras and their modules and discuss how modular and Jacobi forms arise from these algebraic structures. In particular, we will define 1-point functions associated with vertex operator algebras, and explain their currently known and conjectured invariance with respect to the modular and Jacobi groups.

### Geoff Mason, UCSC Professor

We formulate a general conjecture about the modular-invariance of holomorphic vector-valued modular forms. The conjecture has broad application, including the conjectured modular-invariance of partition functions in rational conformal field theory, and conjectures of Atkin-Swinnerton-Dyer concerning the Fourier coefficients of modular forms on noncongruence subgroups. We discuss a novel approach to the conjecture that leads to a proof for almost all irreducible 2-dimensional representations of $\SL_2(\ZZ)$.

### Philip Perepelitsky

Let G be a finite group and let p be a prime. Let (K,R,F) be a p-modular system. That is, R is a complete discrete valuation ring of characteristic zero, K is the field of fractions of R, F is the residue field of R, and char(F)=p. We assume that K and hence F contains a primitive |G|th root of unity, so that both K and F are splitting fields for G. Let O{R,F}. The so-called O-blocks of G correspond bijectively to the central primitive idempotents of the group algebra OG, also called the block idempotents of G. The canonical epimorphism from Z(RG) to Z(FG) induces a bijection between the set of central primitive idempotents of RG and those of FG. With this identification in mind, we may simply refer to the blocks of G when the ring R is understood. The blocks of G partition the irreducible K-characters of G in a very natural way, and this partition as well as the block idempotents themselves can be determined from the character table of G. Associated to each block B of G is a conjugacy class of p-subgroups of G of order p^a, which by abuse of language, is called the "defect group of B." The nonnegative integer a is called the defect of B and can be determined from the degrees of the irreducible K-characters belonging to B. Thus we see that blocks containing precisely one irreducible character have defect zero, and that the defect group of B is a Sylow p-subgroup of G if and only if B contains a character whose degree is coprime to p. In particular, the defect group of the block B_0 of G containing the trivial character is a Sylow p-subgroup of G. The block B_0 is called the principal block of G. Let P be any p-subgroup of G. Brauer's first main theorem states that the blocks of G with defect group P correspond bijectively to the blocks of N_G(P) with defect group P, via the so-called "Brauer correspondence," which can be explicitly described via the so-called "Brauer homomorphism." Finally, suppose P is a Sylow p-subgroup of G. Brauer's third main theorem states that the Brauer correspondent of the principal block of G is the principal block of N_G(P).

### Benjamin Sambale, Universitat Jena

I will continue Philipp's talk about various topics in modular representation theory. In particular I will present Brauer's second main theorem.

### Benjamin Sambale, Universitāt Jena

Many open conjectures in modular representation theory relate invariants of blocks of finite groups with properties of their defect groups. In order to verify these conjectures it is natural to compute the block invariants for a given defect group. This task has a long history which goes back to Brauer and Olsson. In my talk I like to present some new progress on these problems.

### Ted Nitz, UCSC

The Witt vector construction, of level n, takes a characteristic p field as input and outputs a ring in which p^n equals zero. In the limit, when n=\infty, it takes the field of p elements as input and outputs the ring of p-adic integers. The Witt vector construction can be used to directly construct the p-adic numbers and their unramified extensions. In this way, Witt vectors provide a mixed-characteristic'' analogue of the power-series rings that arise from characteristic p local fields. I will describe the Witt vector construction and some moduli spaces of particular interest in my research, endeavoring to make the talk accessible to everyone.

### Mitchell Owen, UCSC

A 1984 paper of Cohen and Lenstra introduced the idea that in a random collection of groups, having more automorphisms makes certain groups less likely to appear. They were then able to define a distribution on the set of finite abelian p-groups which matched nicely with data from quadratic class groups. In this talk I will motivate and define this distribution, consider some contexts where it applies, and discuss what results are known.

### Bryden Cais, University of Arizona

What is the probability that a random abelian variety over Fq is ordinary? Using Dieudonne modules, we will answer an analogue of this question, and explain how our method can be used to answer similar statistical questions about p-rank and a-number. The answers are perhaps surprising, and deviate from what one might expect via naive reasoning. Using these computations and numerical evidence, we formulate several Cohen-Lenstra" heuristics for the structure of the p-torsion on the Jacobian of a random hyperelliptic curve over Fq. These heuristics are the "l=p " analogue of Cohen-Lenstra in the function field setting. This is joint work with Jordan Ellenberg and David Zureick-Brown.

### Robert Boltje, UCSC

According to the celebrated Induction Theorem of Richard Brauer every irreducible character of a finite group can be written as an integral linear combination of induced linear characters. Brauer's motivation came from a question on Artin L-functions that was answered by his theorem. In the talk we introduce a functorial version of Brauer's theorem on the level of Grothendieck groups and a lift to a categorical level as a resolution functor between the homotopy categories of complex representations and of a category of finite equivariant line bundles.