## AWM at JMM 2013

### Special Session and Workshop Abstracts

### Contents

- AMS-AWM Special Session on the Brauer Group in Algebra and Geometry, I
- AMS-AWM Special Session on the Brauer Group in Algebra and Geometry, II
- AWM Workshop on Number Theory, I
- AWM Workshop on Number Theory, II

#### Friday, January 11, 2013, 1:00 p.m.-6:20 p.m., Room 15B, Mezzanine Level, San Diego Convention Center

### AMS-AWM Special Session on the Brauer Group in Algebra and Geometry, I

### 1:00 p.m. Degree three cohomology of function field of surfaces

#### Suresh Venapally*, GA (1086-11-2342)

##### Let F be the function field of a surface X over a finite field. Let l be a prime not equal to the characteristic of F. Suppose that F contains a primitive l^{th} root of unity. We prove a certain local-global principle for elements of H^{3}(F, μ_{l}) in terms of symbols in H^{2}(F, μ_{l}) with respect to the discrete valuations of F. We use this to prove that every element in H^{3}(F, μ_{l}) is a symbol. The local-global principle also leads to the vanishing of certain unramified degree 3 cohomology groups of conic fibrations over X. This has implications towards the validity of the conjecture that Brauer-Manin obstruction is the only obstruction to the existence of zero-cycles of degree one for certain surfaces over global fields of positive characteristic.

### 1:30 p.m. Zero cycles on torsors under groups of low rank

#### Jodi Black*, Bucknell University;

R. Parimala, Emory University (1086-11-1456)

##### Let k be a field and let G be a connected linear algebraic group over k. In a 2004 paper, Totaro asked whether a torsor under G and over k, which admits a zero cycle of degree d, also admits a closed étale point of degree dividing d. We give a positive answer to this question for some semisimple groups of low rank when k is perfect and of characteristic different from 2.

### 2:00 p.m. Vertical Brauer groups and degree 4 del Pezzo surfaces

#### Anthony Várilly-Alvarado, Rice University

Bianca Viray*, Brown University (1086-14-1767)

##### We show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical, that is, that every Brauer class is obtained by pullback from an element of Br k(ℙ^{1}) for some rational map f : X ???- – > ℙ^{1}. As a consequence, we prove that a Brauer class does not obstruct the existence of a rational point if and only if there exists a fiber of f that is locally solvable. The proof is constructive and gives a simple and practical algorithm, distinct from that in [BBFL07], for computing all nonconstant classes in the Brauer group of X.

### 2:30 p.m. Generalized Mordell curves, generalized Fermat curves, and the Hasse principle

#### Dong Quan Ngoc Nguyen*, University of British Columbia (1086-11-1172)

##### We show that for each prime p ≡ 1 (mod 8), there exists a threefold X_{p} such that the existence of certain rational points on X_{p} produces families of generalized Mordell curves and families of generalized Fermat curves violating the Hasse principle explained by the Brauer-Manin obstruction. We also introduce a notion of the descending chain condition for sequences of curves, and prove that there are sequences of generalized Mordell curves and generalized Fermat curves satisfying the descending chain condition.

### 3:00 p.m. Estimating torsion using the twisted gamma-filtration

#### Caroline Junkins*, University of Ottawa (1086-14-1335)

##### For the Grothendieck group of a split simple linear algebraic group, the twisted γ-filtration provides a useful tool for constructing torsion elements in γ-rings of twisted flag varieties. In the present talk, we construct a non-trivial torsion element in the γ-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline.

### 3:30 p.m. Essential Dimension and the Brauer Group

#### Anthony Ruozzi*, Emory University (1086-16-1830)

##### Interest in essential dimension problems has been growing in recent years. This should not be surprising since the essential dimension captures quite elegantly the “least number of parameters” needed to define a wide range of algebraic objects. Calculations of this number require most of our algebraic and geometric machinery. Consequently, what began as a problem in Galois cohomology and representation theory now has connections to versal torsors, stacks, motives, birational geometry, and invariant theory. This talk will survey the basics of essential dimension and how it relates to central simple algebras. I will briefly discuss what is known, more of what is unknown, and how to actually compute some bounds on the essential p-dimension of PGL_{pn}.

### 4:00 p.m. A Decomposition for Idempotents of the Brauer Monoid.

#### Holly E Attenborough*, Indiana University (1086-00-1008)

##### Let G = Gal(K/F) for a Galois field extension K/F. The Brauer Monoid, M^{2}(G,K) is defined by adapting the cocycle construction of the Relative Brauer Group, Br(K/F) = H^{2}(G,K^{*}). We adapt the cocyles by allowing the image to be all of K, these cocycles, f : G × G → K, are called weak 2-cocycles. Let e an idempotent weak 2-cocycle, define the group M_{e}^{2}(G,K) to be {[f] ∈ M^{2}(G,K)|ef = f}. For a specfic ring R_{e} associated with an idempotent e, we have a complex on R_{e} -modules, M_{e} , that gives us that M_{e}^{2}(G, K) ≅ H^{2}(Hom_{ Re} (M^{e}, K^{∗})). The idempotents e are in one-to-one correspondence with lower subtractive partial orders on the group G, e ?↦≤_{e}. For S and T lower subtractive subsets of G, such that S ∪ T = (G, ≤_{e}), there exists a split short exact sequence on the complexes:

^{S∩T}→M

^{S}⊕M

^{T}→M

^{e}→0.