AWM at JMM 2013

Special Session and Workshop Abstracts

Contents

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 lth root of unity. We prove a certain local-global principle for elements of H3(F, μl) in terms of symbols in H2(F, μl) with respect to the discrete valuations of F. We use this to prove that every element in H3(F, μl) 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(ℙ1) for some rational map f : X ???- – > ℙ1. 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 Xp such that the existence of certain rational points on Xp 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 PGLpn.

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, M2(G,K) is defined by adapting the cocycle construction of the Relative Brauer Group, Br(K/F) = H2(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 Me2(G,K) to be {[f] ∈ M2(G,K)|ef = f}. For a specfic ring Re associated with an idempotent e, we have a complex on Re -modules, Me , that gives us that Me2(G, K) ≅ H2(Hom Re (Me, 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:

0→ MS∩T →MS ⊕MT →Me →0.

This gives us a long exact sequence on cohomology, which can aid with the computation of H2(HomRe(Me, K∗)) ≅ Me2(G, K).

4:30 p.m. The Brauer monoid of quaternion rings

John Voight*, University of Vermont (1086-16-1235)

The 2-torsion of the Brauer group of a field of characteristic not 2 is generated by quaternion algebras, by a theorem of Merkurjev. We consider a monoid extension of this subgroup: it is generated by a larger class of quaternion rings, allowing for degeneration, and is therefore closed under taking limits.

5:00 p.m. Noncommutative surfaces and curves of finite representation type

Colin J Ingalls*, University of New Brunswick;
Daniel Chan, University of New South Wales (1086-16-1802)

This is joint work with Daniel Chan. Local orders of global dimension two, over surfaces of finite representation type have been classified geometrically by Artin and by AR quivers by Reiten and Van den Bergh. We present a third classification via central extensions of finite subgroups of . This methods easily allows one to link all three classifications. We further classify noncommutative curves of finite representation type using noncommutative matrix factorizations and the classification of orders.

5:30 p.m. Generically Trivial Azumaya Algebras on a Rational Surface with a Non-rational Singularity

Drake M Harmon*, Florida Atlantic University;
Timothy J Ford, Florida Atlantic University;
Djordje N Bulj, Florida Atlantic University (1086-14-606)

Elementary examples are presented of normal algebraic surfaces X with singular points x such that at the local ring ?X,x there exist Azumaya algebras of all orders in the Brauer group that are split by the field of rational functions on X. These algebra classes correspond to elements of torsion in the class group of the henselian local ring ?hX,x . The surfaces X are affine normal rational and the singularities x are non-rational.

6:00 p.m. Patching and local-global principles for Brauer groups

David Harbater*, University of Pennsylvania
Julia Hartmann, RWTH Aachen University;
Daniel Krashen, University of Georgia (1086-16-2319)

and local-global principles for Brauer groups.
Using patching methods, local-global principles can be obtained for Brauer groups of function fields of curves over complete discretely valued fields. Over such function fields and related fields (such as two-variable Laurent series fields), this leads to results concerning the period-index problem. Motivated by work of Kato, these methods also lead to local-global principles for analogs of the Brauer group in higher cohomology over function fields as above, with applications to torsors and other structures such as Albert algebras.

Saturday, January 12, 2013, 8:00 a.m. − 10:50 a.m., Room 15B, Mezzanine Level, San Diego Convention Center

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

8:00 a.m. Generating the Brauer Group of the function field of a p-adic curve with cyclic algebras

Eric Brussel, California Polytechnic State University;
Kelly McKinnie*, University of Montana;
Eduardo Tengan, Universidade de São Paulo, Inst. de Ciências Matemáticas e de Computação (1086-16-2667)

In this talk we will discuss recent results about the number of cyclic algebras one needs to generate the Brauer group of the function field of a p-adic curve

8:30 a.m. The Brauer group of the function field of a curve over a complete discrete valuation ring

Eric Brussel*, California Polytechnic State University/Emory University (1086-16-1180)

Let F be the function field of a smooth curve over a local field. We prove the following for a number n that is coprime to the residue characteristic: a) The Z/n-cyclic length in the n-torsion of Br(F) is two; and b) If n is prime then all F-division algebras of degree n are cyclic. The second result was first proved by Saltman. We prove some results when the local field is replaced by a field that is henselian with respect to a discrete valuation of rank one.

9:00 a.m. Azumaya maximal orders do not always exist

Benjamin Antieau*, UCLA;
Ben Williams, USC (1086-14-980)

I will explain how to use the homotopy theory of classifying spaces of algebraic groups to construct smooth complex varieties X and Brauer classes α over X with the following property: if D is the division algebra over the function field of X with Brauer class α, then there is no Azumaya algebra on X with class α that restricts to D. In particular, no maximal order over X in D is Azumaya; equivalently, no maximal order over X in D is locally free.

9:30 a.m. Homogeneous Spaces over Function Fields of Dimension Two

Yi Zhu*, University of Utah (1086-14-1177)

Let K be either a global function field or a function field of an algebraic surface. Johan de Jong formulated the following principle: a “rationally simply connected” K-variety admits a rational point if and only if the elementary obstruction vanishes. In this talk, I will discuss how this principle works for projective homogeneous spaces. In particular, it leads to a classification-free result towards the quasi-split case of Serre’s Conjecture II over K

10:00 a.m. Splitting dimension and symbol length in Galois cohomology

Daniel R Krashen*, University of Georgia (1086-12-2020)

In this talk I will discuss how to bound symbol length using information on dimensions of splitting fields for elements in Galois cohomology and constructions of generic cocycles.

10:30 a.m. Finite u Invariant and Bounds on Cohomology Symbol Lengths

David J Saltman*, CCR-Princeton (1086-12-761)

At an AIM workshop in January 2011, Parimala asked whether in a field with finite u invariant there was a bound on the “symbol length” of any element of μ2 cohomology in any degree. We answer this question in the affirmative for fields of characteristic 0, and at the same time get bounds on the Galois groups that realize all the properties of these cohomology elements and show that our results extend to finite field extensions.

Saturday, January 12, 2013, 8:00 a.m. − 10:50 a.m., Room 6F, Upper Level, San Diego Convention Center

AWM Workshop on Number Theory, I

8:00 a.m. Zeta zeroes of Artin–Schreier curves

Chantal David*, Concordia University;
Alina Bucur, UCSD;
Brooke Feigon, CUNY;
Matilde Lalin, University of Montreal;
Kaneenika Sinha, Indian Institute of Science Education and Research, Kolkata (1086-11-681)

We study the distribution of the zeroes of the zeta functions in the family of Artin-Schreier curves over the finite fields Fq, when q is fixed and the genus goes to infinity. More precisely, we show that the distribution of the properly normalized zeroes in intervals of the unit circle follows a Gaussian distribution. This is done by computing the normalized moments of certain approximations of the number of zeroes in intervals given by the Beurling-Selberg polynomials. This is joint work with A. Bucur, B. Feigon, M. Lalin and K. Sinha.

8:30 a.m. How often is #E(?p) squarefree?

Shabnam Akhtari, University of Oregon;
Chantal David, Concordia University;
Heekyoung Hahn, Duke University;
Lola Thompson*, University of Georgia (1086-11-284)

Let E be an elliptic curve over Q. For each prime p of good reduction, E reduces to a curve Ep over the finite field ?p with #Ep (?p ) = p+1−ap , where |ap (E)| ≤ 2√p. In this talk, we discuss the problem of determining how often #E(?p ) is squarefree. Our results in this vein are twofold. For any fixed curve E, we give an asymptotic formula for the number of primes up to X for which #Ep (?p ) is squarefree. This resolves affirmatively a conjecture of David and Urroz. Moreover, we use sieve methods to improve upon a result of Gekeler that computes the average number of primes up to X for which #Ep(?p) is squarefree (over curves E in a suitable box). This talk is based on joint work with Shabnam Akhtari, Chantal David, and Heekyoung Hahn.

9:00 a.m. Parametrizing D4 covers over finite fields

Alina Bucur*, UCSD;
Ling Hoeschler, University of Illinois at Chicago;
Renate Scheidler, University of Calgary;
Melanie Matchett Wood, University of Wisconsin-Madison (1086-11-775)

This talk will report on the WIN2 project of my group. We will focus on the parametrization of D4 covers of the projective line over a finite field by tuples of quadratic forms.

9:30 a.m. Enumerating abelian varieties using matrix groups

Cassie L Williams*, James Madison University (1086-11-252)

The Frobenius endomorphism of an abelian variety A/?q acts as a symplectic similitude on the torsion subgroups A[?n](?q). In 2003, Gekeler used an equidistribution assumption on the elements of GL2(Z/?r) to show that the number of elliptic curves with certain characteristics is related, via results of Sato-Tate and the class number, to the Euler factors of the L-function of a quadratic imaginary field. By determining the sizes of conjugacy classes of Frobenius elements in the groups GSp2g(Z/?r) and applying a theorem of Everett Howe, we will extend Gekeler’s heuristic to higher dimensional abelian varieties.

10:00 a.m. Mahler measure of some singular K3-surfaces

Marie-Jose Bertin, Universite Pierre et Marie Curie;
Amy Feaver, University of Colorado at Boulder;
Jenny Fuselier, High Point University;
Matilde Lalin, University of Montreal;
Michelle Manes*, University of Hawaii at Manoa (1086-11-1019)

Jensen’s formula relates the Mahler measure of a one-variable polynomial to a simple formula depending on the roots of the polynomial. The situation for multivariate polynomials is quite different. Beginning in the 1970s and continuing to the present day, researchers have explored the connection between the Mahler measure of a polynomial defining an elliptic curve and the L-function of that curve. A natural extension of this line of inquiry involves connecting the polynomials whose zeros define Calabi-Yau varieties of dimension greater than one with L-functions associated to those varieties. In two dimensions, that means connecting the Mahler measure of polynomials to the L-functions of elliptic K3-surfaces. Building on previous work of Bertin, we prove three new formulas of this type. The strategy for proving these formulas is as follows:
*Understand the transcendental lattice and the group of sections for the K3-surface.
*Relate the Mahler measure of the polynomial to the L-function of a modular form.
*Relate the L-function of the K3-surface to the L-function of that same modular form.
We will outline each piece of the argument and point out technical difficulties that arise in some cases.

10:30 a.m. Ramanujan-type Supercongruences and complex multiplications on elliptic curves

Sarah Chisholm, University of Calgary, Canada;
Alyson Deines, University of Washington;
Ling Long*, Cornell University/Iowa State University;
Gabriele Nebe, RWTH Aachen University, Germany;
Holly Swisher, Oregon State University (1086-11-380)

In Ramanujan’s work on modular equations and approximation to pi, he gave a list of formulas for 1 over pi in terms of values of special hypergeometric series. It is discovered by van Hamme and later formulated more explicitly by Zudilin that the corresponding truncated hypergeometric series satisfy surprising congruence properties. In this talk, we will discuss some recent progress regarding these so-called Ramanujan-type Supercongruences via the arithmetic of complex multiplications on elliptic curves.

Saturday January 12, 2013, 1:00 p.m.-5:55 p.m., Room 6F, Upper Level, San Diego Convention Center

AWM Workshop on Number Theory, II

1:00 p.m. Newton polygons for a variant of the Kloosterman family

Rebecca Bellovin, Stanford University;
Sharon Anne Garthwaite, Bucknell University;
Ekin Ozman, University of Texas – Austin;
Rachel Pries*, Colorado State University;
Cassandra Williams, James Madison University;
Hui June Zhu, SUNY – Buffalo (1086-11-913)

Given a Laurent polynomial f in n variables defined over a finite field of characteristic p, one can associate to it the L- function of the exponential sum of f. Under a non-degeneracy condition, the L-function or its reciprocal is a polynomial, and the p-adic valuations of the roots of this polynomial can be studied using its Newton polygon NP(f).
Also associated to f is a convex n-dimensional polytope ∆. The Hodge polygon of ∆ is a combinatorial object which is a lower bound for NP(f). In this paper, we apply Wan’s simplicial decomposition theory to determine the Newton polygons for several families of Laurent polynomials f under certain congruence conditions. Specifically, we study non-diagonal reflection and Kloosterman variants of a type of diagonal polynomial.
This project was initiated at the workshop WIN Women in Numbers in November 2011.

1:30 p.m. Newton and Hodge Polygons for a Variant of the Kloosterman Family

Rebecca Bellovin, Stanford University;
Sharon Anne Garthwaite, Bucknell University;
Ekin Ozman*, University of Texas-Austin;
Rachel Pries, Colorado State University;
Cassandra Williams, Colorado State University;
Hui June Zhu, State University of New York (1086-11-907)

In this talk we investigate the behavior of Hodge polygons for the L-functions of a family of exponential sums of Laurent polynomial f in ?q[x1 , . . . , xn, (x1 · · · xn)−1], where f deforms the diagonal polynomial
f0 = x1m + · · · + xnm . We explicitly compute the Hodge numbers for such deformations in lower dimensions. Using these computations, one can determine the corresponding Hodge polygon which is a lower bound for the Newton polygon.

2:00 p.m. The a-numbers of Jacobians of Suzuki Curves

Holley Friedlander, University of Massachusetts, Amherst
Derek Garton, Northwestern University;
Beth Malmskog*, Colorado College;
Rachel Pries, Colorado State University;
Colin Weir, University of Calgary (1086-11-309)

For m ∈ N, let Sm be the Suzuki curve defined over ?22m+1 . It is well-known that Sm is supersingular, but the p-torsion group scheme of its Jacobian is not known. The a-number is an invariant of the isomorphism class of the p-torsion group scheme. In this talk, I will discuss joint work in which we computed a closed formula for the a-number of Sm using the action of the Cartier operator on H0.

2:30 p.m. Simultaneous Prime Values of Pairs of Quadratic Forms

D. R. Heath-Brown, University of Oxford
Lillian B. Pierce*, University of Oxford (1086-11-248)

We prove that under a certain geometric assumption, any two quadratic forms with integer coefficients simultaneously attain values infinitely often in any specified subset of the integers, as long as the elements of the set satisfy certain local conditions, and the set is not too sparse. The sparsity threshold is dependent upon the number of variables in the quadratic forms, as well as on the forms themselves. In particular, we show that 5 variables suffice in the case where the target set is the prime numbers, so that any two quadratic forms in 5 variables or more, which satisfy the relevant geometric condition, simultaneously attain prime values infinitely often. The proof proceeds via an application of the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.

3:00 p.m. Arithmetic of thin groups

Elena Fuchs*, University of California, Berkeley (1086-11-1014)

In the last few years there has been an increase in interest in arithmetic problems connected to so-called thin groups, mainly because tools to tackle such problems have only recently been developed. In this talk, we explore a few such arithmetic problems and discuss some of the questions that remain to be answered in the area.

3:30 p.m. Weierstrass points on the Drinfeld modular curve X0(?)

Christelle Vincent*, Stanford University (1086-11-343)

We prove that under a certain geometric assumption, any two quadratic forms with integer coefficients simultaneously attain values infinitely often in any specified subset of the integers, as long as the elements of the set satisfy certain local conditions, and the set is not too sparse. The sparsity threshold is dependent upon the number of variables in the quadratic forms, as well as on the forms themselves. In particular, we show that 5 variables suffice in the case where the target set is the prime numbers, so that any two quadratic forms in 5 variables or more, which satisfy the relevant geometric condition, simultaneously attain prime values infinitely often. The proof proceeds via an application of the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.

4:00 p.m. Local global compatibility and monodromy

Ana Caraiani*, University of Chicago (1086-11-889)

The global Langlands correspondence for GLn associates to a cuspidal automorphic representation of GLn satisfying certain technical conditions a number theoretic object, namely a Galois representation. This is known in many cases to be compatible with the local Langlands correspondence due to Harris-Taylor and Henniart. This compatibility generalizes the classical compatibility between local and global class field theory. In the case of GLn, I will explain how to prove the compatibility for monodromy operators, when the automorphic representation is regular-algebraic and conjugate-self dual.

4:30 p.m. Comparing the arithmetic intersection formulas of Bruinier-Yang and Lauter-Viray

Jacqueline Anderson, Brown University;
Jennifer S. Balakrishnan*, Harvard University;
Kristin Lauter, Microsoft Research;
Jennifer Park, MIT ;
Bianca Viray, Brown University (1086-11-1103)

Bruinier and Yang gave a conjectural formula for the arithmetic intersection number CM(K).G1 on the Siegel moduli space of abelian surfaces. This intersection number allows one to compute the denominators of Igusa class polynomials and has applications to the construction of genus 2 curves for use in cryptography.
Yang proved this conjecture under certain assumptions on the ramification in the quartic CM field K. More recently, Lauter and Viray gave a seemingly different formula for this intersection for a larger class of primitive quartic CM fields. We discuss each formula and sketch the correspondence between the two formulas in the range where they both apply.

5:00 p.m. On the Independence of Heegner Points

Hatice Sahinoglu*, Max Planck Institute for Mathematics (1086-11-842)

In this talk we are going to describe the construction of a particular set of algebraic points, which will be called as Heegner Points, on elliptic curves. Then we will investigate under which constraints Heegner points are independent. We give a sufficient condition on the class numbers of distinct quadratic imaginary fields so that on a given CM elliptic curve over ℚ with fixed modular parametrisation, the Heegner points associated to (the maximal orders of) the quadratic imaginary fields are linearly independent. This result extends the results of Rosen and Silverman from the non-CM elliptic curves to the CM ones. We will also show how to generalize the independence of Heegner points associated to maximal orders result to the Heegner points associated to orders of an an arbitrary fixed conductor of quadratic imaginary fields. We will look at independence of Heegner points arising from Shimura curve parametrisation and p-adic uniformisations. Finally, if time permits, we will build tools to investigate the independence of the Heegner points on higher dimensional abelian varieties.

5:30 p.m. The computation of anticyclotomic Λ-adic regulators of elliptic curves

Mirela Ciperiani*, The University of Texas at Austin (1086-11-1183)

Let E be an elliptic curve defined over ℚ, K an imaginary quadratic field, and p a prime of good ordinary non-anomalous reduction. Set ? to be the inverse limit of the points of E defined over the layers of the anticyclotomic Zp-extension of K. The image of ? under the cyclotomic p-adic height pairing is generated by the anticyclotomic Λ-adic regulator. If K satisfies the Heegner hypothesis, the elliptic curve has analytic rank 1 over K, and the Heegner point defined over K is not divisible by p, then Heegner points generate ?.
In this talk, we will describe a method that allows us to compute anticyclotomic Λ-adic regulators. We generalize results of Cohen and Watkins, and thereby compute Heegner points defined over different layers of the anticyclotomic Zp-extension of K. We also prove a connection which gives rise to an efficient way of using results of Mazur-Stein-Tate to compute p-adic heights. This is joint work with Jennifer Balakrishnan and William Stein.