AWM at JMM 2015

Special Session Abstracts

Contents

Tuesday January 13, 2015, 8:00am-10:50am Room 007B, Convention Center

AMS-AWM Special Session on Recent Developments in Algebraic Number Theory, I

8:00 am Visualising the arithmetic of quadratic imaginary fields

Katherine E Stange*, University of Colorado, Boulder (1106-11-1783)

We study the orbit of ℝ under the Bianchi group PSL2(?K), where K is an imaginary quadratic field. The orbit, called a Schmidt arrangement SK, is a geometric realisation, as an intricate circle packing, of the arithmetic of K. This paper
presents several examples of this phenomenon. First, we show that the curvatures of the circles are integer multiples √−Δ and describe the curvatures of tangent circles in terms of the norm form of ?K. Second, we show that the circles themselves are in bijection with certain ideal classes in orders of OK, the conductor being a certain multiple of the curvature. This allows us to count circles with class numbers. Third, we show that the arrangement of circles is connected if and only if OK is Euclidean if and only if the tangency graph contains loops. These results are meant as foundational for a study of a new class of thin groups generalising Apollonian groups.

8:30 am Derivatives of p-adic L-functions of Hilbert modular forms

Daniel Barrera, University of Montreal;
Mladen Dimitrov, University of Lille;
Andrei Jorza*, University of Notre Dame (1106-11-2418)

P-adic L-functions are analogues of classical complex L-functions where the variable is a p-adic number instead of a com- plex one. Just like the Birch and Swinnerton-Dyer conjectures (and their generalizations) study the first Taylor coefficient of complex L-functions of Galois representations, so-called exceptional zero conjectures relate the first Taylor coefficients of p-adic L-functions to arithmetic information. This relationship encodes congruences between Hilbert modular forms and has wide-ranging applications. I will present recent results in the case of Hilbert modular forms.

9:00 am Moduli Interpretations for Noncongruence Modular Curves

William Y Chen*, Pennsylvania State University (1106-11-1877)

Let Γ be a subgroup of SL2(ℤ), and let ℋ be the upper half plane. If Γ is a congruence subgroup, then it’s well known that the quotient Γ \ ℋ is a coarse moduli space for isomorphism classes of elliptic curves equipped with some level structure. We will generalize the standard level structures and show that for most noncongruence subgroups Γ, the quotient Γ \ ℋ has a natural interpretation as the coarse moduli space classifying isomorphism classes of elliptic curves together with a generalized level structure. In this generalization the standard level structures associated to congruence subgroups should be considered “abelian”, while those corresponding to noncongruence subgroups should be considered “nonabelian”. We will also discuss applications to the arithmetic of noncongruence modular forms.

9:30 am p-adic q-expansions and families of automorphic forms

Ellen Eischen*, The University of North Carolina at Chapel Hill (1106-11-1801)

One approach to p-adically interpolating special values of certain L-functions relies on constructing p-adic families of automorphic forms. I will explain how to produce such p-adic families for certain unitary groups. In particular, this talk will focus on the q-expansions (and certain analogues of q-expansions) of these forms. I will also mention some applications to number theory and beyond.

10:00 am Local points of supersingular elliptic curves on ℤp-extensions

Mirela Ciperiani*, University of Texas at Austin
(1106-11-2572)

By work of Kobayashi and Iovita-Pollack we know that local points of supersingular elliptic curves on ramified ℤp– extensions of ℚp split into two strands of even and odd points. We will discuss a generalization of this result to ℤ/p-extensions that are localizations of anticyclotomic ℤp-extensions over which the elliptic curve has non-trivial CM points.

10:30 am On the restriction of F-crystalline p-adic Galois representations

Bryden Cais*, University of Arizona
Tong Liu, Purdue University (1106-11-1225)

Let K be a finite extension of ℚp, and let K∞ be the extension of K obtained by adjoining a compatible system of p-power roots of a uniformizer of K. A theorem of Kisin asserts that the restriction of crystalline (p-adic) GK∞-representations to GK∞ is fully faithful. We generalize this theorem to include a large class of infinite, totally wildly ramified strictly APF extensions of K.

Tuesday January 13, 2015, 8:00 a.m.-10:30 a.m. Room 217A, Convention Center

AWM Workshop on Homotopy Theory, I

8:00 a.m. Spaces of long embeddings and right-angled Artin operads

William G. Dwyer, Notre Dame University;
Kathryn Hess*, Ecole Polytechnique Fédérale de Lausanne (1106-55-1557)

(Joint work with Bill Dwyer) Generalizing the notion of a right-angled Artin group or monoid, we define a right-angled Artin operad to be the quotient of a free operad by the operadic ideal generated by a set of “commutator” relations of the form (x;y,…,y) ∼ (y;x,…,x)⋅τ, where x and y are generators, and τ is an appropriate permutation. The Boardman-Vogt tensor product of two free operads is an important example of a right-angled Artin operad.
Explicit resolutions of a right-angled Artin operad as a bimodule or an infinitesimal bimodule over itself are essential tools in our identification of the space of long embeddings of ℝm into ℝn as the (m + 1)-fold loop space on the derived mapping space of operad maps from the little m-balls operad to the little n-balls operad. I will sketch the proof of this identification briefly, emphasizing the role of right-angled Artin operads.

9:00 a.m. Geometric homology classes in the space of knots

Kristine Pelatt*, St. Catherine University (1106-55-156)

Using the calculus of functors, Sinha found spectral sequences converging to the homology and cohomology of knot spaces. These spectral sequences, however, do not immediately give representatives of cycles and cocycles. Generalizing methods of Cattaneo, Cotta-Ramusino, and Longoni, we develop a method of describing representatives of cycles in the space of knots by resolving intersection points on singular knots. The method of resolution is dictated by the combinatorics of the homology spectral sequence. In particular, we describe geometric representatives of non-trivial 3(d − 8)-dimensional cycles and cocycles, which guide our search for additional geometric representatives of cycles and cocycles.

9:30 a.m. Constructing equivariant spectra

Anna Marie Bohmann*, Northwestern University;
Angelica M. Osorno, Reed College (1106-55-189)

Equivariant spectra determine cohomology theories that incorporate a group action on spaces. Such spectra are increasingly important in algebraic topology but can be difficult to understand or construct. In recent work, Angelica Osorno and I have created a machine for building such spectra out of purely algebraic data based symmetric monoidal categories. In this talk I will discuss an extension of our work to the world of Waldhausen categories. This new construction is more flexible and is designed to be suitable for equivariant algebraic K-theory constructions.

10:00 a.m. Models for equivariant (∞, 1)-categories

Julia E Bergner*, University of California, Riverside (1106-55-603)

Recent results of Stephan give conditions under which a cofibrantly generated model category has an equivariant analogue, where the objects have a group action and weak equivalences and fibrations are defined via fixed point objects. We apply his results to several models for (∞, 1)-categories. For discrete groups, all of these models satisfy the required conditions. Applying a result of Bohmann-Mazur-Osorno-Ozornova-Ponto-Yarnall, we get an extension to the equivariant setting of the Quillen equivalences between their respective model categories. For actions of simplicial groups or compact Lie groups, we need to restrict to those models which have the additional structure of a simplicial or topological model category, respectively.

Tuesday January 13, 2015, 1:00 p.m.-5:50 p.m.Room 007B, Convention Center

AMS-AWM Special Session on Recent Developments in Algebraic Number Theory, II

1:00 p.m. An HN-theory for Kisin modules

Brandon Levin*, University of Chicago (1106-11-1061)

We begin by introducing Kisin varieties in the context of Galois deformation rings. We will then describe a generalization of Fargues’ HN-theory for finite flat group schemes to the larger category of Kisin modules. This HN-theory gives rise to stratifications of Kisin varieties. We also discuss partial results towards a tensor product theorem in this context. This is joint work with Carl Wang Erickson.

1:30 p.m. Bad reduction of genus 3 curves with complex multiplication

Irene Bouw, Universitaet Ulm;
Jenny Cooley, University of Warwick;
Kristin E. Lauter*, Microsoft Research;
Elisa Lorenzo Garcia, UPC Barcelona;
Michelle Manes, University of Hawaii;
Rachel Newton, University of Leiden;
Ekin Ozman, University of Texas Austin (1106-11-1017)

Let C be a smooth, absolutely irreducible genus-3 curve over a number field M. Suppose that the Jacobian of C has complex multiplication by a sextic CM-field K. Suppose further that K contains no imaginary quadratic subfield. We give a bound on the primes ? of M such that the stable reduction of C at ? contains three irreducible components of genus 1.

2:00 p.m. Étale π1 obstructions to rational points on Fermat curves

Kirsten Graham Wickelgren*, Georgia Institute of Technology (1106-11-840)

Jordan Ellenberg introduced obstructions to rational points coming from the lower central series of the étale fundamental group. This talk will describe joint work with Rachel Davis, Rachel Pries, and Vesna Stojanoska towards computing Ellenberg’s 2-nilpotent obstruction for Fermat curves, using a description of the homology of Fermat curves due to Greg Anderson.

2:30 p.m. An algebro-geometric theory of vector-valued modular forms of half-integral weight attached to Weil representations

Luca Candelori*, Louisiana State University (1106-11-839)

In this work we give a geometric theory of vector-valued modular forms attached to Weil representations of rank 1 lattices. More specifically, we construct vector bundles ?m,k over the moduli stack ℳ1 of elliptic curves, whose sections over the complex numbers give weight k + 1/2 vector-valued modular forms attached to rank 1 lattices with quadratic form x ?→ mx2/2, for m ∈ 2ℤ>0. The key idea is to construct vector bundles of Schrödinger representations and line bundles of half-forms over appropriate ‘metaplectic stacks’, which are μ2-gerbes over M1, and then show that their tensor products ?m,k descend to ℳ1. We then extend the bundles ?m,k to the cusp ∞ and give an algebraic notion of q-expansions of vector-valued modular forms. We define holomorphic vector-valued modular forms and cusp forms and compute algebraic dimension formulas for these spaces over any algebraically closed field of characteristic ≠ 2, 3, by using the Riemann-Roch theorem for DM stacks. Finally, by specializing the theory to the case m = 2, we obtain an algebro-geometric theory of modular forms of half-integral weight, as defined in the complex-analytic case by Shimura.

3:00 p.m. Counting Square Discriminants

Thomas Hulse, Queen’s University;
Mehmet Kiral, Texas A&M;
Chan Ieong Kuan, University of Maine;
Li-Mei Lim*, Bard College at Simon’s Rock (1106-11-727)

Hee Oh and Nimish Shah prove that the number of integral binary quadratic forms whose coefficients are bounded by a quantity X, and with discriminant a fixed square integer d, is cXlogX + O(X(logX)3/4). This result was obtained by the use of ergodic methods. Here we use the method of shifted convolution sums of Fourier coefficients of certain automorphic forms to obtain a sharpened result of a related asymptotic, obtaining a second main term and an error of O(X1/2).

3:30 p.m. Eulerian multizeta values over function fields

Chieh-Yu Chang, National Tsing Hua University
Matthew A. Papanikolas*, Texas A&M University
Jing Yu, National Taiwan University (1106-11-561)

A classical multiple zeta value (MZV) is said to be Eulerian if it is a rational multiple of a power of π. Examples of Eulerian MZV’s abound and date back at least to Euler. In the setting of function fields over a finite field, Thakur defined multizeta values in direct analogy with classical MZV’s, and Anderson and Thakur showed that they arise as periods of iterated extensions of the Carlitz motive. In this talk we will investigate a new criterion for determining when a function field MZV is Eulerian, in this case meaning that it is a rational multiple of a power of the Carlitz period. Furthermore we will discuss how this criterion can be used effectively to show whether or not a given MZV is Eulerian and present computational findings that confirm conjectures of Thakur and Lara Rodríguez.

4:00 p.m. Weierstrass mock modular forms and elliptic curves

Michael J. Griffin*, Emory University;
Claudia Alfes, Technische Universität Darmstadt;
Ken Ono, Emory University;
Larry Rolen, University of Cologne (1106-11-487)

Mock modular forms, which give the theoretical framework for Ramanujan’s enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves E/ℚ. We show that mock modular forms which arise from Weierstrass ζ-functions encode the central L-values and L-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by Hövel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of E. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain p-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form.

4:30 p.m. Arithmetic Progressions on Curves

Edray Herber Goins*, Purdue University
Alejandra Alvarado, Eastern Illinois University (1106-11-445)

The set {?1, 25, 49}? is a 3-term collection of integers which forms an arithmetic progression; the common difference is 24. Hence the set {?(1, 1), (5, 25), (7, 49)}? is a 3-term collection of rational points on the parabola y = x2 whose y-coordinates form an arithmetic progression. Similarly, the set {?6, 12, 18}? is a 3-term collection of integers which also forms an arithmetic progression; the common difference is 6. Hence the set {?(6, 3), (12, 39), (18, 75)}? is a 3-term collection of rational points on the elliptic curve y2 = x3 − 207 whose x-coordinates form an arithmetic progression. Are there other examples such as these? What is the longest progression of rational points on either a quadratic or cubic curve such that either the x- or y-coordinates form an arithmetic progression? In this talk, we give a survey on what’s known about arithmetic progressions on algebraic curves. We introduce elliptic curves as a means to show the non-existence of certain arithmetic progressions. We also introduce bielliptic curves in order to settle conjectures of Saraju P. Mohanty.

5:00 p.m. The reductions of finite subgroups of CM abelian varieties

Taisong Jing*, Pennsylvania State University (1106-11-403)

Let L be a CM field. CM abelian schemes with L-action over a (0,p) mixed characteristic complete discrete valuation ring with algebraically closed residue field are classified up to L-linear isogeny by the p-adic CM type. If we further require the full ring of integers ?L to act on the abelian scheme, then the p-adic CM type determines the L-linear isomorphic type. Under certain assumptions on such CM abelian schemes, we give a description on the reductions of their finite locally free subgroup schemes. This work has applications in the CM lifting problem for abelian varieties.

5:30 p.m. Almost generic p-divisibility bound

Hui June Zhu*, State University of New York at Buffalo (1106-11-252)

The well-known theorem of Ax and Katz gives a p-divisibility bound for the number of rational points on an algebraic variety V over a finite field of characteristic p in terms of the degree and number of variables of defining polynomials of V. We give a p-divisibility bound in terms of the supporting coefficient sets of the algebraic variety that refines and strengthens Ax-Katz bound; given prescribed sets of nonzero coefficient supports, suppose its combinatorial conditional number is nonzero, we show that any algebraic variety supported on these sets over the rationals achieves our p-divisibility bound at a set of primes p of positive density.

Tuesday January 13, 2015, 1:00 p.m.-5:00 p.m. Room 217A, Convention Center

AWM Workshop on Homotopy Theory, II

1:00 p.m. Commutative K-Theory

Ulrike Tillmann*, Oxford University (1106-55-1059)

Vector bundles over a compact manifold can be defined via transition functions to a linear group. Often one imposes conditions on this structure group. For example for real vector bundles one may ask that all transition functions lie in the special orthogonal group to encode orientability. Commutative K-theory arises when we impose the condition that the transition functions commute with each other whenever they are simultaneously defined. We will introduce commutative K-theory and some natural variants of it, and will show that they give rise to new generalised cohomology theories.
This is joint work with Adem, Gomez and Lind building on previous work by Adem, F. Cohen, and Gomez.

2:00 p.m. New developments in equivariant algebraic K-theory

Mona Merling*, Johns Hopkins University (1106-55-347)

There is a rapidly evolving development of equivariant infinite loop space theory which is expected to have long-range applications to algebraic K-theory. I will give a brief overview of motivations, results, and prospects.

2:30 p.m. An investigation of small model categories

Inna I Zakharevich*, University of Chicago (1106-18-1297)

An investigation of small model categories. Model categories have been widely used since their introduction by Quillen in 1967, and although many techniques exist for constructing model categories the most fundamental question remains open: given a bicomplete category ? together with a subcategory ?, when does there exist a model structure on ? with ? as the subcategory of weak equivalences? This question is fundamentally important, as model categories do not generally arise naturally “in the wild”; instead, one generally has a category with a subcategory of weak equivalences, and must construct the model structure by hand. Although this question is very difficult in general, it turns out that when ?[?−1] is a preorder the question can often be answered. We present some techniques for dealing with this question in general and, in the case when ? is small, give necessary and sufficient conditions for the existence of a model structure.

3:00 p.m. Coffee Break

3:30 p.m. Computations in the K(2)-local category at the prime 2

Irina Bobkova*, University of Rochester (1106-55-351)

Chromatic homotopy theory describes the homotopy of the p-local sphere spectrum S through a family of localizations LK(n)S with respect to Morava K-theories K(n). Considerable information about LK(n)S can be derived from the action of the Morava stabilizer group on the Lubin-Tate theory. One of the major computational tools is breaking up the homotopy of LK(n)S using various finite subgroups of the Morava stabilizer group. We will discuss some recent results and computations in the K(2)-local category at the prime p = 2.

4:00 p.m. Cohomology : A Mirror of Homotopy

Agnes Beaudry*, University of Chicago (1106-55-1269)

The philosophy of chromatic homotopy theory is that the stable homotopy groups of the sphere S can be re-assembled from the homotopy groups of a family of spectra LK(n)S. Roughly, LK(n)S is the n-th chromatic layer of S. There are spectral sequences whose input is the cohomology of a group, the Morava Stabilizer group, and whose output is the homotopy of the n-th chromatic layer. In this talk, I will illustrate of how some of these spectral sequences mirror the homotopy groups of LK(n)S and of S.

4:30 p.m. Computations in Algebraic K-Theory

Vigleik Angeltveit, Australian National University;
Teena Gerhardt*, Michigan State University (1106-55-2045)

In general algebraic K-theory groups are difficult to compute, but in recent years methods in equivariant stable homotopy theory have made some computations more accessible. Using these methods to compute the algebraic K-theory of pointed monoid algebras is particularly interesting, as the full power of equivariant homotopy groups is used. I will recall some successes of these methods and describe how equivariant techniques contribute to a new strategy for answering a classical computational question. In particular, I will discuss a new approach to computing the algebraic K-theory of the group ring Z[C2].