Seminars are on Wednesdays, at 4:00 PM, in Eckhart 203, unless
indicated otherwise.
Talks scheduled for the spring quarter are:
April 6
Tong Liu (U. Michigan): Potentially good reduction of
Barsotti-Tate groups.
Abstract: Let R be a complete discrete valuation ring of
mixed characteristic (0, p) with perfect residue field, K the
fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible
group) defined over K which acquires good reduction over a finite
extension K' of K.
We prove that there exists a constant c\geq 2 which depends on
the absolute ramification index e(K'/ Q_p) and the height of G such
that G has good reduction over K if and only if G[p^c] can be extended
to a finite flat group scheme over R. For abelian varieties with
potentially good reduction, this result generalizes
Grothendieck's ``p-adic N\'eron-Ogg-Shafarevich criterion" to finite
level. We also extend Raynaud's Theorem on finite flat group schemes to
high ramification case.
April 13
Bangere Purnaprajna (U. Kansas): On the ubiquity of
canonical covers of varieties of minimal degree.
The canonical map (that is the map induced by the canonical linear
series) of an algebraic curve $C$ of genus $g>1$ is reasonably well
understood: either it is an embedding or maps $C$ 2:1 onto a rational
normal curve (which is a variety of minimal degree in the sense that
its degree=codim+1.) For an algebraic variety of dimension two or
higher, the canonical map is much more subtle due to, among other
things, the existence of higher degree covers. I will talk about some
recent results (with F. J. Gallego) on the canonical map of a surface
of general type and its connection to the so-called "mapping geography"
of surfaces of general type.
April 20
Kiran Kedlaya (MIT): Slope filtrations for Frobenius modules
The classification of finite dimensional vector spaces over an
algebraically closed field, equipped with a linear transformation, is
well known. The theme of this talk will be to introduce analogues of
this
classification for semilinear transformations of finite free modules
over a ring equipped
with some sort of Frobenius endomorphism. I'll start with the original
such analogue, the classification of rational Dieudonne modules over
the Witt vectors of an algebraically closed field, attributed to
Dieudonne and
Manin. I'll then introduce a generalization of this
classification, to rings resembling the ring of power series (over a
p-adic field) convergent in
an annulus; it bears a formal resemblance to the classification of
vector bundles on the projective line, including the presence of a
complete numerical invariant that looks like a Harder-Narasimhan
polygon.
Finally, I'll make some comments about how this classification shows up
in the
study of p-adic cohomology (via work of Crew, Tsuzuki) and p-adic
Galois representations (via work of Colmez, Cherbonnier, Berger, Kisin).
April 27
Steve Kudla (U. Maryland): Modular generating series for
arithmetic cycles.
I will discuss joint work with Rapoport and Yang on the definition, modularity and relations for certain generating functions for classes in the arithmetic Chow groups of the arithmetic surface attached to a Shimura curve.
Friday, May 6 at 4pm in E312 (Note change of day and room!)
Toby Gee (Imperial College): Companion forms.
Abstract: We prove a companion forms theorem for mod l Hilbert
modular forms. This work generalises results of Gross and
Coleman--Voloch for modular forms over Q, and gives a new and more
conceptual proof of their results in many cases. Applications to
Artin's conjecture and to recent conjectures of Diamond on the weights
of mod l Hilbert modular forms will be discussed.
May 11 Juliana Tymoczko (U. Michigan): Title: Some results in
Schubert calculus
Abstract: Schubert calculus is the study of the
cohomology ring of flag varieties and Grassmannians, which involves a
cluster of tools and
techniques from algebraic geometry, topology, representation theory,
and combinatorics. A few years ago, Knutson and Tao applied the
approach of Goresky, Kottwitz, and MacPherson to compute the
equivariant cohomology of Grassmannians. In the process, they
gave a new description of the ordinary cohomology. Interestingly,
this was a case where the combinatorics led the geometry: their
description had no geometric interpretation until Vakil later
discovered what in retrospect is a very natural way to intersect
Schubert classes using controlled degenerations. In this talk, I
describe a collection of old results, new results, and open questions
in the area.
Monday, May 16 S.M. Bhatwadekar (Tata Institute): Projective modules over real affine varieties.
May 18, Frank Calegari (Harvard): Automorphic
Forms and Rational Homology Spheres
Abstract: Let K be an imaginary quadratic field. Modular forms
for K are related to the cohomology of arithmetic 3-manifolds. This
absence of algebraic geometry makes calculating the dimensions of such
cohomology groups difficult. Using Galois representations, we give an
approach to calculate such spaces in certain explicit towers of
congruence subgroups. (Joint work with Nathan Dunfield)
June 1, Jonathan Hanke (Duke): Finiteness Theorems
and Representing Numbers by Quadratic Forms
Abstract: This talk will describe several finiteness theorems
for quadratic forms, and progress on the question: "Which positive
definite integer-valued quadratic forms represent all positive
integers?". The answer to this question depends on settling the related
question "Which integers are represented by a given quadratic form?"
for finitely many forms. The answer to this question can involve both
arithmetic and analytic techniques, though only recently has the
analytic approach become practical.
We will describe the theory of quadratic forms as it relates to
answering these questions, its connections with the theory of modular
forms, and give an idea of how one can obtain explicit bounds to
describe which numbers are represented by a given quadratic form.