Description
We study Hilbert modular forms in characteristic p and over p-adic
rings. In the characteristic p theory we describe the kernel and
image of the q-expansion map and prove the existence of
filtration for Hilbert modular forms; we define operators U, V
and Theta_chi and study the variation of the filtration under
these operators. Our methods are geometric - comparing
holomorphic Hilbert modular forms with rational functions on a
moduli scheme with level-p structure, whose poles are supported
on the non-ordinary locus.
In the p-adic theory we study congruences between Hilbert modular
forms. This applies to the study of congruences between special
values of zeta functions of totally real fields. It also allows
us to define p-adic Hilbert modular forms "a la Serre"
as p-adic uniform limit of classical modular forms, and compare
them with p-adic modular forms "a la Katz" that are
regular functions on a certain formal moduli scheme. We show that
the two notions agree for cusp forms and for a suitable class of
weights containing all the classical ones. We extend the
operators V and Theta_chi to the p-adic setting.
Contents
Introduction
Notations
Moduli spaces of abelian varieties with real multiplication
Properties of mathcal{G}
Hilbert modular forms
The q-expansion map
The partial Hasse invariants
Reduceness of the partial Hasse invariants
A compactification of mathfrak{M} (k,mu_{pN})^{rm{Kum}}
Congruences mod p^n and Serre's p-adic modular forms
Katz's p-adic Hilbert modular forms
The operators Theta_{mathfrak{P},i}
The operator V
The operator U
Applications to filtrations of modular forms
Theta cycles and parallel filtration (inert case)
Functorialities
Integrality and congruences for values of zeta functions
Numerical examples
Comments regarding values of zeta functions
References
Details:
Series: Memoirs of the American Mathematical Society, Volume: 819
Publication Year: 2005
ISBN: 0-8218-3609-9
Paging: 100 pp.
Binding: Softcover
Description
In the space of polynomials in two variables mathbb{C}[x,y] with
complex coefficients we let the group of automorphisms of the
affine plane mathbb{A}^2_{mathbb{C}} act by composition on the
right. In this paper we investigate the geometry of the orbit
space.
We associate a graph with each polynomial in two variables that
encodes part of its geometric properties at infinity; we define a
partition of mathbb{C}[x,y] imposing that the polynomials in the
same stratum are the polynomials with a fixed associated graph.
The graphs associated with polynomials belong to certain class of
graphs (called behaviour graphs), that has a purely combinatorial
definition. We show that any behaviour graph is actually a graph
associated with a polynomial. Using this we manage to give a
quite precise geometric description of the subsets of the
partition.
We associate a moduli functor with each behaviour graph of the
class, which assigns to each scheme T the set of families of
polynomials with the given graph parametrized over T. Later,
using the language of groupoids, we prove that there exists a
geometric quotient of the subsets of the partition associated
with the given graph by the equivalence relation induced by the
action of Aut(mathbb{C}^2). This geometric quotient is a coarse
moduli space for the moduli functor associated with the graph. We
also give a geometric description of it based on the
combinatorics of the associated graph.
The results presented in this memoir need the development of a
certain combinatorial formalism. Using it we are also able to
reprove certain known theorems in the subject.
Contents
Introduction
Automorphisms of the affine plane
A partition on mathbb{C}[x,y]
The geometry of the partition
The action of Aut(mathbb{C}^2) on mathbb{C}[x,y]
The moduli problem
The moduli spaces
Appendix A. Canonical orders
Bibliography
Details:
Series: Memoirs of the American Mathematical Society, Volume: 817
Publication Year: 2005
ISBN: 0-8218-3593-9
Paging: 136 pp.
Binding: Softcover
Description
Very recently, the classification of Moufang polygons has been
completed by Tits and Weiss. Moufang n-gons exist for n in { 3, 4,
6, 8 } only. For n in { 3, 6, 8 }, the proof is nicely divided
into two parts: first, it is shown that a Moufang n-gon can be
parametrized by a certain interesting algebraic structure, and
secondly, these algebraic structures are classified. The
classification of Moufang quadrangles (n=4) is not organized in
this way due to the absence of a suitable algebraic structure.
The goal of this article is to present such a uniform algebraic
structure for Moufang quadrangles, and to classify these
structures without referring back to the original Moufang
quadrangles from which they arise, thereby also providing a new
proof for the classification of Moufang quadrangles, which does
consist of the division into these two parts. We hope that these
algebraic structures will prove to be interesting in their own
right.
Contents
Introduction
Definition
Some identities
From quadrangular systems to Moufang quadrangles
From Moufang quadrangles to quadrangular systems
Some remarks
Examples
The classification
Appendix A. Abelian quadrangular systems
Bibliography
Details:
Series: Memoirs of the American Mathematical Society, Volume: 818
Publication Year: 2005
ISBN: 0-8218-3608-0
Paging: 99 pp.
Binding: Softcover
Description
The framework of affine and hyperbolic laminations provides a
unifying foundation for many aspects of conformal dynamics and
hyperbolic geometry. The central objects of this approach are an
affine Riemann surface lamination mathcal A and the associated
hyperbolic 3-lamination mathcal H endowed with an action of a
discrete group of isomorphisms. This action is properly
discontinuous on mathcal H, which allows one to pass to the
quotient hyperbolic lamination mathcal M. Our work explores
natural "geometric" measures on these laminations.
We begin with a brief self-contained introduction to the measure
theory on laminations by discussing the relationship between
leafwise, transverse and global measures. The central themes of
our study are: leafwise and transverse "conformal streams"
on an affine lamination mathcal A (analogues of the Patterson-Sullivan
conformal measures for Kleinian groups), harmonic and invariant
measures on the corresponding hyperbolic lamination mathcal H,
the "Anosov--Sinai cocycle", the corresponding "basic
cohomology class" on mathcal A (which provides an
obstruction to flatness), and the Busemann cocycle on mathcal H.
A number of related geometric objects on laminations -- in
particular, the backward and forward Poincare series and the
associated critical exponents, the curvature forms and the Euler
class, currents and transverse invariant measures, lambda-harmonic
functions and the leafwise Brownian motion -- are discussed along
the lines.
The main examples are provided by the laminations arising from
the Kleinian and the rational dynamics. In the former case,
mathcal M is a sublamination of the unit tangent bundle of a
hyperbolic 3-manifold, its transversals can be identified with
the limit set of the Kleinian group, and we show how the
classical theory of Patterson-Sullivan measures can be recast in
terms of our general approach. In the latter case, the
laminations were recently constructed by Lyubich and Minsky in [LM97].
Assuming that they are locally compact, we construct a transverse
delta-conformal stream on mathcal A and the corresponding lambda-harmonic
measure on mathcal M, where lambda=delta(delta-2). We prove that
the exponent delta of the stream does not exceed 2 and that the
affine laminations are never flat except for several explicit
special cases (rational functions with parabolic Thurston
orbifold).
Contents
Introduction
Affine and hyperbolic laminations
Measures and currents on laminations
Laminations associated with rational maps
Measures on laminations associated with rational maps
Appendix A. Laminations associated with Kleinian groups
List of notations
Bibliography
Details:
Series: Memoirs of the American Mathematical Society,Volume: 820
Publication Year: 2005
ISBN: 0-8218-3615-3
Paging: 119 pp.
Binding: Softcover
Description
In recent years, increasingly complex methods have been brought
into play in the treatment of geometric and topological problems
for partial differential operators on manifolds. This collection
of papers, resulting from a Workshop on Spectral Geometry of
Manifolds with Boundary and Decomposition of Manifolds, provides
a broad picture of these methods with new results.
Subjects in the book cover a wide variety of topics, from recent
advances in index theory and the more general theory of spectral
invariants on closed manifolds and manifolds with boundary, to
applications of those invariants in geometry, topology, and
physics.
Papers are grouped into four parts: Part I gives an overview of
the subject from various points of view. Part II deals with
spectral invariants, such as traces, indices, and determinants.
Part III is concerned with general geometric and topological
questions. Part IV deals specifically with problems on manifolds
with singularities. The book is suitable for graduate students
and researchers interested in spectral problems in geometry.
Contents
Part I. Basic material-Reviews
D. V. Vassilevich -- Spectral problems from quantum field theory
G. Esposito -- Euclidean quantum gravity in light of spectral
geometry
G. Grubb -- Analysis of invariants associated with spectral
boundary problems for elliptic operators
Part II. Spectral invariants and asymptotic expansions
G. Grubb -- A resolvent approach to traces and zeta Laurent
expansions
Y. Lee -- Asymptotic expansion of the zeta-determinant of an
invertible Laplacian on a stretched manifold
J. Park and K. P. Wojciechowski -- Agranovich-Dynin formula for
the zeta-determinants of the Neumann and Dirichlet problems
Part III. Geometric and topological problems
H. U. Boden, C. M. Herald, and P. Kirk -- The Calderon projector
for the odd signature operator and spectral flow calculations in
3-dimensional topology
E. Leichtnam and P. Piazza -- Cut-and-paste on foliated bundles
M. Lesch -- The uniqueness of the spectral flow on spaces of
unbounded self-adjoint Fredholm operators
M. Marcolli and B.-L. Wang -- Variants of equivariant Seiberg-Witten
Floer homology
Part IV. Manifolds with singularities
P. Loya -- Dirac operators, boundary value problems, and the b-calculus
V. E. Nazaikinskii, G. Rozenblum, A. Yu. Savin, and B. Yu.
Sternin -- Guillemin transform and Toeplitz representations for
operators on singular manifolds
V. Nistor -- Pseudodifferential operators on non-compact
manifolds and analysis on polyhedral domains
Details:
Series: Contemporary Mathematics, Volume: 366
Publication Year: 2005
ISBN: 0-8218-3536-X
Paging: 328 pp.
Binding: Softcover