57 research outputs found
A Gluing Lemma And Overconvergent Modular Forms
We prove a gluing lemma for sections of line bundles on a rigid analytic
variety. We apply the lemma, in conjunction with a result of Buzzard's, to give
a proof of (a generalization) of Coleman's theorem which states that
overconvergent modular forms of small slope are classical. The proof is
"geometric" in nature, and is suitable for generalization to other PEL Shimura
varieties
Modularity lifting in parallel weight one
We prove an analogue of the main result of Buzzard and Taylor (Annals of
Mathematics 149 (1999), 905-919) for totally real fields in which p is
unramified. This can be used to prove certain cases of the strong Artin
conjecture over totally real fields.Comment: 25 page
Overconvergence and classicality: the case of curves
Given our set-up of a system of curves and maps between them satisfying
certain assumptions, we prove a classicality criterion for overconvergent
sections of line bundles over these curves. As a result, we prove such criteria
for overconvergent modular forms over various Shimura curves. In particular, we
provide a classicality criterion for overconvergent modular forms studied in
[Kassaei: P-adic modular forms over Shimura curves over totally real fields,
Compositio Math. 140 (2004), no 2, 359-395] and their higher-level
generalizations
The canonical subgroup: a "subgroup-free" approach
Beyond the crucial role they play in the foundations of the theory of
overconvergent modular forms, canonical subgroups have found new applications
to analytic continuation of overconvergent modular forms. For such
applications, it is essential to understand various ``numerical'' aspects of
the canonical subgroup, and in particular, the precise extent of its
overconvergence.
We develop a theory of canonical subgroups for a general class of curves
(including the unitary and quaternionic Shimura curves), using formal and rigid
geometry. In our approach, we use the common geometric features of these curves
rather than their (possible) specific moduli-theoretic description.Comment: 16 pages, 1 figur
Companion Forms in Parallel Weight One
Let be prime, and let be a totally real field in which is
unramified. We give a sufficient criterion for a mod Galois representation
to arise from a mod Hilbert modular form of parallel weight one, by proving
a "companion forms" theorem in this case. The techniques used are a mixture of
modularity lifting theorems and geometric methods. As an application, we show
that Serre's conjecture for implies Artin's conjecture for totally odd
two-dimensional representations over .Comment: 12 page
The cone of minimal weights for mod Hilbert modular forms
We prove that all mod Hilbert modular forms arise via multiplication by
generalized partial Hasse invariants from
forms whose weight falls within a certain minimal cone. This answers a
question posed by Andreatta and Goren, and generalizes our previous results
which treated the case where is unramified in the totally real field.
Whereas our previous work made use of deep Jacquet-Langlands type results on
the Goren-Oort stratification (not yet available when is ramified), here we
instead use properties of the stratification at Iwahori level which are more
readily generalizable to other Shimura varieties
Virtual backbone formation in wireless ad hoc networks
We study the problem of virtual backbone formation in wireless ad hoc networks. A virtual backbone provides a hierarchical infrastructure that can be used to address important challenges in ad hoc networking such as efficient routing, multicasting/broadcasting, activity-scheduling, and energy efficiency. Given a wireless ad hoc network with symmetric links represented by a unit disk graph G = (V, E ), one way to construct this backbone is by finding a Connected Dominating Set (CDS) in G , which is a subset V' ✹ V such that for every node u, u is either in V' or has a neighbor in V' and the subgraph induced by V' is connected. In a wireless ad hoc network with asymmetric links represented by a directed graph G = (V, E ), finding such a backbone translates to constructing a Strongly Connected Dominating and Absorbent Set (SCDAS) in G . An SCDAS is a subset of nodes V' ✹ V such that every node u is either in V' or has an outgoing and an incoming neighbor in V' , and the subgraph induced by V' is strongly connected. Based on most of its applications, minimizing the size of the virtual backbone is an important objective. Therefore, we are interested in constructing CDSs and SCDASs of minimal size. We give efficient distributed algorithms with linear time and message complexities for the construction of the CDS in ad hoc networks with symmetric links. Since topology changes are quite frequent in most ad hoc networks, we propose schemes to locally maintain the CDS in the face of such changes. We also give a distributed algorithm for the construction of the SCDAS in ad hoc networks with asymmetric links. Extensive simulations show that our algorithms outperform all previously known algorithms in terms of the size of the constructed sets
p-adic modular forms over Shimura curves over Q
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999.Includes bibliographical references (p. 60-61).by Payman L. Kassaei.Ph.D
Canonical Subgroups over Hilbert Modular Varieties
We obtain new results on the geometry of Hilbert modular varieties in
positive characteristic and morphisms between them. Using these results and
methods of rigid geometry, we develop a theory of canonical subgroups for
abelian varieties with real multiplication.Comment: 56 page
Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case
We extend the modularity lifting result of the arXiv:1111.2804 to allow
Galois representations with some ramification at p. We also prove modularity
mod 2 and 5 of certain Galois representations. We use these results to prove
many new cases of the strong Artin conjecture over totally real fields in which
5 is unramified. As an ingredient of the proof, we provide a general result on
the automatic analytic continuation of overconvergent p-adic Hilbert modular
forms of finite slope which substantially generalizes a similar result in
arXiv:1111.2804.Comment: 47 page
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