We use the method of Ash and Stevens to prove the existence of
small slope $p$-adic families of cohomological modular forms
for an indefinite quaternion algebra $B$. We prove that the
Jacquet-Langlands correspondence relating modular forms on
$\textbf{GL}_2/\mathbb{Q}$
and cohomomological modular forms for $B$ is compatible with
the formation of $p$-adic families. This result is an analogue
of a theorem of Chenevier concerning definite quaternion algebras.
We study unitary quotients of the free product unitary pivotal
category $A_2*T_2$.
We show that such quotients are parametrized by an integer $n\geq
1$ and an $2n$-th root of unity $\omega$.
We show that for $n=1,2,3$, there is exactly one quotient and
$\omega=1$.
For $4\leq n\leq 10$, we show that there are no such quotients.
Our methods also apply to quotients of $T_2*T_2$, where we have
a similar result.
The essence of our method is a consistency check on jellyfish
relations.
While we only treat the specific cases of $A_2 * T_2$ and $T_2
* T_2$, we anticipate that our technique can be extended to a
general method for proving nonexistence of planar algebras with
a specified principal graph.
During the preparation of this manuscript, we learnt of Liu's
independent result on composites of $A_3$ and $A_4$ subfactor
planar algebras
(arxiv:1308.5691).
In 1994, Bisch-Haagerup showed that the principal graph of a
composite of $A_3$ and $A_4$ must fit into a certain family,
and Liu has classified all such subfactor planar algebras.
We explain the connection between the quotient categories and
the corresponding composite subfactor planar algebras.
As a corollary of Liu's result, there are no such quotient categories
for $n\geq 4$.
We formulate and prove a ``winding number'' index
theorem for certain ``Toeplitz'' operators in the same spirit
as Gohberg-Krein, Lesch and others. The ``number'' is replaced
by a self-adjoint operator in a subalgebra $Z\subseteq Z(A)$
of a unital $C^*$-algebra, $A$. We assume a faithful $Z$-valued
trace $\tau$ on $A$ left invariant under an action $\alpha:{\mathbf
R}\to Aut(A)$ leaving $Z$ pointwise fixed.If $\delta$ is the
infinitesimal generator of $\alpha$ and $u$ is invertible in
$\operatorname{dom}(\delta)$ then the
``winding operator'' of $u$ is $\frac{1}{2\pi i}\tau(\delta(u)u^{-1})\in
Z_{sa}.$ By a careful choice of representations we extend $(A,Z,\tau,\alpha)$
to a von Neumann setting
$(\mathfrak{A},\mathfrak{Z},\bar\tau,\bar\alpha)$ where $\mathfrak{A}=A^{\prime\prime}$
and $\mathfrak{Z}=Z^{\prime\prime}.$
Then $A\subset\mathfrak{A}\subset \mathfrak{A}\rtimes{\bf R}$, the von
Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$-trace
on $\mathfrak{A}\rtimes{\bf R}$. If $P$ is the projection in $\mathfrak{A}\rtimes{\bf
R}$
corresponding to the non-negative spectrum of the generator of
$\mathbf R$ inside $\mathfrak{A}\rtimes{\mathbf R}$ and
$\tilde\pi:A\to\mathfrak{A}\rtimes{\mathbf R}$
is the embedding then we define for $u\in A^{-1}$, $T_u=P\tilde\pi(u)
P$
and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$-valued
index of $T_u$ is the negative of the winding operator.
In outline the proof follows the proof of the scalar case done
previously by the authors. The main difficulty is making sense
of the constructions with the scalars replaced by $\mathfrak{Z}$ in
the von Neumann setting. The construction of the dual $\mathfrak{Z}$-trace
on $\mathfrak{A}\rtimes{\mathbf R}$ required the nontrivial development
of a $\mathfrak{Z}$-Hilbert Algebra theory. We show that certain of
these Fredholm operators fiber as a ``section'' of Fredholm operators
with scalar-valued index and the centre-valued index fibers as
a section of the scalar-valued indices.
The notion of positive-definite functions over locally compact
quantum
groups was recently introduced and studied by Daws and Salmi.
Based
on this work, we generalize various well-known results about
positive-definite
functions over groups to the quantum framework. Among these are
theorems
on "square roots" of positive-definite functions, comparison
of
various topologies, positive-definite measures and characterizations
of amenability, and the separation property with respect to compact
quantum subgroups.
This article provides an account of the functorial correspondence
between irreducible singular $G$-monopoles on $S^1\times \Sigma$
and $\vec{t}$-stable meromorphic pairs on $\Sigma$.
A theorem of B. Charbonneau and J. Hurtubise
is thus generalized here from unitary to arbitrary
compact, connected gauge groups. The required distinctions and
similarities for unitary versus arbitrary gauge are clearly outlined
and many parallels are drawn for easy transition. Once the correspondence
theorem is complete, the spectral decomposition is addressed.
Assuming Lang's conjectured lower bound on the heights of non-torsion
points on an elliptic curve, we show that there exists an absolute
constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion
point $P \in E(\mathbb{Q})$, there is at most one integral multiple
$[n]P$ such that $n \gt C$. The proof is a modification of a proof
of Ingram giving an unconditional but not uniform bound. The
new ingredient is a collection of explicit formulae for the
sequence $v(\Psi_n)$ of valuations of the division polynomials.
For $P$ of non-singular reduction, such sequences are already
well described in most cases, but for $P$ of singular reduction,
we are led to define a new class of sequences called elliptic
troublemaker sequences, which measure the failure of the Néron
local height to be quadratic. As a corollary in the spirit of
a conjecture of Lang and Hall, we obtain a uniform upper bound
on $\widehat{h}(P)/h(E)$ for integer points having two large
integral multiples.
In this work type II Hermite-Padé approximants for a vector
of Cauchy transforms of smooth Jacobi-type densities are considered.
It is assumed that densities are supported on mutually disjoint
intervals (an Angelesco system with complex weights). The formulae
of strong asymptotics are derived for any ray sequence of multi-indices.