We prove the Gross-Deligne conjecture on CM periods for motives
associated with $H^2$ of certain surfaces fibered over the projective
line. Then we prove for the same motives a formula which expresses
the $K_1$-regulators in terms of hypergeometric functions ${}_3F_2$,
and obtain a new example of non-trivial regulators.
The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager
invariant subspace theorem for operators on certain Banach spaces
of functions on a multiply connected domain in $\mathbb{C}$. The norms
for these spaces are either the usual Lebesgue and Hardy space
norms or certain continuous gauge norms.
In the Hardy space case the expected corollaries include the
characterization of the cyclic vectors as the outer functions
in this context, a demonstration that the set of analytic multiplication
operators is maximal abelian and reflexive, and a determination
of the closed operators that commute with all analytic multiplication
operators.
The paper constitutes a comprehensive study of ten classes of
self-maps on metric spaces $\langle X,d\rangle$ with the local
and pointwise (a.k.a. local radial) contraction properties.
Each of those classes appeared previously in the literature in
the context of fixed point theorems.
We begin with presenting an overview
of these fixed point results, including concise self contained
sketches of their proofs.
Then, we proceed with a discussion of the relations among the
ten classes of self-maps with domains $\langle X,d\rangle$
having various topological properties which often appear in the
theory of fixed point theorems: completeness, compactness, (path)
connectedness, rectifiable path connectedness, and $d$-convexity.
The bulk of the results presented in this part consists of examples
of maps that show non-reversibility of the previously established
inclusions between theses classes.
Among these examples, the most striking is a differentiable auto-homeomorphism
$f$ of a compact perfect subset $X$ of $\mathbb R$
with $f'\equiv 0$, which constitutes also a minimal dynamical
system.
We finish with discussing a few remaining open problems on weather
the maps with specific pointwise contraction
properties must have the fixed points.
We characterize the Local Langlands Correspondence (LLC) for
inner forms of $\operatorname{GL}_n$ via the Jacquet-Langlands Correspondence
(JLC) and compatibility with the Langlands Classification. We
show that LLC satisfies a natural compatibility with parabolic
induction and characterize LLC for inner forms as a unique family
of bijections $\Pi(\operatorname{GL}_r(D)) \to \Phi(\operatorname{GL}_r(D))$ for each $r$,
(for a fixed $D$) satisfying certain properties. We construct
a surjective map of Bernstein centers $\mathfrak{Z}(\operatorname{GL}_n(F))\to
\mathfrak{Z}(\operatorname{GL}_r(D))$
and show this produces pairs of matching distributions in the
sense of Haines. Finally, we construct explicit Iwahori-biinvariant
matching functions for unit elements in the parahoric Hecke
algebras
of $\operatorname{GL}_r(D)$, and thereby produce many explicit pairs of matching
functions.
We investigate the smoothing problem of limit linear series of
rank one on an enrichment of the notions of nodal curves and
metrized complexes called saturated metrized complexes. We give
a finitely verifiable full criterion for smoothability of a limit
linear series of rank one on saturated metrized complexes, characterize
the space of all such smoothings, and extend the criterion to
metrized complexes. As applications, we prove that all limit
linear series of rank one are smoothable on saturated metrized
complexes corresponding to curves of compact-type, and prove
an analogue for saturated metrized complexes of a theorem of
Harris and Mumford on the characterization of nodal curves contained
in a given gonality stratum. In addition, we give a full combinatorial
criterion for smoothable limit linear series of rank one on saturated
metrized complexes corresponding to nodal curves whose dual graphs
are made of separate loops.
We study a relation between distinction and special values of
local invariants for representations of the general linear group
over a quadratic extension of $p$-adic fields.
We show that the local Rankin-Selberg root number of any pair
of distinguished representation is trivial and as a corollary
we obtain an analogue for the global root number of any pair
of distinguished cuspidal representations. We further study the
extent to which the gamma factor at $1/2$ is trivial for distinguished
representations as well as the converse problem.
The $\operatorname{SL}(2,\mathbb C)$-representation varieties of punctured surfaces
form natural families parameterized by monodromies at the punctures.
In this paper, we compute the loci where these varieties are
singular for the cases of one-holed and two-holed tori and the
four-holed sphere. We then compute the de Rham cohomologies
of these varieties of the one-holed torus and the four-holed
sphere when the varieties are smooth via the Grothendieck theorem.
Furthermore, we produce the explicit Gauss-Manin connection
on the natural family of the smooth $\operatorname{SL}(2,\mathbb C)$-representation
varieties of the one-holed torus.