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961  Low Frequency Estimates for Long Range Perturbations in Divergence Form Bouclet, JeanMarc
We prove a uniform control as $ z \rightarrow 0 $ for the resolvent $
(Pz)^{1} $ of long range perturbations $ P $ of the Euclidean
Laplacian in divergence form by combining positive commutator
estimates and properties of Riesz transforms. These estimates hold in
dimension $d \geq 3 $ when $ P $ is defined on $ \mathbb{R}^d $ and in dimension $ d \geq 2 $ when $ P $ is defined outside a compact obstacle with Dirichlet boundary conditions.


992  The Arithmetic of Genus Two Curves with (4,4)Split Jacobians Bruin, Nils; Doerksen, Kevin
In this paper we study genus $2$ curves whose Jacobians admit a
polarized $(4,4)$isogeny to a product of elliptic curves. We consider
base fields of characteristic different from $2$ and $3$, which we do
not assume to be algebraically closed.
We obtain a full classification of all principally polarized abelian
surfaces that can arise from gluing two elliptic curves along their
$4$torsion, and we derive the relation their absolute invariants
satisfy.


1025  Universal Series on a Riemann Surface Clouâtre, Raphaël
Every holomorphic function on a compact subset of a Riemann surface can
be uniformly approximated by partial sums of a given series of functions.
Those functions behave locally like the classical fundamental solutions
of the CauchyRiemann operator in the plane.


1038  Critical Points and Resonance of Hyperplane Arrangements Cohen, D.; Denham, G.; Falk, M.; Varchenko, A.
If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement
$\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship
between the critical set of $\Phi_\lambda$, the variety defined by the vanishing
of the oneform $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$.
For arrangements satisfying certain conditions, we show that if $\lambda$ is
resonant in dimension $p$, then the critical set
of $\Phi_\lambda$ has codimension
at most $p$. These include all free arrangements and all rank $3$ arrangements.


1058  $S_3$covers of Schemes Easton, Robert W.
We analyze flat $S_3$covers of schemes, attempting to create
structures parallel to those found in the abelian and triple cover
theories. We use an initial local analysis as a guide in finding a
global description.


1083  Decomposition of Splitting Invariants in Split Real Groups Kaletha, Tasho
For a maximal torus in a quasisplit semisimple simplyconnected group over a local field of characteristic $0$,
Langlands and Shelstad constructed a
cohomological invariant called the splitting invariant, which is an important
component of their endoscopic transfer factors. We study this invariant in the
case of a split real group and prove a
decomposition theorem which expresses this invariant for a general torus as a product of the corresponding
invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants
between different tori in the given real group.


1107  Genericity of Representations of pAdic $Sp_{2n}$ and Local Langlands Parameters Liu, Baiying
Let $G$ be the $F$rational points of the symplectic group $Sp_{2n}$,
where $F$ is a nonArchimedean local field
of characteristic
$0$. Cogdell, Kim, PiatetskiShapiro, and Shahidi
constructed local Langlands functorial lifting from irreducible
generic representations of $G$ to irreducible representations of
$GL_{2n+1}(F)$.
Jiang and Soudry constructed the descent map from irreducible
supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$,
showing that the local Langlands functorial lifting from the
irreducible supercuspidal generic representations is surjective. In
this paper, based on above results, using the same descent method of
studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest
of local Langlands functorial lifting is also surjective, and for any
local Langlands parameter $\phi \in \Phi(G)$, we construct a
representation $\sigma$ such that $\phi$ and $\sigma$ have the same
twisted local factors. As one application, we prove the $G$case of a
conjecture of
GrossPrasad and Rallis, that is, a local Langlands parameter $\phi
\in \Phi(G)$ is generic, i.e., the representation attached to
$\phi$ is generic, if and only if the adjoint $L$function of $\phi$
is holomorphic at $s=1$. As another application, we prove for each
Arthur parameter $\psi$, and the corresponding local Langlands
parameter
$\phi_{\psi}$, the representation attached to $\phi_{\psi}$
is generic if and only if $\phi_{\psi}$ is tempered.


1137  Distribution Algebras on padic Groups and Lie Algebras Moy, Allen
When $F$ is a $p$adic field, and $G={\mathbb
G}(F)$ is the group of $F$rational points of a connected algebraic
$F$group, the complex vector space ${\mathcal H}(G)$ of compactly
supported locally constant distributions on $G$ has a natural
convolution product that makes it into a ${\mathbb C}$algebra
(without an identity) called the Hecke algebra. The Hecke algebra is a
partial analogue for $p$adic groups of the enveloping algebra of a
Lie group. However, $\mathcal{H}(G)$ has drawbacks such as the lack of
an identity element, and the process $G \mapsto \mathcal{H}(G)$
is not a functor. Bernstein introduced an enlargement
$\mathcal{H}\,\hat{\,}(G)$
of $\mathcal{H}(G)$. The algebra
$\mathcal{H}\,\hat{\,} (G)$ consists of the distributions that are left
essentially compact. We show that the process $G \mapsto
\mathcal{H}\,\hat{\,} (G)$ is a functor. If $\tau \colon G \rightarrow
H$ is a morphism of $p$adic groups, let $F(\tau) \colon
\mathcal{H}\,\hat{\,} (G) \rightarrow \mathcal{H}\,\hat{\,} (H)$ be
the morphism of $\mathbb{C}$algebras. We identify the kernel of
$F(\tau)$ in terms of $\textrm{Ker}(\tau)$. In the setting of $p$adic
Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra,
$\mathfrak{m}$ a Levi, and $\tau \colon \mathfrak{g} \to \mathfrak{m}$ the
natural projection, we show that $F(\tau)$ maps $G$invariant distributions
on $\mathcal{G}$ to $N_G (\mathfrak{m})$invariant distributions on
$\mathfrak{m}$. Finally, we exhibit a natural family of $G$invariant
essentially compact distributions on $\mathfrak{g}$ associated with a
$G$invariant nondegenerate symmetric bilinear form on ${\mathfrak g}$
and in the case of $SL(2)$ show how certain members of the family can
be moved to the group.


1161  Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group Neuwirth, Stefan; Ricard, Éric
We inspect the relationship between relative Fourier
multipliers on noncommutative LebesgueOrlicz spaces of a discrete
group $\varGamma$ and relative ToeplitzSchur multipliers on
SchattenvonNeumannOrlicz classes. Four applications are given:
lacunary sets, unconditional Schauder bases for the subspace of a
Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the
norm of the Hilbert transform and the Riesz projection on
SchattenvonNeumann classes with exponent a power of 2, and the norm of
Toeplitz Schur multipliers on SchattenvonNeumann classes with
exponent less than 1.


1188  On Complemented Subspaces of NonArchimedean Power Series Spaces Śliwa, Wiesław; Ziemkowska, Agnieszka
The nonarchimedean power series spaces, $A_1(a)$ and $A_\infty(b)$, are the
best known and most important examples of nonarchimedean nuclear Fréchet spaces.
We prove that the range of every continuous linear map from $A_p(a)$ to $A_q(b)$
has a Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{b,a}$ of all
bounded limit points of the double sequence
$(b_i/a_j)_{i,j\in\mathbb{N}}$ is bounded. It
follows that every complemented subspace of a power series space $A_p(a)$ has a
Schauder basis if either $p=1$ or $p=\infty$ and the set $M_{a,a}$ is bounded.

