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481  Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields Breuer, Florian; Im, BoHae
Let $k$ be a global field, $\overline{k}$ a separable
closure of $k$, and $G_k$ the absolute Galois group
$\Gal(\overline{k}/k)$ of $\overline{k}$ over $k$. For every
$\sigma\in G_k$, let $\ks$ be the fixed subfield of $\overline{k}$
under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known
that the MordellWeil group $E(\ks)$ has infinite rank. We present a
new proof of this fact in the following two cases. First, when $k$
is a global function field of odd characteristic and $E$ is
parametrized by a Drinfeld modular curve, and secondly when $k$ is a
totally real number field and $E/k$ is parametrized by a Shimura
curve. In both cases our approach uses the nontriviality of a
sequence of Heegner points on $E$ defined over ring class fields.


491  A MultiFrey Approach to Some MultiParameter Families of Diophantine Equations Bugeaud, Yann; Mignotte, Maurice; Siksek, Samir
We solve several multiparameter families of binomial Thue equations of arbitrary
degree; for example, we solve the equation
\[
5^u x^n2^r 3^s y^n= \pm 1,
\]
in nonzero integers $x$, $y$ and positive integers $u$, $r$, $s$ and $n \geq 3$.
Our approach uses several Frey curves simultaneously, Galois representations
and levellowering, new lower bounds for linear
forms in $3$ logarithms due to Mignotte and a famous theorem of Bennett on binomial
Thue equations.


520  Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences Chen, ChangPao; Huang, HaoWei; Shen, ChunYen
Let $A=(a_{j,k})_{j,k \ge 1}$ be a nonnegative matrix. In this
paper, we characterize those $A$ for which $\A\_{E, F}$ are
determined by their actions on decreasing sequences, where $E$ and
$F$ are suitable normed Riesz spaces of sequences. In particular,
our results can apply to the following spaces: $\ell_p$, $d(w,p)$,
and $\ell_p(w)$. The results established here generalize
ones given by Bennett; Chen, Luor, and Ou; Jameson; and
Jameson and Lashkaripour.


532  Local Bounds for Torsion Points on Abelian Varieties Clark, Pete L.; Xarles, Xavier
We say that an abelian variety over a $p$adic field $K$ has
anisotropic reduction (AR) if the special fiber of its N\'eron minimal
model does not contain a nontrivial split torus. This includes all
abelian varieties with potentially good reduction and, in particular,
those with complex or quaternionic multiplication. We give a bound for
the size of the $K$rational torsion subgroup of a $g$dimensional AR
variety depending only on $g$ and the numerical invariants of $K$ (the
absolute ramification index and the cardinality of the residue
field). Applying these bounds to abelian varieties over a number field
with everywhere locally anisotropic reduction, we get bounds which, as
a function of $g$, are close to optimal. In particular, we determine
the possible cardinalities of the torsion subgroup of an AR abelian
surface over the rational numbers, up to a set of 11 values which are
not known to occur. The largest such value is 72.


556  Polarization of Separating Invariants Draisma, Jan; Kemper, Gregor; Wehlau, David
We prove a characteristic free version of Weyl's theorem on
polarization. Our result is an exact analogue of Weyl's theorem, the
difference being that our statement is about separating invariants
rather than generating invariants. For the special case of finite
group actions we introduce the concept of cheap polarization,
and show that it is enough to take cheap polarizations of invariants
of just one copy of a representation to obtain separating vector
invariants for any number of copies. This leads to upper bounds on
the number and degrees of separating vector invariants of finite
groups.


572  NonSelfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point Hitrik, Michael; Sj{östrand, Johannes
This is the third in a series of works devoted to spectral
asymptotics for nonselfadjoint
perturbations of selfadjoint $h$pseudodifferential operators in dimension 2, having a
periodic classical flow. Assuming that the strength $\epsilon$
of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$
(and may sometimes reach even smaller values), we
get an asymptotic description of the eigenvalues in rectangles
$[1/C,1/C]+i\epsilon [F_01/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point
value of the flow average of the leading perturbation.


658  Inverse Pressure Estimates and the Independence of Stable Dimension for NonInvertible Maps Mihailescu, Eugen; Urba\'nski, Mariusz
We study the case of an Axiom A holomorphic nondegenerate
(hence noninvertible) map $f\from\mathbb P^2
\mathbb C \to \mathbb P^2 \mathbb C$, where $\mathbb P^2 \mathbb C$
stands for the complex
projective space of dimension 2. Let $\Lambda$ denote a basic set for
$f$ of unstable index 1, and $x$ an arbitrary point of $\Lambda$; we
denote by $\delta^s(x)$ the Hausdorff dimension of $W^s_r(x) \cap
\Lambda$, where $r$ is some fixed positive number and $W^s_r(x)$ is
the local stable manifold at $x$ of size $r$; $\delta^s(x)$ is called
the stable dimension at $x$. Mihailescu and
Urba\'nski introduced a notion of inverse topological pressure,
denoted by $P^$, which takes into consideration preimages of points.
Manning and McCluskey study the case of hyperbolic diffeomorphisms on
real surfaces and give formulas for Hausdorff dimension. Our
noninvertible situation is different here since the local unstable
manifolds are not uniquely determined by their base point, instead
they depend in general on whole prehistories of the base points. Hence
our methods are different and are based on using a sequence of inverse
pressures for the iterates of $f$, in order to give upper and lower
estimates of the stable dimension. We obtain an estimate of the
oscillation of the stable dimension on $\Lambda$. When each point $x$
from $\Lambda$ has the same number $d'$ of preimages in $\Lambda$,
then we show that $\delta^s(x)$ is independent
of $x$; in fact $\delta^s(x)$ is shown to be equal in this case with
the unique zero of the map $t \to P(t\phi^s  \log d')$. We also
prove the Lipschitz continuity of the stable vector spaces over
$\Lambda$; this proof is again different than the one for
diffeomorphisms (however, the unstable distribution is not always
Lipschitz for conformal noninvertible maps). In the end we include
the corresponding results for a real conformal setting.


685  Closed and Exact Functions in the Context of GinzburgLandau Models Savu, Anamaria
For a general vector field we exhibit two Hilbert spaces, namely
the space of so called closed functions and the space of exact functions
and we calculate the codimension of the space of exact functions
inside the larger space of closed functions.
In particular we provide a new approach for the known cases:
the Glauber field and the secondorder GinzburgLandau field
and for the case of the fourthorder GinzburgLandau field.


703  $\mathcal{Z}$Stable ASH Algebras Toms, Andrew S.; Winter, Wilhelm
The JiangSu algebra $\mathcal{Z}$ has come to prominence in the
classification program for nuclear $C^*$algebras of late, due
primarily to the fact that Elliott's classification conjecture in its
strongest form predicts that all simple, separable, and nuclear
$C^*$algebras with unperforated $\mathrm{K}$theory will absorb
$\mathcal{Z}$ tensorially, i.e., will be $\mathcal{Z}$stable. There
exist counterexamples which suggest that the conjecture will only hold
for simple, nuclear, separable and $\mathcal{Z}$stable
$C^*$algebras. We prove that virtually all classes of nuclear
$C^*$algebras for which the Elliott conjecture has been confirmed so
far consist of $\mathcal{Z}$stable $C^*$algebras. This
follows in large part from the following result, also proved herein:
separable and approximately divisible $C^*$algebras are
$\mathcal{Z}$stable.

