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241  On Rational Equivalence in Tropical Geometry Allermann, Lars; Hampe, Simon; Rau, Johannes
This article discusses the concept of rational equivalence
in tropical
geometry
(and replaces an older and imperfect version).
We give the basic definitions in the context of tropical varieties
without boundary points and prove some basic properties.
We then compute the ``bounded'' Chow groups of $\mathbb{R}^n$ by showing
that they are isomorphic
to the group of fan cycles. The main step in the proof is of
independent interest:
We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated)
fan cycles. This also
proves that the intersection ring of tropical cycles is generated
in codimension 1 (by hypersurfaces).


258  Equivariant Map Queer Lie Superalgebras Calixto, Lucas; Moura, Adriano; Savage, Alistair
An equivariant map queer Lie superalgebra is the Lie superalgebra
of regular maps from an algebraic variety (or scheme) $X$ to
a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect
to the action of a finite group $\Gamma$ acting on $X$ and $\mathfrak{q}$.
In this paper, we classify all irreducible finitedimensional
representations of the equivariant map queer Lie superalgebras
under the assumption that $\Gamma$ is abelian and acts freely
on $X$. We show that such representations are parameterized
by a certain set of $\Gamma$equivariant finitely supported maps
from $X$ to the set of isomorphism classes of irreducible finitedimensional
representations of $\mathfrak{q}$. In the special case where $X$ is the
torus, we obtain a classification of the irreducible finitedimensional
representations of the twisted loop queer superalgebra.


280  Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising from Mirror Symmetry and Middle Convolution da Silva, Genival Jr.; Kerr, Matt; Pearlstein, Gregory
We collect evidence in support of a conjecture of Griffiths,
Green
and Kerr
on the arithmetic of extension classes of
limiting
mixed Hodge structures arising from semistable degenerations
over
a number field. After briefly summarizing how a result of Iritani
implies this conjecture for a collection of hypergeometric
CalabiYau threefold examples studied by Doran and Morgan,
the authors investigate a sequence of (nonhypergeometric) examples
in dimensions $1\leq d\leq6$ arising from Katz's theory of the
middle
convolution.
A crucial role is played by the MumfordTate
group (which is $G_{2}$) of the family of 6folds, and the theory
of boundary components of MumfordTate domains.


309  Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups Daws, Matthew
We show that the assignment of the (left) completely bounded
multiplier algebra
$M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group
$\mathbb G$, and
the assignment of the intrinsic group, form functors between
appropriate
categories. Morphisms of locally compact quantum
groups can be described by Hopf $*$homomorphisms between universal
$C^*$algebras, by bicharacters, or by special sorts of coactions.
We show that the whole
theory of completely bounded multipliers can be lifted to the
universal
$C^*$algebra level, and that then the different pictures of
both multipliers
(reduced, universal, and as centralisers)
and morphisms interact in extremely natural ways. The intrinsic
group of a
quantum group can be realised as a class of multipliers, and
so our techniques
immediately apply. We also show how to think of the intrinsic
group using
the universal $C^*$algebra picture, and then, again, show how
the differing
views on the intrinsic group interact naturally with morphisms.
We show that
the intrinsic group is the ``maximal classical'' quantum subgroup
of a locally
compact quantum group, show that it is even closed in the strong
Vaes sense,
and that the intrinsic group functor is an adjoint to the inclusion
functor
from locally compact groups to quantum groups.


334  Kernels in the Category of Formal Group Laws Demchenko, Oleg; Gurevich, Alexander
Fontaine described the category of formal groups over the ring
of Witt vectors over a finite field
of characteristic $p$ with the aid of triples consisting of the
module of logarithms,
the Dieudonné module and the morphism from the former to the
latter. We propose
an explicit construction for the kernels in this category in
term of Fontaine's triples.
The construction is applied to the formal norm homomorphism in
the case of an unramified extension
of $\mathbb{Q}_p$ and of a totally ramified extension of degree less
or equal than $p$. A similar
consideration applied to a global extension allows us to establish
the existence of a strict
isomorphism between the formal norm torus and a formal group
law coming from $L$series.


361  Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent Fité, Francesc; González, Josep; Lario, Joan Carles
Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime
exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$
of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians
$\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell2$, where
$\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by
certain subgroups of automorphisms of $\mathcal{C}$. It is well known
that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely
simple abelian variety $B_k$ with complex multiplication. We call
degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM
type. For a nondegenerate pair $(\ell,k)$, we compute the SatoTate
group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized
SatoTate Conjecture for it, and give an explicit method to compute
the moments and measures of the involved distributions. Regardless of
$(\ell,k)$ being degenerate or not, we also obtain Frobenius
equidistribution results for primes of certain residue degrees in the
$\ell$th cyclotomic field. Key to our results is a detailed study of
the rank of certain generalized Demjanenko matrices.


395  Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups Garibaldi, Skip; Nakano, Daniel K.
The representation theory of semisimple algebraic groups over
the complex numbers (equivalently, semisimple complex Lie algebras
or Lie groups, or real compact Lie groups) and the question of
whether a
given complex representation is symplectic or orthogonal has
been solved since at least the 1950s. Similar results for Weyl
modules of split reductive groups over fields of characteristic
different from 2 hold by
using similar proofs. This paper considers analogues of these
results for simple, induced and tilting modules of split reductive
groups over fields of prime characteristic as well as a complete
answer for Weyl modules over fields of characteristic 2.


422  Heegner Points on Cartan Nonsplit Curves Kohen, Daniel; Pacetti, Ariel
Let $E/\mathbb{Q}$ be an elliptic curve of conductor
$N$, and
let $K$ be an imaginary quadratic field such that the root
number of
$E/K$ is $1$. Let $\mathscr{O}$ be an order in $K$ and assume that
there
exists an odd prime $p$, such that $p^2 \mid\mid N$, and $p$
is inert in
$\mathscr{O}$. Although there are no Heegner points on $X_0(N)$
attached to $\mathscr{O}$, in this article we construct such points on
Cartan nonsplit curves. In order to do that we
give a method to compute Fourier expansions for forms on Cartan
nonsplit curves, and prove that the constructed points form a
Heegner system as in the classical case.


445  Geometric Invariants of Cuspidal Edges Martins, Luciana de Fátima; Saji, Kentaro
We give a normal form of the cuspidal edge
which uses only diffeomorphisms on the source
and isometries on the target.
Using this normal form, we study differential
geometric invariants of
cuspidal edges which determine them up to order three.
We also
clarify relations between these invariants.


463  The Weak bprinciple: Mumford Conjecture Sadykov, Rustam
In this note we introduce and study a new class of maps called
oriented colored broken submersions. This is the simplest class
of maps that satisfies a version of the bprinciple and in dimension
$2$ approximates the class of oriented submersions well in the
sense that
every oriented colored broken submersion of dimension $2$ to
a closed simply connected manifold is bordant to a submersion.
We show that the MadsenWeiss theorem (the standard Mumford Conjecture)
fits a general setting of the bprinciple. Namely, a version
of the bprinciple for
oriented colored broken submersions together with the Harer
stability theorem and MillerMorita theorem implies the MadsenWeiss
theorem.

