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1. CJM Online first
Faithfulness of Actions on Riemann-Roch Spaces Given a faithful action of a finite group $G$ on an algebraic
curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for
the induced action of~$G$ on the Riemann-Roch space~$H^0(X,\mathcal{O}_X(D))$
to be faithful, where $D$ is a $G$-invariant divisor on $X$ of
degree at least~$2g_X-2$. This leads to a concise answer to the
question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes
m})$ of global holomorphic polydifferentials of order $m$ is
faithful. If $X$ is hyperelliptic, we furthermore provide an
explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we
give applications in deformation theory and in coding theory
and we discuss the analogous problem for the action of~$G$ on
the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.
Keywords:faithful action, Riemann-Roch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology Categories:14H30, 30F30, 14L30, 14D15, 11R32 |
2. CJM 2008 (vol 60 pp. 208)
Constructing Galois Representations with Very Large Image Starting with a 2-dimensional mod $p$ Galois representation, we
construct a deformation to a power series ring in infinitely many
variables over the $p$-adics. The image of this representation is full
in the sense that it contains $\SL_2$ of this power series
ring. Furthermore, all ${\mathbb Z}_p$ specializations of this
deformation are potentially semistable at $p$.
Keywords:Galois representation, deformation Category:11f80 |
3. CJM 2006 (vol 58 pp. 64)
Multiplicity Results for Nonlinear Neumann Problems In this paper we study nonlinear elliptic problems of Neumann type driven by the
$p$-Laplac\-ian differential operator. We look for situations guaranteeing the existence
of multiple solutions. First we study problems which are strongly resonant at infinity
at the first (zero) eigenvalue. We prove five multiplicity results, four for problems
with nonsmooth potential and one for problems with a $C^1$-potential. In the last part,
for nonsmooth problems in which the potential eventually exhibits a strict
super-$p$-growth under a symmetry condition, we prove the existence of infinitely
many pairs of nontrivial solutions. Our approach is variational based on the critical
point theory for nonsmooth functionals. Also we present some results concerning the first
two elements of the spectrum of the negative $p$-Laplacian with Neumann boundary condition.
Keywords:Nonsmooth critical point theory, locally Lipschitz function,, Clarke subdifferential, Neumann problem, strong resonance,, second deformation theorem, nonsmooth symmetric mountain pass theorem,, $p$-Laplacian Categories:35J20, 35J60, 35J85 |