2,120 research outputs found
Isogeny graphs of ordinary abelian varieties
Fix a prime number . Graphs of isogenies of degree a power of
are well-understood for elliptic curves, but not for higher-dimensional abelian
varieties. We study the case of absolutely simple ordinary abelian varieties
over a finite field. We analyse graphs of so-called -isogenies,
resolving that they are (almost) volcanoes in any dimension. Specializing to
the case of principally polarizable abelian surfaces, we then exploit this
structure to describe graphs of a particular class of isogenies known as
-isogenies: those whose kernels are maximal isotropic subgroups
of the -torsion for the Weil pairing. We use these two results to write
an algorithm giving a path of computable isogenies from an arbitrary absolutely
simple ordinary abelian surface towards one with maximal endomorphism ring,
which has immediate consequences for the CM-method in genus 2, for computing
explicit isogenies, and for the random self-reducibility of the discrete
logarithm problem in genus 2 cryptography.Comment: 36 pages, 4 figure
On the Pauli graphs of N-qudits
A comprehensive graph theoretical and finite geometrical study of the
commutation relations between the generalized Pauli operators of N-qudits is
performed in which vertices/points correspond to the operators and edges/lines
join commuting pairs of them. As per two-qubits, all basic properties and
partitionings of the corresponding Pauli graph are embodied in the geometry of
the generalized quadrangle of order two. Here, one identifies the operators
with the points of the quadrangle and groups of maximally commuting subsets of
the operators with the lines of the quadrangle. The three basic partitionings
are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part
and (c) a maximum independent set and the Petersen graph. These factorizations
stem naturally from the existence of three distinct geometric hyperplanes of
the quadrangle, namely a set of points collinear with a given point, a grid and
an ovoid, which answer to three distinguished subsets of the Pauli graph,
namely a set of six operators commuting with a given one, a Mermin's square,
and set of five mutually non-commuting operators, respectively. The generalized
Pauli graph for multiple qubits is found to follow from symplectic polar spaces
of order two, where maximal totally isotropic subspaces stand for maximal
subsets of mutually commuting operators. The substructure of the (strongly
regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic
to a graph of a partial geometry. Finally, the (not strongly regular) Pauli
graph of a two-qutrit system is introduced; here it turns out more convenient
to deal with its dual in order to see all the parallels with the two-qubit case
and its surmised relation with the generalized quadrangle Q(4, 3), the dual
ofW(3).Comment: 17 pages. Expanded section on two-qutrits, Quantum Information and
Computation (2007) accept\'
Entanglement frustration for Gaussian states on symmetric graphs
We investigate the entanglement properties of multi-mode Gaussian states,
which have some symmetry with respect to the ordering of the modes. We show how
the symmetry constraints the entanglement between two modes of the system. In
particular, we determine the maximal entanglement of formation that can be
achieved in symmetric graphs like chains, 2d and 3d lattices, mean field models
and the platonic solids. The maximal entanglement is always attained for the
ground state of a particular quadratic Hamiltonian. The latter thus yields the
maximal entanglement among all quadratic Hamiltonians having the considered
symmetry.Comment: 5 pages, 1 figur
Representation Growth and Rational Singularities of the Moduli Space of Local Systems
We relate the asymptotic representation theory of and
the singularities of the moduli space of -local systems on a smooth
projective curve, proving new theorems about both. Regarding the former, we
prove that, for every d, the number of n-dimensional representations of
grows slower than , confirming a conjecture of
Larsen and Lubotzky. Regarding the latter, we prove that the moduli space of
-local systems on a smooth projective curve of genus at least 12 has
rational singularities. Most of our results apply more generally to semi-simple
algebraic groups.
For the proof, we study the analytic properties of push forwards of smooth
measures under algebraic maps. More precisely, we show that such push forwards
have continuous density if the algebraic map is flat and all of its fibers have
rational singularities.Comment: preliminary version, comments are welcome. v2. Revised version, now
covering all semi simple group
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