3,187 research outputs found
On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields
Given any family of cubic fields defined by local conditions at finitely many
primes, we determine the mean number of 2-torsion elements in the class groups
and narrow class groups of these cubic fields when ordered by their absolute
discriminants.
For an order in a cubic field, we study the three groups: , the group of ideal classes of of order 2; , the group of narrow ideal classes of of order 2; and
, the group of ideals of of order 2. We prove that
the mean value of the difference is always equal to , whether one averages over the maximal orders in
real cubic fields, over all orders in real cubic fields, or indeed over any
family of real cubic orders defined by local conditions. For the narrow class
group, we prove that the mean value of the difference is equal to for any such family. For any family
of complex cubic orders defined by local conditions, we prove similarly that
the mean value of the difference is always equal to , independent of the family.
The determination of these mean numbers allows us to prove a number of
further results as by-products. Most notably, we prove---in stark contrast to
the case of quadratic fields---that: 1) a positive proportion of cubic fields
have odd class number; 2) a positive proportion of real cubic fields have
isomorphic 2-torsion in the class group and the narrow class group; and 3) a
positive proportion of real cubic fields contain units of mixed real signature.
We also show that a positive proportion of real cubic fields have narrow class
group strictly larger than the class group, and thus a positive proportion of
real cubic fields do not possess units of every possible real signature.Comment: 17 page
Computing special values of partial zeta functions
We discuss computation of the special values of partial zeta functions
associated to totally real number fields. The main tool is the \emph{Eisenstein
cocycle} , a group cocycle for ; the special values are
computed as periods of , and are expressed in terms of generalized
Dedekind sums. We conclude with some numerical examples for cubic and quartic
fields of small discriminant.Comment: 10 p
Electric field and exciton structure in CdSe nanocrystals
Quantum Stark effect in semiconductor nanocrystals is theoretically
investigated, using the effective mass formalism within a
Baldereschi-Lipari Hamiltonian model for the hole states. General expressions
are reported for the hole eigenfunctions at zero electric field. Electron and
hole single particle energies as functions of the electric field
() are reported. Stark shift and binding energy of the
excitonic levels are obtained by full diagonalization of the correlated
electron-hole Hamiltonian in presence of the external field. Particularly, the
structure of the lower excitonic states and their symmetry properties in CdSe
nanocrystals are studied. It is found that the dependence of the exciton
binding energy upon the applied field is strongly reduced for small quantum dot
radius. Optical selection rules for absorption and luminescence are obtained.
The electric-field induced quenching of the optical spectra as a function of
is studied in terms of the exciton dipole matrix element. It
is predicted that photoluminescence spectra present anomalous field dependence
of the emission lines. These results agree in magnitude with experimental
observation and with the main features of photoluminescence experiments in
nanostructures.Comment: 9 pages, 7 figures, 1 tabl
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