5,346 research outputs found
Moduli interpretation of Eisenstein series
Let L >= 3. Using the moduli interpretation, we define certain elliptic
modular forms of level Gamma(L) over any field k where 6L is invertible and k
contains the Lth roots of unity. These forms generate a graded algebra R_L,
which, over C, is generated by the Eisenstein series of weight 1 on Gamma(L).
The main result of this article is that, when k=C, the ring R_L contains all
modular forms on Gamma(L) in weights >= 2. The proof combines algebraic and
analytic techniques, including the action of Hecke operators and nonvanishing
of L-functions. Our results give a systematic method to produce models for the
modular curve X(L) defined over the Lth cyclotomic field, using only exact
arithmetic in the L-torsion field of a single Q-rational elliptic curve E^0.Comment: 29 pages, amslatex. Version 6: corrected a sign misprint in equation
(4.6) (thanks to N. Mascot for pointing it out). Final accepted versio
Neutrino dispersion in external magnetic fields
We calculate the neutrino self-energy operator Sigma (p) in the presence of a
magnetic field B. In particular, we consider the weak-field limit e B <<
m_\ell^2, where m_\ell is the charged-lepton mass corresponding to the neutrino
flavor \nu_\ell, and we consider a "moderate field" m_\ell^2 << e B << m_W^2.
Our results differ substantially from the previous literature. For a moderate
field, we show that it is crucial to include the contributions from all Landau
levels of the intermediate charged lepton, not just the ground-state. For the
conditions of the early universe where the background medium consists of a
charge-symmetric plasma, the pure B-field contribution to the neutrino
dispersion relation is proportional to (e B)^2 and thus comparable to the
contribution of the magnetized plasma.Comment: 9 pages, 1 figure, revtex. Version to appear in Phys. Rev. D
(presentation improved, reference list revised, numerical error in Eq.(41)
corrected, conclusions unchanged
On the full, strongly exceptional collections on toric varieties with Picard number three
We investigate full strongly exceptional collections on smooth, com- plete
toric varieties. We obtain explicit results for a large family of varieties
with Picard number three, containing many of the families already known. We
also describe the relations between the collections and the split of the push
forward of the trivial line bundle by the toric Frobenius morphism
A JOURNEY FROM THE OCTONIONIC â„™2 TO A FAKE â„™2
We discover a family of surfaces of general type with K2 = 3 and pg = q = 0 as free C13 quotients of special linear cuts of the octonionic projective plane Oâ„™2. A special member of the family has 3 singularities of type A2, and is a quotient of a fake projective plane. We use the techniques of earlier work of Borisov and Fatighenti to define this fake projective plane by explicit equations in its bicanonical embedding
Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs
Bound states of the Hamiltonian describing a quantum particle living on three
dimensional straight strip of width are investigated. We impose the Neumann
boundary condition on the two concentric windows of the radii and
located on the opposite walls and the Dirichlet boundary condition on the
remaining part of the boundary of the strip. We prove that such a system
exhibits discrete eigenvalues below the essential spectrum for any .
When and tend to the infinity, the asymptotic of the eigenvalue is
derived. A comparative analysis with the one-window case reveals that due to
the additional possibility of the regulating energy spectrum the anticrossing
structure builds up as a function of the inner radius with its sharpness
increasing for the larger outer radius. Mathematical and physical
interpretation of the obtained results is presented; namely, it is derived that
the anticrossings are accompanied by the drastic changes of the wave function
localization. Parallels are drawn to the other structures exhibiting similar
phenomena; in particular, it is proved that, contrary to the two-dimensional
geometry, at the critical Neumann radii true bound states exist.Comment: 25 pages, 7 figure
- …