11,619 research outputs found
Compactification of Drinfeld modular varieties and Drinfeld Modular Forms of Arbitrary Rank
We give an abstract characterization of the Satake compactification of a
general Drinfeld modular variety. We prove that it exists and is unique up to
unique isomorphism, though we do not give an explicit stratification by
Drinfeld modular varieties of smaller rank which is also expected. We construct
a natural ample invertible sheaf on it, such that the global sections of its
-th power form the space of (algebraic) Drinfeld modular forms of weight
. We show how the Satake compactification and modular forms behave under all
natural morphisms between Drinfeld modular varieties; in particular we define
Hecke operators. We give explicit results in some special cases
A Novel Long Range Spin Chain and Planar N=4 Super Yang-Mills
We probe the long-range spin chain approach to planar N=4 gauge theory at
high loop order. A recently employed hyperbolic spin chain invented by
Inozemtsev is suitable for the SU(2) subsector of the state space up to three
loops, but ceases to exhibit the conjectured thermodynamic scaling properties
at higher orders. We indicate how this may be bypassed while nevertheless
preserving integrability, and suggest the corresponding all-loop asymptotic
Bethe ansatz. We also propose the local part of the all-loop gauge transfer
matrix, leading to conjectures for the asymptotically exact formulae for all
local commuting charges. The ansatz is finally shown to be related to a
standard inhomogeneous spin chain. A comparison of our ansatz to semi-classical
string theory uncovers a detailed, non-perturbative agreement between the
corresponding expressions for the infinite tower of local charge densities.
However, the respective Bethe equations differ slightly, and we end by refining
and elaborating a previously proposed possible explanation for this
disagreement.Comment: 48 pages, 1 figure. v2, further results added: discussion of the
relationship to an inhomogeneous spin chain, normalization in sec 3 unified,
v3: minor mistakes corrected, published versio
Holomorphic automorphic forms and cohomology
We investigate the correspondence between holomorphic automorphic forms on
the upper half-plane with complex weight and parabolic cocycles. For integral
weights at least 2 this correspondence is given by the Eichler integral. Knopp
generalized this to real weights. We show that for weights that are not an
integer at least 2 the generalized Eichler integral gives an injection into the
first cohomology group with values in a module of holomorphic functions, and
characterize the image. We impose no condition on the growth of the automorphic
forms at the cusps.
For real weights that are not an integer at least 2 we similarly characterize
the space of cusp forms and the space of entire automorphic forms. We give a
relation between the cohomology classes attached to holomorphic automorphic
forms of real weight and the existence of harmonic lifts.
A tool in establishing these results is the relation to cohomology groups
with values in modules of "analytic boundary germs", which are represented by
harmonic functions on subsets of the upper half-plane. Even for positive
integral weights cohomology with these coefficients can distinguish all
holomorphic automorphic forms, unlike the classical Eichler theory.Comment: 150 pages. An earlier version appeared as an Oberwolfach Preprint
(OWP 2014-07
The p-adic L-functions of Evil Eisenstein Series
We compute the -adic -functions of evil Eisenstein series, showing that
they factor as products of two Kubota--Leopoldt -adic -functions times a
logarithmic term. This proves in particular a conjecture of Glenn Stevens.Comment: 49 page
Charge density and electric charge in quantum electrodynamics
The convergence of integrals over charge densities is discussed in relation
with the problem of electric charge and (non-local) charged states in Quantum
Electrodynamics (QED). Delicate, but physically relevant, mathematical points
like the domain dependence of local charges as quadratic forms and the time
smearing needed for strong convergence of integrals of charge densities are
analyzed. The results are applied to QED and the choice of time smearing is
shown to be crucial for the removal of vacuum polarization effects responible
for the time dependence of the charge (Swieca phenomenon). The possibility of
constructing physical charged states in the Feynman-Gupta-Bleuler gauge as
limits of local states vectors is discussed, compatibly with the vanishing of
the Gauss charge on local states. A modification by a gauge term of the Dirac
exponential factor which yields the physical Coulomb fields from the
Feynman-Gupta-Bleuler fields is shown to remove the infrared divergence of
scalar products of local and physical charged states, allowing for a
construction of physical charged fields with well defined correlation functions
with local fields
Global estimates for nonlinear parabolic equations
We consider nonlinear parabolic equations of the type under standard growth
conditions on , with only assumed to be integrable. We prove general
decay estimates up to the boundary for level sets of the solutions and the
gradient which imply very general estimates in Lebesgue and Lorentz
spaces. Assuming only that the involved domains satisfy a mild exterior
capacity density condition, we provide global regularity results.Comment: To appear in J. Evol. Equation
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