518,717 research outputs found
Primitive geodesic lengths and (almost) arithmetic progressions
In this article, we investigate when the set of primitive geodesic lengths on
a Riemannian manifold have arbitrarily long arithmetic progressions. We prove
that in the space of negatively curved metrics, a metric having such arithmetic
progressions is quite rare. We introduce almost arithmetic progressions, a
coarsification of arithmetic progressions, and prove that every negatively
curved, closed Riemannian manifold has arbitrarily long almost arithmetic
progressions in its primitive length spectrum. Concerning genuine arithmetic
progressions, we prove that every non-compact, locally symmetric, arithmetic
manifold has arbitrarily long arithmetic progressions in its primitive length
spectrum. We end with a conjectural characterization of arithmeticity in terms
of arithmetic progressions in the primitive length spectrum. We also suggest an
approach to a well known spectral rigidity problem based on the scarcity of
manifolds with arithmetic progressions.Comment: v3: 23 pages. To appear in Publ. Ma
Zariski decompositions on arithmetic surfaces
In this paper, we establish the Zariski decompositions of arithmetic
R-divisors of continuous type on arithmetic surfaces and investigate several
properties. We also develop the general theory of arithmetic R-divisors on
arithmetic varieties.Comment: 81 pages. Rewrite several part
Modular subvarieties of arithmetic quotients of bounded symmetric domains
Arithmetic quotients are quotients of bounded symmetric domains by arithmetic
groups, and modular subvarieties of arithmetic quotients are themselves
arithmetic quotients of lower dimension which live on arithmetic quotients, by
an embedding induced from an inclusion of groups of hermitian type. We show the
existence of such modular subvarieties, drawing on earlier work of the author.
If is a fixed arithmetic subgroup, maximal in some sense, then we
introduce the notion of ``-integral symmetric'' subgroups, which in
turn defines a notion of ``integral modular subvarieties'', and we show that
there are finitely many such on an (isotropic, i.e, non-compact) arithmetic
variety.Comment: 48 pages, also available at http://www.mathematik.uni-kl.de/~wwwagag/
LaTeX (e-mail: [email protected]
On the concavity of the arithmetic volumes
In this note, we study the differentiability of the arithmetic volumes along
arithmetic R-divisors, and give some equality conditions for the
Brunn-Minkowski inequality for arithmetic volumes over the cone of nef and big
arithmetic R-divisors.Comment: 35 page
Abelian arithmetic Chern-Simons theory and arithmetic linking numbers
Following the method of Seifert surfaces in knot theory, we define arithmetic
linking numbers and height pairings of ideals using arithmetic duality
theorems, and compute them in terms of n-th power residue symbols. This
formalism leads to a precise arithmetic analogue of a 'path-integral formula'
for linking numbers
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