3,118 research outputs found
Rational Tate classes
In despair, as Deligne (2000) put it, of proving the Hodge and Tate
conjectures, we can try to find substitutes. For abelian varieties in
characteristic zero, Deligne (1982) constructed a theory of Hodge classes
having many of the properties that the algebraic classes would have if the
Hodge conjecture were known. In this article I investigate whether there exists
a theory of "rational Tate classes" on varieties over finite fields having the
properties that the algebraic classes would have if the Hodge and Tate
conjectures were known. v3. Submitted version
The Tate conjecture over finite fields (AIM talk)
These are my notes for a talk at the The Tate Conjecture workshop at the
American Institute of Mathematics in Palo Alto, CA, July 23--July 27, 2007,
somewhat revised and expanded. The intent of the talk was to review what is
known and to suggest directions for research. v2: Revised expanded (24 pages).Comment: A pdf file with better fonts, style file, etc. is available at
http://www.jmilne.org/math/ v2: Revised expanded (24 pages
Points on Shimura varieties over finite fields: the conjecture of Langlands and Rapoport
We state an improved version of the conjecture of Langlands and Rapoport, and
we prove the conjecture for a large class of Shimura varieties. In particular,
we obtain the first proof of the (original) conjecture for Shimura varieties of
PEL-type
On Perverse Equivalences and Rationality
We show that perverse equivalences between module categories of
finite-dimensional algebras preserve rationality. As an application, we give a
connection between some famous conjectures from the modular representation
theory of finite groups, namely Brou\'e's Abelian Defect Group conjecture and
Donovan's Finiteness conjectures
Rational conformal field theory with matrix level and strings on a torus
Study of the matrix-level affine algebra is motivated by conformal
field theory and the fractional quantum Hall effect. Gannon completed the
classification of modular-invariant partition functions. Here we
connect the algebra to strings on 2-tori describable by rational
conformal field theories. As Gukov and Vafa proved, rationality selects the
complex-multiplication tori. We point out that the rational conformal field
theories describing strings on complex-multiplication tori have characters and
partition functions identical to those of the matrix-level algebra .
This connection makes obvious that the rational theories are dense in the
moduli space of strings on , and may prove useful in other ways.Comment: 13 page
Rationality and Chow-Kuenneth decompositions for some moduli stacks of curves
In this paper, we show the existence of a Chow--Kuenneth decomposition for
the moduli stack of stable curves of genus g with r marked points, for low
values of g,r. We also look at the moduli space R of double covers of genus 3
curves, branched along 4 distinct points. We obtain a birational model of the
moduli space R as a group quotient of a product of two Grassmanian varieties.
This provides a Chow-Kuenneth decomposition over an open subset of R. The
question of rationality of R is also discussed.Comment: Some errors are corrected and new remarks are include
The integral Hodge conjecture for two-dimensional Calabi-Yau categories
We formulate a version of the integral Hodge conjecture for categories, prove
the conjecture for two-dimensional Calabi-Yau categories which are suitably
deformation equivalent to the derived category of a K3 or abelian surface, and
use this to deduce cases of the usual integral Hodge conjecture for varieties.
Along the way, we prove a version of the variational integral Hodge conjecture
for families of two-dimensional Calabi-Yau categories, as well as a general
smoothness result for relative moduli spaces of objects in such families. Our
machinery also has applications to the structure of intermediate Jacobians,
such as a criterion in terms of derived categories for when they split as a sum
of Jacobians of curves.Comment: 45 pages, minor update
Noether's problem and descent
We study Noether's problem from the perspective of torsors under linear
algebraic groups and descent.Comment: 14 page
Fibonacci Sequences And Real Quadratic p-Rational Fields
We study the p-rationality of real quadratic fields in terms of generalized
Fibonacci numbers and their periods modulo positive integers
Poisson algebras via model theory and differential-algebraic geometry
Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds
for any complex affine Poisson algebra; that is, whether the sets of Poisson
rational ideals, Poisson primitive ideals, and Poisson locally closed ideals
coincide. In this article a complete answer is given to this question using
techniques from differential-algebraic geometry and model theory. In
particular, it is shown that while the sets of Poisson rational and Poisson
primitive ideals do coincide, in every Krull dimension at least four there are
complex affine Poisson algebras with Poisson rational ideals that are not
Poisson locally closed. These counterexamples also give rise to counterexamples
to the classical (noncommutative) Dixmier-Moeglin equivalence in finite
dimension. A weaker version of the Poisson Dixmier-Moeglin
equivalence is proven for all complex affine Poisson algebras, from which it
follows that the full equivalence holds in Krull dimension three or less.
Finally, it is shown that everything, except possibly that rationality implies
primitivity, can be done over an arbitrary base field of characteristic zero.Comment: 28 pages; v2: minor changes; accepted for publication in Journal of
the European Mathematical Societ
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