1,802 research outputs found
Independence Structures on the Submodules of a Module
Two definitions of dimension of a module are each shown to be the rank of an independence structure on a certain set of submodules of the module. This applies to Varadarajan's dual Goldie dimension and to Fleury's spanning dimension; the dualization of the latter is also discussed
Cokernels of random matrices satisfy the Cohen-Lenstra heuristics
Let A be an n by n random matrix with iid entries taken from the p-adic
integers or Z/NZ. Then under mild non-degeneracy conditions the cokernel of A
has a universal probability distribution. In particular, the p-part of an iid
random matrix over the integers has cokernel distributed according to the
Cohen-Lenstra measure up to an exponentially small error.Comment: 21 pages; submitte
Galois module structure of Galois cohomology for embeddable cyclic extensions of degree p^n
Let p>2 be prime, and let n,m be positive integers. For cyclic field
extensions E/F of degree p^n that contain a primitive pth root of unity, we
show that the associated F_p[Gal(E/F)]-modules H^m(G_E,mu_p) have a sparse
decomposition. When E/F is additionally a subextension of a cyclic, degree
p^{n+1} extension E'/F, we give a more refined F_p[Gal(E/F)]-decomposition of
H^m(G_E,mu_p)
An Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications
We fix data consisting of a Galois extension of
characteristic global fields with arbitrary abelian Galois group and a
Drinfeld module defined over a certain Dedekind subring of . For this
data, we define a -equivariant -function and prove an
equivariant Tamagawa number formula for certain Euler-completed versions of its
special value . This generalizes Taelman's class number
formula for the value of the Goss zeta function
associated to the pair . Taelman's result is obtained from our result
by setting . As a consequence, we prove a perfect Drinfeld module analogue
of the classical (number field) refined Brumer--Stark conjecture, relating a
certain -Fitting ideal of Taelman's class group to the special
value in question
Extensions of differential representations of SL(2) and tori
Linear differential algebraic groups (LDAGs) measure differential algebraic
dependencies among solutions of linear differential and difference equations
with parameters, for which LDAGs are Galois groups. The differential
representation theory is a key to developing algorithms computing these groups.
In the rational representation theory of algebraic groups, one starts with
SL(2) and tori to develop the rest of the theory. In this paper, we give an
explicit description of differential representations of tori and differential
extensions of irreducible representation of SL(2). In these extensions, the two
irreducible representations can be non-isomorphic. This is in contrast to
differential representations of tori, which turn out to be direct sums of
isotypic representations.Comment: 21 pages; few misprints corrected; Lemma 4.6 adde
Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions
We show that noncommutative gauge theory in two dimensions is an exactly
solvable model. A cohomological formulation of gauge theory defined on the
noncommutative torus is used to show that its quantum partition function can be
written as a sum over contributions from classical solutions. We derive an
explicit formula for the partition function of Yang-Mills theory defined on a
projective module for arbitrary noncommutativity parameter \theta which is
manifestly invariant under gauge Morita equivalence. The energy observables are
shown to be smooth functions of \theta. The construction of noncommutative
instanton contributions to the path integral is described in some detail. In
general, there are infinitely many gauge inequivalent contributions of fixed
topological charge, along with a finite number of quantum fluctuations about
each instanton. The associated moduli spaces are combinations of symmetric
products of an ordinary two-torus whose orbifold singularities are not resolved
by noncommutativity. In particular, the weak coupling limit of the gauge theory
is independent of \theta and computes the symplectic volume of the moduli space
of constant curvature connections on the noncommutative torus.Comment: 52 pages LaTeX, 1 eps figure, uses espf. V2: References added and
repaired; V3: Typos corrected, some clarifying explanations added; version to
be published in Communications in Mathematical Physic
- …