173 research outputs found
Denominators of Eisenstein cohomology classes for GL_2 over imaginary quadratic fields
We study the arithmetic of Eisenstein cohomology classes (in the sense of G.
Harder) for symmetric spaces associated to GL_2 over imaginary quadratic
fields. We prove in many cases a lower bound on their denominator in terms of a
special L-value of a Hecke character providing evidence for a conjecture of
Harder that the denominator is given by this L-value. We also prove under some
additional assumptions that the restriction of the classes to the boundary of
the Borel-Serre compactification of the spaces is integral. Such classes are
interesting for their use in congruences with cuspidal classes to prove
connections between the special L-value and the size of the Selmer group of the
Hecke character.Comment: 37 pages; strengthened integrality result (Proposition 16), corrected
statement of Theorem 3, and revised introductio
L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields
The goal of this paper is to explain how a simple but apparently new fact of
linear algebra together with the cohomological interpretation of L-functions
allows one to produce many examples of L-functions over function fields
vanishing to high order at the center point of their functional equation. The
main application is that for every prime p and every integer g>0 there are
absolutely simple abelian varieties of dimension g over Fp(t) for which the BSD
conjecture holds and which have arbitrarily large rank.Comment: To appear in Inventiones Mathematica
Comments on Condensates in Non-Supersymmetric Orbifold Field Theories
Non-supersymmetric orbifolds of N=1 super Yang-Mills theories are conjectured
to inherit properties from their supersymmetric parent. We examine this
conjecture by compactifying the Z_2 orbifold theories on a spatial circle of
radius R. We point out that when the orbifold theory lies in the weakly coupled
vacuum of its parent, fractional instantons do give rise to the conjectured
condensate of bi-fundamental fermions. Unfortunately, we show that quantum
effects render this vacuum unstable through the generation of twisted
operators. In the true vacuum state, no fermion condensate forms. Thus, in
contrast to super Yang-Mills, the compactified orbifold theory undergoes a
chiral phase transition as R is varied.Comment: 10 Pages. Added clarifying comments, computational steps and a nice
pretty pictur
On higher congruences between cusp forms and Eisenstein series
In this paper we present several finite families of congruences between cusp
forms and Eisenstein series of higher weights at powers of prime ideals. We
formulate a conjecture which describes properties of the prime ideals and their
relation to the weights. We check the validity of the conjecture on several
numerical examples.Comment: 20 page
Introduction to Arithmetic Mirror Symmetry
We describe how to find period integrals and Picard-Fuchs differential
equations for certain one-parameter families of Calabi-Yau manifolds. These
families can be seen as varieties over a finite field, in which case we show in
an explicit example that the number of points of a generic element can be given
in terms of p-adic period integrals. We also discuss several approaches to
finding zeta functions of mirror manifolds and their factorizations. These
notes are based on lectures given at the Fields Institute during the thematic
program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics
On some congruence properties of elliptic curves
In this paper, as a result of a theorem of Serre on congruence properties, a
complete solution is given for an open question (see the text) presented
recently by Kim, Koo and Park. Some further questions and results on similar
types of congruence properties of elliptic curves are also presented and
discussed.Comment: 11 pages, The title is changed. Thanks to a result of J.-P. Serre
from his letter on June 15, 2009 to the author, a complete solution for an
open question of Kim, Koo and Park is obtained in this fifth revised version.
Some related questions and results are also presented and discusse
The Tate conjecture for K3 surfaces over finite fields
Artin's conjecture states that supersingular K3 surfaces over finite fields
have Picard number 22. In this paper, we prove Artin's conjecture over fields
of characteristic p>3. This implies Tate's conjecture for K3 surfaces over
finite fields of characteristic p>3. Our results also yield the Tate conjecture
for divisors on certain holomorphic symplectic varieties over finite fields,
with some restrictions on the characteristic. As a consequence, we prove the
Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite
fields of characteristic p>3.Comment: 20 pages, minor changes. Theorem 4 is stated in greater generality,
but proofs don't change. Comments still welcom
Middle Convolution and Harnad Duality
We interpret the additive middle convolution operation in terms of the Harnad
duality, and as an application, generalize the operation to have a
multi-parameter and act on irregular singular systems.Comment: 50 pages; v2: Submitted version once revised according to referees'
comment
Sur la p-dimension des corps
Let A be an excellent integral henselian local noetherian ring, k its residue
field of characteristic p>0 and K its fraction field. Using an algebraization
technique introduced by the first named author, and the one-dimension case
already proved by Kazuya KATO, we prove the following formula: cd_p(K) = dim(A)
+ p-rank(k), if k is separably closed and K of characteristic zero. A similar
statement is valid without those assumptions on k and K
A Note on Domain Walls and the Parameter Space of N=1 Gauge Theories
We study the spectrum of BPS domain walls within the parameter space of N=1
U(N) gauge theories with adjoint matter and a cubic superpotential. Using a low
energy description obtained by compactifying the theory on R^3 x S^1, we
examine the wall spectrum by combining direct calculations at special points in
the parameter space with insight drawn from the leading order potential between
minimal walls, i.e those interpolating between adjacent vacua. We show that the
multiplicity of composite BPS walls -- as characterised by the CFIV index --
exhibits discontinuities on marginal stability curves within the parameter
space of the maximally confining branch. The structure of these marginal
stability curves for large N appears tied to certain singularities within the
matrix model description of the confining vacua.Comment: 33 pages, LaTeX, 6 eps figures; v2: references adde
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