5,240 research outputs found
Infinitely -divisible points on abelian varieties defined over function fields of characteristic
In this article we consider some questions raised by F. Benoist, E. Bouscaren
and A. Pillay. We prove that infinitely -divisible points on abelian
varieties defined over function fields of transcendence degree one over a
finite field are necessarily torsion points. We also prove that when the
endomorphism ring of the abelian variety is \mZ then there are no infinitely
-divisible points of order a power of
Strongly semistable sheaves and the Mordell-Lang conjecture over function fields
We give a new proof of the Mordell-Lang conjecture in positive
characteristic, in the situation where the variety under scrutiny is a smooth
subvariety of an abelian variety. Our proof is based on the theory of
semistable sheaves in positive characteristic, in particular on Langer's
theorem that the Harder-Narasimhan filtration of sheaves becomes strongly
semistable after a finite number of iterations of Frobenius pull-backs. The
interest of this proof is that it provides simple effective bounds (depending
on the degree of the canonical line bundle) for the degree of the isotrivial
finite cover whose existence is predicted by the Mordell-Lang conjecture. We
also present a conjecture on the Harder-Narasimhan filtration of the cotangent
bundle of a smooth projective variety of general type in positive
characteristic and a conjectural refinement of the Bombieri-Lang conjecture in
positive characteristic
Conjectures on the logarithmic derivatives of Artin L-functions II
We formulate a general conjecture relating Chern classes of subbundles of
Gauss-Manin bundles in Arakelov geometry to logarithmic derivatives of Artin
L-functions of number fields. This conjecture may be viewed as a far-reaching
generalisation of the (Lerch-)Chowla-Selberg formula computing logarithms of
periods of elliptic curves in terms of special values of the -function.
We prove several special cases of this conjecture in the situation where the
involved Artin characters are Dirichlet characters. This article contains the
computations promised in the article {\it Conjectures sur les d\'eriv\'ees
logarithmiques des fonctions L d'Artin aux entiers n\'egatifs}, where our
conjecture was announced. We also give a quick introduction to the
Grothendieck-Riemann-Roch theorem and to the geometric fixed point formula,
which form the geometric backbone of our conjecture.Comment: 54 page
On the determinant bundles of abelian schemes
Let \pi:\CA\ra S be an abelian scheme over a scheme which is
quasi-projective over an affine noetherian scheme and let \CL be a symmetric,
rigidified, relatively ample line bundle on \CA. We show that there is an
isomorphism
\det(\pi_*\CL)^{\o times 24}\simeq\big(\pi_*\omega_{\CA}^{\vee}\big)^{\o
times 12d}
of line bundles on , where is the rank of the (locally free) sheaf
\pi_*\CL. We also show that the numbers 24 and are sharp in the
following sense: if is a common divisor of 12 and 24, then there are data
as above such that
\det(\pi_*\CL)^{\o times
(24/N)}\not\simeq\big(\pi_*\omega_{\CA}^{\vee}\big)^{\o times (12d/N)}.Comment: 8 page
On a canonical class of Green currents for the unit sections of abelian schemes
We show that on any abelian scheme over a complex quasi-projective smooth
variety, there is a Green current for the zero-section, which is axiomatically
determined up to and -exact differential forms. This
current generalizes the Siegel functions defined on elliptic curves. We prove
generalizations of classical properties of Siegel functions, like distribution
relations, limit formulae and reciprocity laws.Comment: 42 page
Finite Volume at Two-loops in Chiral Perturbation Theory
We calculate the finite volume corrections to meson masses and decay
constants in two and three flavour Chiral Perturbation Theory to two-loop
order. The analytical results are compared with the existing result for the
pion mass in two-flavour ChPT and the partial results for the other quantities.
We present numerical results for all quantities.Comment: 26 pages, a number of minor misprints correcte
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