Infinitely pp-divisible points on abelian varieties defined over function fields of characteristic p>0p>0

In this article we consider some questions raised by F. Benoist, E. Bouscaren and A. Pillay. We prove that infinitely $p$-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 $p$-divisible points of order a power of $p$

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 $\Gamma$-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 $S$ 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 $S$, where $d$ is the rank of the (locally free) sheaf \pi_*\CL. We also show that the numbers 24 and $12d$ are sharp in the following sense: if $N>1$ 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 $\partial$ and $\bar\partial$-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