Ill-distributed sets over global fields and exceptional sets in Diophantine Geometry

Let $K\subseteq \mathbb{R}$ be a number field. Using techniques of discrete analysis, we prove that for definable sets $X$ in $\mathbb{R}_{\exp}$ of dimension at most $2$ a conjecture of Wilkie about the density of rational points is equivalent to the fact that $X$ is badly distributed at the level of residue classes for many primes of $K$. This provides a new strategy to prove this conjecture of Wilkie. In order to prove this result, we are lead to study an inverse problem as in the works \cite{Walsh2, Walsh}, but in the context of number fields, or more generally global fields. Specifically, we prove that if $K$ is a global field, then every subset $S\subseteq \mathbb{P}^{n}(K)$ consisting of rational points of projective height bounded by $N$, occupying few residue classes modulo $\mathfrak{p}$ for many primes $\mathfrak{p}$ of $K$, must essentially lie in the solution set of a polynomial equation of degree $\ll (\log(N))^{C}$, for some constant $C$

Inverse Compton e-p pair cascade model for the gamma-ray production in massive binary LSI +61^o 303

We apply an inverse Compton $e^\pm$ pair cascade model for $\gamma$-ray production in massive binary system LSI +61$^{\rm o}$ 303 assuming that electrons are accelerated already inside the inner part of the jet launched by the compact object. $\gamma$-ray spectra, affected by the cascade process, and lower energy spectra, from the synchrotron cooling of the highest energy electrons in the jet, are calculated as a function of the phase of this binary system. $\gamma$-ray spectra expected in such model have different shape than that ones produced by electrons in the jet directly to observer. Moreover, the model predicts clear anti-correlation between $\gamma$-ray fluxes in the GeV (1-10 GeV) and TeV ($>200$ GeV) energy ranges with the peak of the TeV emission at the phase $\sim$0.5 (the peak half width ranges between the phases $\sim$0.4-0.9 for the inclination of the binary system equal to $60^{\rm o}$, and $\sim$0.4-0.1 for $30^{\rm o}$). The fine features of TeV $\gamma$-ray emission (fluxes and spectral shapes) as a function of the phase of the binary system are consistent with recent observations reported by the MAGIC collaboration. Future simultaneous observations in the GeV energies (by the GLAST and AGILE telescopes) and in the TeV energies (by the MAGIC and VERITAS telescopes) should test other predictions of the considered model supporting or disproving the hypothersis of acceleration of electrons already in the inner part of the microquasar jets.Comment: 8 pages, 4 figures, version accepted to MNRA

Microquasars as high-energy gamma-ray sources

Galactic microquasars are certainly one of the most recent additions to the field of high energy astrophysics and have attracted increasing interest over the last decade. However, the high energy part of the spectrum of microquasars is the most poorly known, mainly due the lack of sensitive instrumentation in the past. Microquasars are now primary targets for all of the observatories working in the X-ray and gamma-ray domains. They also appear as the possible counterparts for some of the unidentified sources of high-energy gamma-rays detected by the experiment EGRET on board the satellite COMPTON-GRO. This paper provides a general review of the main observational results obtained up to now as well as a summary of the scenarios for production of high-energy gamma-rays at the present moment.Comment: Invited talk presented at the V Microquasar Workshop, Beijing, June 2004. Accepted for publication in the Chinese Journal of Astronomy and Astrophysics. 14 pages, 9 figure

High energy processes in microquasars

Microquasars are X-ray binary stars with the capability to generate relativisticjets. It is expected that microquasars are gamma-ray sources, because of the analogy with quasars and because the theoretical models predict emission at such energy range. In addition, from observational arguments, there are two microquasars that appear as the possible counterparts for two unidentified high-energy gamma-ray sources.Comment: Universitat de Barcelona, Departament d'Astronomia i Meteorologia, 12 pages, 5 figures. Invited talk presented at the International Symposium "High-Energy Gamma-Ray Astronomy", 26-30 July 2004, Heidelberg (Germany). To be published by AIP Proceedings Serie