Quantum Drinfeld Modules II: Quantum Exponential and Ray Class Fields

This is the second in a series of two papers presenting a solution to Manin's Real Multiplication program \cite{Man} in positive characteristic. If $K$ is a quadratic and real extension of $\mathbb{F}_{q}(T)$ and $\mathcal{O}_{K}$ is the integral closure of $\mathbb{F}_{q}[T]$ in $K$, we associate to each modulus $\mathfrak{M}\subset \mathcal{O}_{K}$ the {\it unit narrow ray class field} $K^{\mathfrak{M}}$: a class field containing the narrow ray class field, whose class group contains an additional contribution coming from $\mathcal{O}^{\times}_{K}$. For $f\in K$ a fundamental unit, we introduce the associated {\it quantum Drinfeld module} $\rho^{\rm qt}_{f}$ of $f$: a generalization of Drinfeld module whose elements are multi-points. The main theorem of the paper is that $$K^{\mathfrak{M}}=H_{\mathcal{O}_{K}} ( {\sf Tr}(\rho^{\rm qt}_{f}[\mathfrak{M}]), {\sf Tr}(\rho^{\rm qt}_{f^{-1}}[\mathfrak{M}]))$$ where $H_{\mathcal{O}_{K}}$ is the Hilbert class field of $\mathcal{O}_{K}$ and ${\sf Tr}(\rho^{\rm qt}_{f}[\mathfrak{M}])$, ${\sf Tr}(\rho^{\rm qt}_{f^{-1}}[\mathfrak{M}])$ are the groups of traces of $\mathfrak{M}$ torsion points of $\rho^{\rm qt}_{f}$, $\rho^{\rm qt}_{f^{-1}}$.Comment: 41 page

Modular Invariant of Rank 1 Drinfeld Modules and Class Field Generation

The modular invariant of rank 1 Drinfeld modules is introduced and used to formulate and prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is a version of Shimura's Main Theorem of Complex Multiplication for global function fields, which is also proved here.Comment: 20 page

A probable giant planet imaged in the Beta Pictoris disk

Since the discovery of its dusty disk in 1984, Beta Pictoris has become the prototype of young early-type planetary systems, and there are now various indications that a massive Jovian planet is orbiting the star at ~ 10 AU. However, no planets have been detected around this star so far. Our goal was to investigate the close environment of Beta Pic, searching for planetary companion(s). Deep adaptive-optics L'-band images of Beta Pic were recorded using the NaCo instrument at the Very Large Telescope. A faint point-like signal is detected at a projected distance of ~ 8 AU from the star, within the North-East side of the dust disk. Various tests were made to rule out with a good confidence level possible instrumental or atmospheric artifacts. The probability of a foreground or background contaminant is extremely low, based in addition on the analysis of previous deep Hubble Space Telescope images. The object L'=11.2 apparent magnitude would indicate a typical temperature of ~1500 K and a mass of ~ 8 Jovian masses. If confirmed, it could explain the main morphological and dynamical peculiarities of the Beta Pic system. The present detection is unique among A-stars by the proximity of the resolved planet to its parent star. Its closeness and location inside the Beta Pic disk suggest a formation process by core accretion or disk instabilities rather than a binary-like formation process.Comment: 5 pages, 3 figures, 1 table. A&A Letters, in pres

LP 714-37: A wide pair of ultracool dwarfs actually is a triple

LP 714-37 was identified by Phan-Bao et al. (2005) as one of the very few wide pairs of very low mass (VLM) stars known to date, with a separation of 33 AU. Here we present adaptive optics imaging which resolves the secondary of the wide pair into a tighter binary, with a projected angular separation of 0.36 arcsec, or 7 AU. The estimated spectral types of LP 714-37B and LP 714-37C are M8.0 and M8.5. We discuss the implications of this finding for brown dwarf formation scenarios.Comment: Accepted by ApJ Letter

The FALCON concept: multi-object spectroscopy combined with MCAO in near-IR

A large fraction of the present-day stellar mass was formed between z=0.5 and z~3 and our understanding of the formation mechanisms at work at these epochs requires both high spatial and high spectral resolution: one shall simultaneously} obtain images of objects with typical sizes as small as 1-2kpc(~0''.1), while achieving 20-50 km/s (R >= 5000) spectral resolution. The obvious instrumental solution to adopt in order to tackle the science goal is therefore a combination of multi-object 3D spectrograph with multi-conjugate adaptive optics in large fields. A partial, but still competitive correction shall be prefered, over a much wider field of view. This can be done by estimating the turbulent volume from sets of natural guide stars, by optimizing the correction to several and discrete small areas of few arcsec2 selected in a large field (Nasmyth field of 25 arcmin) and by correcting up to the 6th, and eventually, up to the 60th Zernike modes. Simulations on real extragalactic fields, show that for most sources (>80%), the recovered resolution could reach 0".15-0".25 in the J and H bands. Detection of point-like objects is improved by factors from 3 to >10, when compared with an instrument without adaptive correction. The proposed instrument concept, FALCON, is equiped with deployable mini-integral field units (IFUs), achieving spectral resolutions between R=5000 and 20000. Its multiplex capability, combined with high spatial and spectral resolution characteristics, is a natural ground based complement to the next generation of space telescopes.Comment: ESO Workshop Proceedings: Scientific Drivers for ESO Future VLT/VLTI Instrumentation, 10 pages and 5 figure

Adaptive Optics for Astronomy

Adaptive Optics is a prime example of how progress in observational astronomy can be driven by technological developments. At many observatories it is now considered to be part of a standard instrumentation suite, enabling ground-based telescopes to reach the diffraction limit and thus providing spatial resolution superior to that achievable from space with current or planned satellites. In this review we consider adaptive optics from the astrophysical perspective. We show that adaptive optics has led to important advances in our understanding of a multitude of astrophysical processes, and describe how the requirements from science applications are now driving the development of the next generation of novel adaptive optics techniques.Comment: to appear in ARA&A vol 50, 201