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

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

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

GRAVITY: getting to the event horizon of Sgr A*

We present the second-generation VLTI instrument GRAVITY, which currently is in the preliminary design phase. GRAVITY is specifically designed to observe highly relativistic motions of matter close to the event horizon of Sgr A*, the massive black hole at center of the Milky Way. We have identified the key design features needed to achieve this goal and present the resulting instrument concept. It includes an integrated optics, 4-telescope, dual feed beam combiner operated in a cryogenic vessel; near infrared wavefront sensing adaptive optics; fringe tracking on secondary sources within the field of view of the VLTI and a novel metrology concept. Simulations show that the planned design matches the scientific needs; in particular that 10 microarcsecond astrometry is feasible for a source with a magnitude of K=15 like Sgr A*, given the availability of suitable phase reference sources.Comment: 13 pages, 11 figures, to appear in the conference proceedings of SPIE Astronomical Instrumentation, 23-28 June 2008, Marseille, Franc

The ADAMTS (A Disintegrin and Metalloproteinase with Thrombospondin motifs) family

The ADAMTS (A Disintegrin and Metalloproteinase with Thrombospondin motifs) enzymes are secreted, multi-domain matrix-associated zinc metalloendopeptidases that have diverse roles in tissue morphogenesis and patho-physiological remodeling, in inflammation and in vascular biology. The human family includes 19 members that can be sub-grouped on the basis of their known substrates, namely the aggrecanases or proteoglycanases (ADAMTS1, 4, 5, 8, 9, 15 and 20), the procollagen N-propeptidases (ADAMTS2, 3 and 14), the cartilage oligomeric matrix protein-cleaving enzymes (ADAMTS7 and 12), the von-Willebrand Factor proteinase (ADAMTS13) and a group of orphan enzymes (ADAMTS6, 10, 16, 17, 18 and 19). Control of the structure and function of the extracellular matrix (ECM) is a central theme of the biology of the ADAMTS, as exemplified by the actions of the procollagen-N-propeptidases in collagen fibril assembly and of the aggrecanases in the cleavage or modification of ECM proteoglycans. Defects in certain family members give rise to inherited genetic disorders, while the aberrant expression or function of others is associated with arthritis, cancer and cardiovascular disease. In particular, ADAMTS4 and 5 have emerged as therapeutic targets in arthritis. Multiple ADAMTSs from different sub-groupings exert either positive or negative effects on tumorigenesis and metastasis, with both metalloproteinase-dependent and -independent actions known to occur. The basic ADAMTS structure comprises a metalloproteinase catalytic domain and a carboxy-terminal ancillary domain, the latter determining substrate specificity and the localization of the protease and its interaction partners; ancillary domains probably also have independent biological functions. Focusing primarily on the aggrecanases and proteoglycanases, this review provides a perspective on the evolution of the ADAMTS family, their links with developmental and disease mechanisms, and key questions for the future

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

LP 349-25: a new tight M8V binary

We present the discovery of a tight M8V binary, with a separation of only 1.2 astronomical units, obtained with the PUEO and NACO adaptive optics systems, respectively at the CFHT and VLT telescopes. The estimated period of LP 349-25 is approximately 5 years, and this makes it an excellent candidate for a precise mass measurement.Comment: Accepted by Astronomy and Astrophysics Letter