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We study a quantum particle coupled to a scalar quantized field, such as the regularized Nelson model. We show that there are situations where the Hamiltonian of the particle alone does not have a bound state, while the particle-field Hamiltonian has a ground state. This complements results for other models such as the Pauli--Fierz model, where such results have been shown. When the uncoupled quantum particle already has a bound state, we provide a new technique to show that this bound state persists when coupling the particle to the field, for any coupling strength. We especially do not use the binding condition. Our results use the functional integral representation and rely on a recent method using the Gaussian correlation inequality., Comment: 24 pages

Self-supervised learning (SSL) approaches have brought tremendous success across many tasks and domains. It has been argued that these successes can be attributed to a link between SSL and identifiable representation learning: Temporal structure and auxiliary variables ensure that latent representations are related to the true underlying generative factors of the data. Here, we deepen this connection and show that SSL can perform system identification in latent space. We propose DynCL, a framework to uncover linear, switching linear and non-linear dynamics under a non-linear observation model, give theoretical guarantees and validate them empirically.

Chemical evolution models predict a gradual build-up of $^{13}$C in the universe, based on empirical nuclear reaction rates and assumptions on the properties of stellar populations. However, old metal-poor stars within the Galaxy contain more $^{13}$C than is predicted, suggesting that further refinements to the models are necessary. Gas at high redshift provides important supplementary information at metallicities $-2\lesssim$ [Fe/H] $\lesssim-1$, for which there are only a few measurements in the Galaxy. We obtained new, high-quality, VLT/ESPRESSO observations of the QSO B1331+170 and used them to measure $^{12}$C/$^{13}$C in the damped Lyman-$\alpha$ system (DLA) at $z_{abs}=1.776$, with [Fe/H]=-1.27. AI-VPFIT, an Artificial Intelligence tool based on genetic algorithms and guided by a spectroscopic information criterion, was used to explore different possible kinematic structures of the carbon gas. Three hundred independent AI-VPFIT models of the absorption system were produced using pre-set $^{12}$C/$^{13}$C values, ranging from 4 to 500. Our results show that $^{12}$C/$^{13}$C$=28.5^{+51.5}_{-10.4}$, suggesting a possibility of $^{13}$C production at low metallicity., Comment: 11 pages, 6 figures. Three appendices. To appear in MNRAS

We present here the preliminary design of the RIZ module, one of the visible spectrographs of the ANDES instrument 1. It is a fiber-fed high-resolution, high-stability spectrograph. Its design follows the guidelines of successful predecessors such as HARPS and ESPRESSO. In this paper we present the status of the spectrograph at the preliminary design stage. The spectrograph will be a warm, vacuum-operated, thermally controlled and fiber-fed echelle spectrograph. Following the phase A design, the huge etendue of the telescope will be reformed in the instrument with a long slit made of smaller fibers. We discuss the system design of the spectrographs system., Comment: Paper submitted to the SPIE astronomical telescope and instrumentation 2024, conference title : Ground-based and Airborne Instrumentation for Astronomy X, paper reference number : 13096-171

Ground-based astronomy is unavoidably subject to the adverse effect of atmospheric turbulence, a.k.a. the seeing, which blurs the images and limits the achievable spatial resolution. For spectroscopic observations, it leads to slit or fiber-injection losses, since not all photons distributed over the extended seeing disk can be captured. These losses might have a very substantial impact on the overall efficiency of a spectrograph and are naturally highly variable. Assessing the fiber-injection losses requires accurate information about the image quality (IQ) delivered by the telescope to the instrument over the course of the observations, which, however, is often not directly available. ESPRESSO provides acquisition and field-stabilization images attached to the science data and thus offers the opportunity for a post-processing analysis. Here, we present a novel method to infer the IQ profile and fiber-injection losses from the integrated field-stabilization images, utilizing the spill-over light that does not get injected into the fiber. We validate these measurements against the IQ observed in the acquisition images and determine that our method delivers unbiased estimates with a scatter of 0.11" for the FWHM of the profile and 15% in terms of fiber-injection losses. This compares favorably to the estimates derived from either the differential image motion monitor (DIMM) or the telescope guide probe sensors and therefore represents a valuable tool to characterize the instrument efficiency and to correct raw spectra for fiber-injection losses., Comment: Accepted for publication with MNRAS

Achieving a truly accurate wavelength calibration of high-dispersion echelle spectrographs is a challenging task but crucially needed for certain science cases, e.g. to test for a possible variation of the fine-structure constant in quasar spectra. One of the spectrographs best suited for this mission is VLT/ESPRESSO. Nevertheless, previous studies have identified significant discrepancies between the classical wavelength solutions and the one derived independently from the laser frequency comb. The dominant parts of these systematics were intra-order distortions, most-likely related to a deviation of the instrumental line-spread function from the assumed Gaussian shape. Here, we therefore present a study focused on a detailed modeling of the ESPRESSO instrumental line-spread function. We demonstrate that it is strongly asymmetric, non-Gaussian, different for the two slices and fibers, and varies significantly along the spectral orders. Incorporating the determined non-parametric model in the wavelength calibration process drastically improves the wavelength calibration accuracy, reducing the discrepancies between the two independent wavelength solutions from 50m/s to about 10m/s. The most striking success is, however, that the different fibers and slices now provide fully consistent measurements with a scatter of just a couple m/s. This demonstrates that the instrument-related systematics can be nearly eliminated over most of the spectral range by properly taking into account the complex shape of the instrumental line-spread function and paves the way for further optimizations of the wavelength calibration process., Comment: Accepted for publication in MNRAS

In arXiv:2212.14023 a decomposition of Gaussian measures on finite-dimensional spaces was introduced, which turned out to be a central technical tool to improve currently known bounds on a long standing conjecture in statistical mechanics called the Polaron problem. This note slightly generalizes this decomposition and provides numerical values for all occurring constants., Comment: 6 pages

Primordial abundances of light elements are sensitive to the physics of the early Universe and can directly constrain cosmological quantities, such as the baryon-to-photon ratio $\eta_{10}$, the baryon density and the number of neutrino families. Deuterium is especially suited for these studies: its primordial abundance is sensitive and monotonically dependent on $\eta_{10}$, allowing an independent measurement of the cosmic baryon density that can be compared, for instance, against the Planck satellite data. The primordial deuterium abundance can be measured in high $H_I$ column density absorption systems towards distant quasars. We report here a new measurement, based on high-resolution ESPRESSO data, of the primordial $D_I$ abundance of a system at redshift $z \sim 3.572$, towards PKS1937-101. Using only ESPRESSO data, we find a D/H ratio of $2.638\pm0.128 \times 10^{-5}$, while including the available UVES data improves the precision, leading to a ratio of $2.608 \pm 0.102 \times 10^{-5}$. The results of this analysis agree with those of the most precise existing measurements. We find that the relatively low column density of this system ($\log{N_{\rm H_I}/ {\rm cm}^{-2}}\sim18 $) introduces modelling uncertainties, which become the main contributor to the error budget., Comment: Accepted for publication in MNRAS. 16 pages, 10 figures and 6 tables (4 in main text, 2 in Appendix). Full model is available at the end of the manuscript

The gravitationally lensed quasar J014516.6-094517 at z=2.719 has been observed with the ESPRESSO instrument at the ESO VLT to obtain high-fidelity spectra of the two images A and B with a resolving power R=70000. At the redshifts under investigation (2.1 < z < 2.7), the Lyman forests along the two sightlines are separated by sub-kiloparsec physical distances and exhibit a strong correlation. We find that the two forests are indistinguishable at the present level of signal-to-noise ratio and do not show any global velocity shift, with the cross-correlation peaking at $\Delta v = 12 \pm 48$ m/s. The distribution of the difference in velocity of individual Lyman-$\alpha$ features is compatible with a null average and a mean absolute deviation of 930 m/s. Significant differences in NHI column density are not detected, putting a limit to the RMS fluctuation in the baryon density on $\leq 1$ proper kpc scales of $\Delta \rho / \rho < 3$%. On the other hand, metal lines show significant differences both in velocity structure and in column density. A toy model shows that the difference in velocity of the metal features between the two sightlines is compatible with the the motions of the baryonic component associated to dark matter halos of typical mass $M\simeq 2\times 10^{10} M_\odot$, also compatible with the observed incidence of the metal systems. The present observations confirm the feasibility of the Sandage test of the cosmic redshift drift with high-fidelity spectroscopy of the Lyman forest of distant, bright quasars, but also provide an element of caution about the intrinsic noise associated to the usage of metal features for the same purpose., Comment: 17 pages, 10 figures, 5 tables, accepted for publication in MNRAS

We study Brownian paths perturbed by semibounded pair potentials and prove upper bounds on the mean square displacement. As a technical tool we derive infinite dimensional versions of key inequalities that were first used in [Sellke; arXiv:2212.14023] in order to study the effective mass of the Fr\"ohlich polaron., Comment: 17 pages

Astronomical precision spectroscopy underpins searches for life beyond Earth, direct observation of the expanding Universe and constraining the potential variability of physical constants across cosmological scales. Laser frequency combs can provide the critically required accurate and precise calibration to the astronomical spectrographs. For cosmological studies, extending the calibration with such astrocombs to the ultraviolet spectral range is highly desirable, however, strong material dispersion and large spectral separation from the established infrared laser oscillators have made this exceedingly challenging. Here, we demonstrate for the first time astronomical spectrograph calibrations with an astrocomb in the ultraviolet spectral range below 400 nm. This is accomplished via chip-integrated highly nonlinear photonics in periodically-poled, nano-fabricated lithium niobate waveguides in conjunction with a robust infrared electro-optic comb generator, as well as a chip-integrated microresonator comb. These results demonstrate a viable route towards astronomical precision spectroscopy in the ultraviolet and may contribute to unlocking the full potential of next generation ground- and future space-based astronomical instruments.

Let $p$ be an odd prime. Let $F$ be a non-archimedean local field of residue characteristic $p$, and let $\mathbb{F}_q$ be its residue field. Let $\mathcal{H}^{(1)}_{\mathbb{F}_q}$ be the pro-$p$-Iwahori-Hecke algebra of the $p$-adic group ${\textrm GL_2}(F)$ with coefficients in $\mathbb{F}_q$, and let $Z(\mathcal{H}^{(1)}_{\mathbb{F}_q})$ be its center. We define a scheme $X(q)_{\mathbb{F}_q}$ whose geometric points parametrize the semisimple two-dimensional Galois representations of ${\textrm Gal}(\overline{F}/F)$ over $\overline{\mathbb{F}}_q$. Then we construct a morphism from the spectrum of $Z(\mathcal{H}^{(1)}_{\mathbb{F}_q})$ to $X(q)_{\mathbb{F}_q}$ generalizing the morphism appearing in \cite{PS2} for $F=\mathbb{Q}_p$. In the case $F/\mathbb{Q}_p$, we show that the induced map from Hecke modules to Galois representations, when restricted to supersingular modules, coincides with Grosse-Kl\"onne's bijection \cite{GK18}. For this, we determine the Lubin-Tate $(\varphi,\Gamma)$-modules associated to absolutely irreducible Galois representations., Comment: The note arXiv:1911.11820 has been included as an Appendix of the present preprint

Global Navigation Satellite Systems (GNSSs), such as GPS and Galileo, provide precise time and space coordinates globally and constitute part of the critical infrastructure of modern society. To reliably operate GNSS, a highly accurate and stable system time is required, such as the one provided by several independent clocks hosted in Precise Timing Facilities (PTFs) around the world. Periodically, the relative clock offset between PTFs is measured to have a fallback system to synchronize the GNSS satellite clocks. The security and integrity of the communication between PTFs is of paramount importance: if compromised, it could lead to disruptions to the GNSS service. Therefore, it is a compelling use-case for protection via Quantum Key Distribution (QKD), since this technology provides information-theoretic security. We have performed a field trial demonstration of such use-case by sharing encrypted time synchronization information between two PTFs, one located in Oberpfaffenhofen (Germany) and one in Matera (Italy) - more than 900km apart as the crow flies. To bridge this large distance, a satellite-QKD system is required, plus a "last-mile" terrestrial link to connect the optical ground station (OGS) to the actual location of the PTF. In our demonstration we have deployed two full QKD systems to protect the last-mile connection at both the locations and have shown via simulation that upcoming QKD satellites will be able to distribute keys between Oberpfaffenhofen and Matera exploiting already existing OGSs.

In the standard picture, episodes of luminous quasar activity are directly related to supermassive black hole (SMBH) growth. The ionising radiation emitted over a quasar's lifetime alters the ionisation state of the surrounding intergalactic medium (IGM), enhancing the Ly$\alpha$ forest transmission -- so-called proximity effect -- which can be observed in absorption spectra of background sources. Owing to the finite speed of light, the transverse direction of the proximity effect is sensitive to the quasar's radiative history, resulting in `light echoes' that encode the growth history of the SMBH on Myr-timescales. In this paper, we introduce a new technique to photometrically map this quasar light echoes using Ly$\alpha$ forest tomography by using a carefully selected pair of narrow-band filters. A foreground narrow-band filter is used to measure Ly$\alpha$ forest transmission along background galaxies selected as Ly$\alpha$ emitters by a background narrow-band filter. This novel double narrow-band tomographic technique utilises the higher throughput and wider field of view of imaging over spectroscopy to efficiently reconstruct a two-dimensional map of Ly$\alpha$ forest transmission around a quasar. We present a fully Bayesian framework to measure the luminous quasar lifetime of a SMBH from photometric IGM tomography, and examine the observational requirements. This new technique provides an efficient strategy to map a large area of the sky with a modest observing time and to identify interesting regions to be examined by further deep 3D follow-up spectroscopic Ly$\alpha$ forest tomography., Comment: 21 pages, 14 figures, accepted for publication in MNRAS

Dynamical scalar fields in an effective four-dimensional field theory are naturally expected to couple to the rest of the theory's degrees of freedom, unless some new symmetry is postulated to suppress these couplings. In particular, a coupling to the electromagnetic sector will lead to spacetime variations of the fine-structure constant, $\alpha$. Astrophysical tests of the space-time stability of $\alpha$ are therefore a powerful probe of new physics. Here we use ESPRESSO and other contemporary measurements of $\alpha$, together with background cosmology data, local laboratory atomic clock and Weak Equivalence Principle measurements, to place stringent constraints on the simplest examples of the two broad classes of varying $\alpha$ models: Bekenstein models and quintessence-type dark energy models, both of which are parametric extensions of the canonical $\Lambda$CDM model. In both cases, previously reported constraints are improved by more than a factor of ten. This improvement is largely due to the very strong local constraints, but astrophysical measurements can help to break degeneracies between cosmology and fundamental physics parameters., Comment: 8 pages, 2 figures, 2 tables; in press at Phys. Rev. D

In the last decade, white-light illuminated Fabry-P\'erot interferometers wave been established as a widely used, relatively simple, reliable, and cost-effective way to precisely calibrate high-resolution echelle spectrographs. However, Terrien et al. (2021) recently reported a chromatic drift of the Fabry-P\'erot interferometer installed at the Habitable-zone Planet Finder spectrograph. In particular, they found that the variation of the etalon effective gap size is not achromatic as usually assumed but in fact depends on wavelength. Here, we present a similar study of the Espresso Fabry-P\'erot interferometer. Using daily calibrations spanning a period of over 2.5 years, we also find clear evidence for a chromatic drift with an amplitude of a few cm/s per day that has a characteristic, quasi-oscillatory dependence on wavelength. We conclude that this effect is probably caused by an aging of the dielectric mirror coatings and expect that similar chromatic drifts might affect all Fabry-P\'erot interferometers used for calibration of astronomical spectrographs. However, we also demonstrate that the chromatic drift can be measured and in principle corrected using only standard calibrations based on hollow cathode lamp spectra., Comment: 8 pages, 3 figures; accepted for publication in A&A

The spectrograph ESPRESSO recently obtained a limit on the variation of the fine-structure constant, $\alpha$, through measurements along the line of sight of a bright quasar with a precision of $1.36$ ppm at $1\sigma$ level. This imposes new constraints on cosmological models with a varying $\alpha$. We assume such a model where the electromagnetic sector is coupled to a scalar field dark energy responsible for the current acceleration of the Universe. We parametrise the variation of $\alpha$ with two extra parameters, one defining the cosmological evolution of the quintessence component and the other fixing the coupling with the electromagnetic field. The objective of this work is to constrain these parameters with both astrophysical and local probes. We also carried out a comparative analysis of how each data probe may constrain our parametrisation. We performed a Bayesian analysis by comparing the predictions of the model with observations. The astrophysical datasets are composed of quasar spectra measurements, including the latest ESPRESSO data point, as well as Planck observations of the cosmic microwave background. We combined these with local results from atomic clocks and the MICROSCOPE experiment. The constraints placed on the quintessence parameter are consistent with a null variation of the field, and are therefore compatible with a $\Lambda$CDM cosmology. The constraints on the coupling to the electromagnetic sector are dominated by the E\"otv\"os parameter local bound. More precise measurements with ESPRESSO will be extremely important to study the cosmological evolution of $\alpha$ as it probes an interval of redshift not accessible to other types of observations. However, for this particular model, current available data favour a null variation of $\alpha$ resulting mostly from the strong MICROSCOPE limits., Comment: 9 pages, 7 figures, as published in A&A

The strong intervening absorption system at redshift 1.15 towards the very bright quasar HE 0515$-$4414 is the most studied absorber for measuring possible cosmological variations in the fine-structure constant, $\alpha$. We observed HE 0515$-$4414 for 16.1$\,$h with the Very Large Telescope and present here the first constraint on relative variations in $\alpha$ with parts-per-million (ppm) precision from the new ESPRESSO spectrograph: $\Delta\alpha/\alpha = 1.3 \pm 1.3_{\rm stat} \pm 0.4_{\rm sys}\,{\rm ppm}$. The statistical uncertainty (1$\sigma$) is similar to the ensemble precision of previous large samples of absorbers, and derives from the high S/N achieved ($\approx$105 per 0.4$\,$km$\,$s$^{-1}$ pixel). ESPRESSO's design, and calibration of our observations with its laser frequency comb, effectively removed wavelength calibration errors from our measurement. The high resolving power of our ESPRESSO spectrum ($R=145000$) enabled the identification of very narrow components within the absorption profile, allowing a more robust analysis of $\Delta\alpha/\alpha$. The evidence for the narrow components is corroborated by their correspondence with previously detected molecular hydrogen and neutral carbon. The main remaining systematic errors arise from ambiguities in the absorption profile modelling, effects from redispersing the individual quasar exposures, and convergence of the parameter estimation algorithm. All analyses of the spectrum, including systematic error estimates, were initially blinded to avoid human biases. We make our reduced ESPRESSO spectrum of HE 0515$-$4414 publicly available for further analysis. Combining our ESPRESSO result with 28 measurements, from other spectrographs, in which wavelength calibration errors have been mitigated, yields a weighted mean $\Delta\alpha/\alpha = -0.5 \pm 0.5_{\rm stat} \pm 0.4_{\rm sys}\,$ppm at redshifts 0.6-2.4., Comment: Accepted by A&A. The ESPRESSO quasar spectrum, reduced contributing exposures, and absorption profile fits are available at https://github.com/MTMurphy77/ESPRESSO_HE0515-4414

Graph Neural Networks (GNNs) are increasingly important given their popularity and the diversity of applications. Yet, existing studies of their vulnerability to adversarial attacks rely on relatively small graphs. We address this gap and study how to attack and defend GNNs at scale. We propose two sparsity-aware first-order optimization attacks that maintain an efficient representation despite optimizing over a number of parameters which is quadratic in the number of nodes. We show that common surrogate losses are not well-suited for global attacks on GNNs. Our alternatives can double the attack strength. Moreover, to improve GNNs' reliability we design a robust aggregation function, Soft Median, resulting in an effective defense at all scales. We evaluate our attacks and defense with standard GNNs on graphs more than 100 times larger compared to previous work. We even scale one order of magnitude further by extending our techniques to a scalable GNN., Comment: 39 pages, 22 figures, 17 tables NeurIPS 2021

We obtained polarimetric differential imaging of a gas-rich debris disk around HD 141569A with SPHERE in the H-band to compare the scattering properties of the innermost ring at 44 au with former observations in total intensity with the same instrument. In polarimetric imaging, we observed that the intensity of the ring peaks in the south-east, mostly in the forward direction, whereas in total intensity imaging, the ring is detected only at the south. This noticeable characteristic suggests a non-uniform dust density in the ring. We implemented a density function varying azimuthally along the ring and generated synthetic images both in polarimetry and in total intensity, which are then compared to the actual data. We find that the dust density peaks in the south-west at an azimuthal angle of $220^{\circ} \sim 238^{\circ}$ with a rather broad width of $61^{\circ} \sim 127^{\circ}$. Although there are still uncertainties that remain in the determination of the anisotropic scattering factor, the implementation of an azimuthal density variation to fit the data proved to be robust. Upon elaborating on the origin of this dust density distribution, we conclude that it could be the result of a massive collision when we account for the effect of the high gas mass that is present in the system on the dynamics of grains. Using the outcome of this modelization, we further measured the polarized scattering phase function for the observed scattering angle between 33$^{\circ}$ and 147$^{\circ}$ as well as the spectral reflectance of the southern part of the ring between 0.98 $\mu$m and 2.1 $\mu$m. We tentatively derived the grain properties by comparing these quantities with MCFOST models and assuming Mie scattering. Our preliminary interpretation indicates a mixture of porous sub-micron sized astro-silicate and carbonaceous grains., Comment: 21 pages, 18 figures, Accepted for publication in Astronomy & Astrophysics journal

We establish a version of Kashiwara's theorem for twisted sheaves of Berthelot's arithmetic differential operators for a closed immersion between smooth p-adic formal schemes. As an application, we construct simple modules for crystalline distribution algebras of reductive groups using the twisted version of the crystalline localisation theorem., Comment: 28 pages, corrected some minor typos in the first version, added the application to crystalline distribution algebras

Observations of metal absorption systems in the spectra of distant quasars allow to constrain a possible variation of the fine-structure constant throughout the history of the Universe. Such a test poses utmost demands on the wavelength accuracy and previous studies were limited by systematics in the spectrograph wavelength calibration. A substantial advance in the field is therefore expected from the new ultra-stable high-resolution spectrograph Espresso, recently installed at the VLT. In preparation of the fundamental physics related part of the Espresso GTO program, we present a thorough assessment of the Espresso wavelength accuracy and identify possible systematics at each of the different steps involved in the wavelength calibration process. Most importantly, we compare the default wavelength solution, based on the combination of Thorium-Argon arc lamp spectra and a Fabry-P\'erot interferometer, to the fully independent calibration obtained from a laser frequency comb. We find wavelength-dependent discrepancies of up to 24m/s. This substantially exceeds the photon noise and highlights the presence of different sources of systematics, which we characterize in detail as part of this study. Nevertheless, our study demonstrates the outstanding accuracy of Espresso with respect to previously used spectrographs and we show that constraints of a relative change of the fine-structure constant at the $10^{-6}$ level can be obtained with Espresso without being limited by wavelength calibration systematics., Comment: 27 pages, accepted for publication in A&A

Let G be a $p$-adic Lie group. We develop a dimension theory for coadmissible G-equivariant $\mathcal{D}$-modules on smooth rigid analytic spaces. We introduce the category of weakly holonomic G-equivariant $\mathcal{D}$-modules, study its duality and its preservation under various operations.

This is the sequel to arXiv:2007.01364v1. Let $F$ be any local field with residue characteristic $p>0$, and $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$ be the mod $p$ pro-$p$-Iwahori Hecke algebra of $\mathbf{GL_2}(F)$. In arXiv:2007.01364v1 we have constructed a parametrization of the $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$-modules by certain $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Satake parameters, together with an antispherical family of $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_p}$-modules. Here we let $F=\mathbb{Q}_p$ (and $p\geq 5$) and construct a morphism from $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Satake parameters to $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-Langlands parameters. As a result, we get a version in families of Breuil's semisimple mod $p$ Langlands correspondence for $\mathbf{GL_2}(\mathbb{Q}_p)$ and of Pa\v{s}k\={u}nas' parametrization of blocks of the category of mod $p$ locally admissible smooth representations of $\mathbf{GL_2}(\mathbb{Q}_p)$ having a central character. The formulation of these results is possible thanks to the Emerton-Gee moduli space of semisimple $\widehat{\mathbf{GL_2}}(\overline{\mathbb{F}}_p)$-representations of the Galois group ${\rm Gal}(\overline{\mathbb{Q}}_p/ \mathbb{Q}_p)$., Comment: 23 pages