Search

We establish a new Iwasawa theory for the kernel of Frobenius on Jacobians of curves in geometric $\mathbf{Z}_p$-towers over the projective line in characteristic $p$, thereby proving several of the main conjectures of [arXiv:2107.12555]., Comment: 49 pages, comments welcome

Let $Y\to X$ be an unramified Galois cover of curves over a perfect field $k$ of characteristic $p>0$ with $\mathrm{Gal}(Y/X)\cong\mathbb{Z}/p\mathbb{Z}$, and let $J_X$ and $J_Y$ be the Jacobians of $X$ and $Y$ respectively. We consider the $p$-torsion subgroup schemes $J_X[p]$ and $J_Y[p]$, analyze the Galois-module structure of $J_Y[p]$, and find restrictions this structure imposes on $J_Y[p]$ (for example, as manifested in its Ekedahl--Oort type) taking $J_X[p]$ as given., Comment: v1: 45 pages. v2: Corrections and major additions about the local-local factor, sections on explicit geometry and calculations removed. 43 pages

Let $k$ be a perfect field of characteristic $p$ and $\Gamma$ an infinite, first countable pro-$p$ group. We study the behavior of the $p$-primary part of the "motivic class group", i.e. the full $p$-divisible group of the Jacobian, in any $\Gamma$-tower of function fields over $k$ that is unramified outside a finite (possibly empty) set of places $\Sigma$, and totally ramified at every place of $\Sigma$. When $\Sigma=\emptyset$ and $\Gamma$ is a torsion free $p$-adic Lie group, we obtain asymptotic formulae which show that the $p$-torsion class group schemes grow in a remarkably regular manner. In the ramified setting $\Sigma\neq\emptyset$, we obtain a similar asymptotic formula for the $p$-torsion in "physical class groups", i.e. the $k$-rational points of the Jacobian, which generalizes the work of Mazur and Wiles, who studied the case $\Gamma=\mathbf{Z}_p$., Comment: Corollaries 1.4 and 1.5 slightly weakened due to potential mistake in key reference in proof of these results in v1. Remark 3.11 (most of which appeared as a footnote in v1) added, and Remark 3.16 adjusted. Introduction modified to to reflect these changes, and a few typos corrected

We investigate a novel geometric Iwasawa theory for $\mathbf{Z}_p$-extensions of function fields over a perfect field $k$ of characteristic $p>0$ by replacing the usual study of $p$-torsion in class groups with the study of $p$-torsion class group schemes. That is, if $\cdots \to X_2 \to X_1 \to X_0$ is the tower of curves over $k$ associated to a $\mathbf{Z}_p$-extension of function fields totally ramified over a finite non-empty set of places, we investigate the growth of the $p$-torsion group scheme in the Jacobian of $X_n$ as $n\rightarrow \infty$. By Dieudonn\'e theory, this amounts to studying the first de Rham cohomology groups of $X_n$ equipped with natural actions of Frobenius and of the Cartier operator $V$. We formulate and test a number of conjectures which predict striking regularity in the $k[V]$-module structure of the space $M_n:=H^0(X_n, \Omega^1_{X_n/k})$ of global regular differential forms as $n\rightarrow \infty.$ For example, for each tower in a basic class of $\mathbf{Z}_p$-towers we conjecture that the dimension of the kernel of $V^r$ on $M_n$ is given by $a_r p^{2n} + \lambda_r n + c_r(n)$ for all $n$ sufficiently large, where $a_r, \lambda_r$ are rational constants and $c_r : \mathbf{Z}/m_r \mathbf{Z} \to \mathbf{Q}$ is a periodic function, depending on $r$ and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on $\mathbf{Z}_p$-towers of curves, and we prove our conjectures in the case $p=2$ and $r=1$., Comment: 48 pages; MAGMA code at https://github.com/jeremybooher/MAGMA-Towers

Let $\pi : Y \to X$ be a branched $\mathbf{Z}/p \mathbf{Z}$-cover of smooth, projective, geometrically connected curves over a perfect field of characteristic $p>0$. We investigate the relationship between the $a$-numbers of $Y$ and $X$ and the ramification of the map $\pi$. This is analogous to the relationship between the genus (respectively $p$-rank) of $Y$ and $X$ given the Riemann-Hurwitz (respectively Deuring--Shafarevich) formula. Except in special situations, the $a$-number of $Y$ is not determined by the $a$-number of $X$ and the ramification of the cover, so we instead give bounds on the $a$-number of $Y$. We provide examples showing our bounds are sharp. The bounds come from a detailed analysis of the kernel of the Cartier operator.

For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic \'etale cohomology $H^i_{\mathrm{\'et}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.

We extend the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. In particular, for a finite and totally ramified extension $F/\mathbb Q_p$, and an arbitrary finite extension $K/F$, we construct a general class of infinite and totally wildly ramified extensions $K_\infty/K$ so that the functor $V\mapsto V|_{G_{K_\infty}}$ is fully-faithfull on the category of $F$-crystalline representations $V$. We also establish a new classification of $F$-Barsotti-Tate groups via Kisin modules of height 1 which allows more general lifts of Frobenius., Comment: 39 pages

Let $k$ be a perfect field of characteristic $p>2$ and $K$ an extension of $F=\mathrm{Frac} W(k)$ contained in some $F(\mu_{p^r})$. Using crystalline Dieudonn\'e theory, we provide a classification of $p$-divisible groups over $\mathscr{O}_K$ in terms of finite height $(\varphi,\Gamma)$-modules over $\mathfrak{S}:=W(k)[[u]]$. Although such a classification is a consequence of (a special case of) the theory of Kisin--Ren, our construction gives an independent proof and allows us to recover the Dieudonn\'e crystal of a $p$-divisible group from the Wach module associated to its Tate module by Berger--Breuil or by Kisin--Ren.

LOFT (Large Observatory for X-ray Timing) is one of the five candidates that were considered by ESA as an M3 mission (with launch in 2022-2024) and has been studied during an extensive assessment phase. It is specifically designed to perform fast X-ray timing and probe the status of the matter near black holes and neutron stars. Its pointed instrument is the Large Area Detector (LAD), a 10 m 2 -class instrument operating in the 2-30keV range, which holds the capability to revolutionise studies of variability from X-ray sources on the millisecond time scales. The LAD instrument has now completed the assessment phase but was not down-selected for launch. However, during the assessment, most of the trade-offs have been closed leading to a robust and well documented design that will be re- proposed in future ESA calls. In this talk, we will summarize the characteristics of the LAD design and give an overview of the expectations for the instrument capabilities., Comment: Proc. SPIE 9144, Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray, 91442W

We report on the development and characterization of the low-noise, low power, mixed analog-digital SIRIUS ASICs for both the LAD and WFM X-ray instruments of LOFT. The ASICs we developed are reading out large area silicon drift detectors (SDD). Stringent requirements in terms of noise (ENC of 17 e- to achieve an energy resolution on the LAD of 200 eV FWHM at 6 keV) and power consumption (650 {\mu}W per channel) were basis for the ASICs design. These SIRIUS ASICs are developed to match SDD detectors characteristics: 16 channels ASICs adapted for the LAD (970 microns pitch) and 64 channels for the WFM (145 microns pitch) will be fabricated. The ASICs were developed with the 180nm mixed technology of TSMC., Comment: Proc. SPIE 9144, Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray, 914463

The Large Observatory For x-ray Timing (LOFT) was studied within ESA M3 Cosmic Vision framework and participated in the final down-selection for a launch slot in 2022-2024. Thanks to the unprecedented combination of effective area and spectral resolution of its main instrument, LOFT will study the behaviour of matter under extreme conditions, such as the strong gravitational field in the innermost regions of accretion flows close to black holes and neutron stars, and the supra-nuclear densities in the interior of neutron stars. The science payload is based on a Large Area Detector (LAD, 10 m 2 effective area, 2-30 keV, 240 eV spectral resolution, 1 deg collimated field of view) and a WideField Monitor (WFM, 2-50 keV, 4 steradian field of view, 1 arcmin source location accuracy, 300 eV spectral resolution). The WFM is equipped with an on-board system for bright events (e.g. GRB) localization. The trigger time and position of these events are broadcast to the ground within 30 s from discovery. In this paper we present the status of the mission at the end of its Phase A study., Comment: Proc. SPIE 9144, Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray, 91442T

We construct the $\Lambda$-adic crystalline and Dieudonn\'e analogues of Hida's ordinary $\Lambda$-adic \'etale cohomology, and employ integral $p$-adic Hodge theory to prove $\Lambda$-adic comparison isomorphisms between these cohomologies and the $\Lambda$-adic de Rham cohomology studied in the prequel to this paper as well as Hida's $\Lambda$-adic \'etale cohomology. As applications of our work, we provide a "cohomological" construction of the family of $(\varphi,\Gamma)$-modules attached to Hida's ordinary $\Lambda$-adic \'etale cohomology by the work of Dee, and we give a new and purely geometric proof of Hida's finitenes and control theorems. We also prove suitable $\Lambda$-adic duality theorems for each of the cohomologies we construct., Comment: This paper is a continuation of our previous paper "The Geometry of Hida Families I: $\Lambda$-adic de Rham cohomology", and is a revised version of part of the paper arXiv:1209.0046

We construct the $\Lambda$-adic de Rham analogue of Hida's ordinary $\Lambda$-adic \'etale cohomology and of Ohta's $\Lambda$-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of $\mathbf{Q}_p$, we give a purely geometric proof of the expected finiteness, control, and $\Lambda$-adic duality theorems. Following Ohta, we then prove that our $\Lambda$-adic module of differentials is canonically isomorphic to the space of ordinary $\Lambda$-adic cuspforms. In the sequel to this paper, we construct the crystalline counterpart to Hida's ordinary $\Lambda$-adic \'etale cohomology, and employ integral $p$-adic Hodge theory to prove $\Lambda$-adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and the sequel, we will be able to provide a "cohomological" construction of the family of $(\varphi,\Gamma)$-modules attached to Hida's ordinary $\Lambda$-adic \'etale cohomology by the work of Dee, as well as a new and purely geometric proof of Hida's finitenes and control theorems. We are also able to prove refinements of theorems of Mazur-Wiles and of Ohta., Comment: This article is a revised version of part of arXiv:1209.0046

Let K denote a finite extension of Qp. We give necessary and sufficient conditions for an infinite totally wildly ramified extension L/K to be strictly APF in the sense of Fontaine-Wintenberger. Our conditions are phrased in terms of the existence of a certain tower of intermediate subfields. These conditions are well-suited to producing examples of strictly APF extensions, and in particular, our main theorem proves that the phi-iterate extensions previously considered by the first two authors are strictly APF.

Fix $K/\mathbf{Q}_p$ a finite extension and let $L/K$ be an infinite, strictly APF extension in the sense of Fontaine--Wintenberger. Let $X_K(L)$ denote its associated norm field. The goal of this paper is to associate to $L/K$, in a canonical and functorial way, a $p$-adically complete subring $\mathbf{A}_{L/K}^+ \subset \widetilde{\mathbf{A}}^+$ whose reduction modulo~$p$ is contained in the valuation ring of $X_K(L)$. When the extension $L/K$ is of a special form, which we call a $\varphi$-iterate extension, we prove that $X_K(L)$ is (at worst) a finite purely inseparable extension of the fraction field of $\mathbf{A}_{L/K}^+/(p)$. The class of $\varphi$-iterate extensions includes all Lubin--Tate extensions, as well as many other extensions such as the non-Galois ``Kummer" extension occurring in work of Faltings, Breuil, and Kisin. In particular, our work provides a canonical and functorial construction of every characteristic zero lift of the norm fields that have thus far played a foundational role in (integral) $p$-adic Hodge theory, as well as many other cases which have yet to be studied.

For a perfect field \kappa of characteristic p>0, a positive ingeger N not divisible by p, and an arbitrary subgroup \Gamma of GL_2(Z/NZ), we prove (with mild additional hypotheses when p\le 3) that the U-operator on the space M_k(\Gamma/\kappa) of (Katz) modular forms for \Gamma over \kappa induces a surjection U:M_{k}(\Gamma/\kappa)\rightarrow M_{k'}(\Gamma/\kappa) for all k\ge p+2, where k'=(k-k_0)/p + k_0 with 2\le k_0\le p+1 the unique integer congruent to k modulo p. When \kappa=F_p, p\ge 5, N\neq 2,3, and \Gamma is the subgroup of upper-triangular or upper-triangular unipotent matrices, this recovers a recent result of Dewar.

The LOFT mission concept is one of four candidates selected by ESA for the M3 launch opportunity as Medium Size missions of the Cosmic Vision programme. The launch window is currently planned for between 2022 and 2024. LOFT is designed to exploit the diagnostics of rapid X-ray flux and spectral variability that directly probe the motion of matter down to distances very close to black holes and neutron stars, as well as the physical state of ultra-dense matter. These primary science goals will be addressed by a payload composed of a Large Area Detector (LAD) and a Wide Field Monitor (WFM). The LAD is a collimated (<1 degree field of view) experiment operating in the energy range 2-50 keV, with a 10 m^2 peak effective area and an energy resolution of 260 eV at 6 keV. The WFM will operate in the same energy range as the LAD, enabling simultaneous monitoring of a few-steradian wide field of view, with an angular resolution of <5 arcmin. The LAD and WFM experiments will allow us to investigate variability from submillisecond QPO's to year-long transient outbursts. In this paper we report the current status of the project., Comment: Proceedings of SPIE, Vol. 8443, Paper No. 8443-85, 2012

The Large Observatory for X-ray Timing (LOFT) is one of the four candidate ESA M3 missions considered for launch in the 2022 time-frame. It is specifically designed to perform fast X-ray timing and probe the status of the matter near black holes and neutron stars. The LOFT scientific payload is composed of a Large Area Detector (LAD) and a Wide Field Monitor (WFM). The LAD is a 10 m2-class pointed instrument with 20 times the collecting area of the best past timing missions (such as RXTE) over the 2-30 keV range, which holds the capability to revolutionize studies of X-ray variability down to the millisecond time scales. Its ground-breaking characteristic is a low mass per unit surface, enabling an effective area of ~10 m^2 (@10 keV) at a reasonable weight. The development of such large but light experiment, with low mass and power per unit area, is now made possible by the recent advancements in the field of large-area silicon detectors - able to time tag an X-ray photon with an accuracy <10 {\mu}s and an energy resolution of ~260 eV at 6 keV - and capillary-plate X-ray collimators. In this paper, we will summarize the characteristics of the LAD instrument and give an overview of its capabilities., Comment: Proceedings of SPIE, Vo. 8443, Paper No. 8443-87

We construct \Lambda-adic de Rham and crystalline analogues of Hida's ordinary \Lambda-adic etale cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \Q_p, we prove appropriate finiteness and control theorems in each case. We then employ integral p-adic Hodge theory to prove \Lambda-adic comparison isomorphisms between our cohomologies and Hida's etale cohomology. As applications of our work, we provide a "cohomological" construction of the family of (\phi,\Gamma)-modules attached to Hida's ordinary \Lambda-adic etale cohomology by Dee, and we give a new and purely geometric proof of Hida's finitenes and control theorems. We are also able to prove refinements of theorems of Mazur-Wiles and of Ohta; in particular, we prove that there is a canonical isomorphism between the module of ordinary \Lambda-adic cuspforms and the part of the crystalline cohomology of the Igusa tower on which Frobenius acts invertibly.

We describe a probability distribution on isomorphism classes of principally quasi-polarized p-divisible groups over a finite field k of characteristic p which can reasonably be thought of as "uniform distribution," and we compute the distribution of various statistics (p-corank, a-number, etc.) of p-divisible groups drawn from this distribution. It is then natural to ask to what extent the p-divisible groups attached to a randomly chosen hyperelliptic curve (resp. curve, resp. abelian variety) over k are uniformly distributed in this sense. For instance, one can ask whether the proportion of genus-g curves over F_p whose Jacobian is ordinary approaches the limit that such a heuristic would predict. This heuristic is analogous to conjectures of Cohen-Lenstra type for fields k of characteristic other than p, in which case the random p-divisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over F_3 appear substantially less likely to be ordinary than hyperelliptic curves over F_3., Comment: 28 pages, relevant code attached

The exact evaluation of the disconnected diagram contributions to the flavor-singlet pseudoscalar meson mass, the nucleon sigma term and the nucleon electromagnetic form factors, is carried out utilizing GPGPU technology with the NVIDIA CUDA platform. The disconnected loops are also computed using stochastic methods with several noise reduction techniques. Various dilution schemes as well as the truncated solver method are studied. We make a comparison of these stochastic techniques to the exact results and show that the number of noise vectors depends on the operator insertion in the fermionic loop., Comment: Version accepted for publication in Comp. Phys. Commun. References added. 13 pages, 12 figures

In contrast with numerous studies on the physical and chemical structure of low- and high-mass protostars, much less is known about their intermediate-mass counterparts, a class of objects that could help to elucidate the mechanisms of star formation on both ends of the mass range. We present the first results from a rich HIFI spectral dataset on an intermediate-mass protostar, OMC2-FIR4, obtained in the CHESS (Chemical HErschel SurveyS of star forming regions) key programme. The more than 100 methanol lines detected between 554 and 961 GHz cover a range in upper level energy of 40 to 540 K. Our physical interpretation focusses on the hot core, but likely the cold envelope and shocked regions also play a role in reality, because an analysis of the line profiles suggests the presence of multiple emission components. An upper limit of 10^-6 is placed on the methanol abundance in the hot core, using a population diagram, large-scale source model and other considerations. This value is consistent with abundances previously seen in low-mass hot cores. Furthermore, the highest energy lines at the highest frequencies display asymmetric profiles, which may arise from infall around the hot core., Comment: 2 figures, accepted for publication in the A&A Herschel-HIFI special issue

We present results for the negative parity low-lying state of the nucleon, $N{\frac{1}{2}}^{-}$ (1535 MeV) ${\rm S}_{11}$, from a variational analysis method. The analysis is performed in quenched QCD with the FLIC fermion action. The principal focus of this paper is to explore the level ordering between the Roper (${\rm{P}}_{11}$) and the negative parity ground (${\rm{S}}_{11}$) states of the nucleon. Evidence of the physical level ordering is observed at light quark masses. A wide variety of smeared-smeared correlation functions are used to construct correlation matrices. A comprehensive correlation matrix analysis is performed to ensure an accurate isolation of the $N{\frac{1}{2}}^{-}$ state., Comment: 9 pages

Despite the low elemental deuterium abundance in the Galaxy, enhanced molecular D/H ratios have been found in the environments of low-mass star forming regions, and in particular the Class 0 protostar IRAS 16293-2422. The CHESS (Chemical HErschel Surveys of Star forming regions) Key Program aims at studying the molecular complexity of the interstellar medium. The high sensitivity and spectral resolution of the HIFI instrument provide a unique opportunity to observe the fundamental 1,1,1 - 0,0,0 transition of the ortho-D2O molecule, inaccessible from the ground, and to determine the ortho-to-para D2O ratio. We have detected the fundamental transition of the ortho-D2O molecule at 607.35 GHz towards IRAS 16293-2422. The line is seen in absorption with a line opacity of 0.62 +/- 0.11 (1 sigma). From the previous ground-based observations of the fundamental 1,1,0 - 1,0,1 transition of para-D2O seen in absorption at 316.80 GHz we estimate a line opacity of 0.26 +/- 0.05 (1 sigma). We show that the observed absorption is caused by the cold gas in the envelope of the protostar. Using these new observations, we estimate for the first time the ortho to para D2O ratio to be lower than 2.6 at a 3 sigma level of uncertainty, to be compared with the thermal equilibrium value of 2:1., Comment: 5 pages, 5 figures, accepted the A&A HIFI Special Issue as a letter

The HIFI instrument on board the Herschel Space Observatory has been used to observe interstellar nitrogen hydrides along the sight-line towards G10.6-0.4 in order to improve our understanding of the interstellar chemistry of nitrogen. We report observations of absorption in NH N=1-0, J=2-1 and ortho-NH2 1_1,1-0_0,0. We also observed ortho-NH3 1_0-0_0, and 2_0-1_0, para-NH3 2_1-1_1, and searched unsuccessfully for NH+. All detections show emission and absorption associated directly with the hot-core source itself as well as absorption by foreground material over a wide range of velocities. All spectra show similar, non-saturated, absorption features, which we attribute to diffuse molecular gas. Total column densities over the velocity range 11-54 km/s are estimated. The similar profiles suggest fairly uniform abundances relative to hydrogen, approximately 6*10^-9, 3*10^-9, and 3*10^-9 for NH, NH2, and NH3, respectively. These abundances are discussed with reference to models of gas-phase and surface chemistry., Comment: 5 pages, 3 figures, 2 online pages with 2 figures. Accepted for publication in A&A July 6 (Herschel/HIFI special issue)

We report the detection of strong absorption by interstellar hydrogen fluoride along the sight-line to the submillimeter continuum source G10.6-0.4 (W31C). We have used Herschel's HIFI instrument, in dual beam switch mode, to observe the 1232.4763 GHz J=1-0 HF transition in the upper sideband of the Band 5a receiver. The resultant spectrum shows weak HF emission from G10.6-0.4 at LSR velocities in the range -10 to -3 km/s, accompanied by strong absorption by foreground material at LSR velocities in the range 15 to 50 km/s. The spectrum is similar to that of the 1113.3430 GHz 1(11)-0(00) transition of para-water, although at some frequencies the HF (hydrogen fluoride) optical depth clearly exceeds that of para-H2O. The optically-thick HF absorption that we have observed places a conservative lower limit of 1.6E+14 cm-2 on the HF column density along the sight-line to G10.6-0.4. Our lower limit on the HF abundance, 6E-9 relative to hydrogen nuclei, implies that hydrogen fluoride accounts for between ~ 30 and 100% of the fluorine nuclei in the gas phase along this sight-line. This observation corroborates theoretical predictions that - because the unique thermochemistry of fluorine permits the exothermic reaction of F atoms with molecular hydrogen - HF will be the dominant reservoir of interstellar fluorine under a wide range of conditions., Comment: Accepted for publication in Astronomy and Astrophysics (Herschel special issue). This revised version corrects a typographic error in the HTML abstract, in which the lower limit on the HF abundance (should be 6E-9) was previously misstated. The abstract in the PDF version is correct and the latter has not been modified