Eisenstein series and automorphic representations

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of the Langlands constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker function associated to unramified automorphic representations of G(Q_p). In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore also introduce some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In particular, we provide a detailed treatment of supersymmetry constraints on string amplitudes which enforce differential equations of the same type that are satisfied by automorphic forms. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics which go beyond the scope of this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with highlighted applications to string theory. v2: 375 pages. Substantially extended and small correction

A reduction principle for Fourier coefficients of automorphic forms

In this paper we analyze a general class of Fourier coefficients of automorphic forms on reductive adelic groups $\mathbf{G}(\mathbb{A}_\mathbb{K})$ and their covers. We prove that any such Fourier coefficient is expressible through integrals and sums involving 'Levi-distinguished' Fourier coefficients. By the latter we mean the class of Fourier coefficients obtained by first taking the constant term along the nilradical of a parabolic subgroup, and then further taking a Fourier coefficient corresponding to a $\mathbb{K}$-distinguished nilpotent orbit in the Levi quotient. In a follow-up paper we use this result to establish explicit formulas for Fourier expansions of automorphic forms attached to minimal and next-to-minimal representations of simply-laced reductive groups.Comment: 35 pages. v2: Extended results and paper split into two parts with second part appearing soon. New title to reflect new focus of this part. v3: Minor corrections and updated reference to the second part that has appeared as arXiv:1908.08296. v4: Minor corrections and reformulation

Multi-elemental speciation analysis of barley genotypes diering in tolerance to cadmium toxicity using SEC-ICP-MS and ESI-TOF-MS

Plants respond to Cd exposure by synthesizing heavy-metal-binding oligopeptides, called phytochelatins (PCs). These peptides reduce the activity of Cd2+ ions in the plant tissues by forming Cd chelates. The main objective of the present work was to develop an analytical technique, which allowed identication of the most prominent Cd species in plant tissue by SEC-ICP-MS and ESI-TOF-MS. An integrated part of the method development was to test the hypothesis that dierential Cd tolerance between two barley genotypes was linked to dierences in Cd speciation. Only one fraction of Cd species, ranging from 7001800 Da, was detected in the shoots of both genotypes. In the roots, two additional fractions ranging from 29004600 and 670015 000 Da were found. The Cd-rich SEC fractions were heart-cut, de-salted and demetallized using reversed-phase chromatography (RPC), followed by ESI-MS-TOF to identify the ligands. Three dierent families of PCs, viz. (gGlu-Cys)n-Gly (PCn), (gGlu-Cys)n-Ser (iso-PCn) and Cys-(gGlu-Cys)n-Gly (des-gGlu-PCn), the last lacking the N-terminal amino acid, were identied. The PCs induced by Cd toxicity also bound several essential trace elements in plants, including Zn, Cu, and Ni, whereas no Mn species were detected. Zn, Cu and Ni-species were distributed between the 7001800 Da and 670015 000 Da fractions, whereas only Cd species were found in the 29004600 Da fraction dominated by PC3 ligands. Although the total tissue concentration of Cd was similar for the two species, the tolerant barley genotype synthesized signicantly more CdPC3 species with a high Cd specicity than the intolerant genotype, clearly indicating a correlation between Cd tolerance and the CdPC speciation

Eulerianity of Fourier coefficients of automorphic forms

We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a `hidden' invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier-Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory.Comment: 28 pages. v2: Clarified connection to Fourier-Jacobi coefficients and references added. v3: Minor correction

Birth weight and grip strength in young Swedish males: a longitudinal matched sibling analysis and across all body mass index ranges

Low birth weight is associated with a lower grip strength later in life. However, associations between birth weight among infants born at-term and factors driving associations between birth weight and grip strength are largely unknown. A cohort of 144,369 young men born at-term, including 10,791 individuals who had at least one male sibling/s, were followed until conscription where they performed a grip strength test. We used linear and non-linear regression analyses in the full cohort, and fixed-effects regression analyses in the sibling cohort, to address confounding by factors that are shared between siblings. After adjustment, each unit increase in birth weight z-score was associated with increases of 17.7 (95% CI, 17.2-18.2) and 13.4 (10.1-16.6) newton grip strength, which converts to approximately 1.8 and 1.4 kilogram-force in the full and within-families cohorts, respectively. The associations did not vary with young adulthood BMI. Birth weight, within the at-term range, is robustly positively associated with grip strength in young adulthood among men across all BMI categories and associations appears to be mainly driven by factors that are not shared between siblings. These findings underline the importance of recognizing the influence of low birth weight, also within the at-termrange, on young adulthood muscle strength

Micro-scaled high-throughput digestion of plant tissue samples for multi-elemental analysis

Quantitative multi-elemental analysis by inductively coupled plasma (ICP) spectrometry depends on a complete digestion of solid samples. However, fast and thorough sample digestion is a challenging analytical task which constitutes a bottleneck in modern multi-elemental analysis. Additional obstacles may be that sample quantities are limited and elemental concentrations low. In such cases, digestion in small volumes with minimum dilution and contamination is required in order to obtain high accuracy data. Results We have developed a micro-scaled microwave digestion procedure and optimized it for accurate elemental profiling of plant materials (1-20 mg dry weight). A commercially available 64-position rotor with 5 ml disposable glass vials, originally designed for microwave-based parallel organic synthesis, was used as a platform for the digestion. The novel micro-scaled method was successfully validated by the use of various certified reference materials (CRM) with matrices rich in starch, lipid or protein. When the micro-scaled digestion procedure was applied on single rice grains or small batches of Arabidopsis seeds (1 mg, corresponding to approximately 50 seeds), the obtained elemental profiles closely matched those obtained by conventional analysis using digestion in large volume vessels. Accumulated elemental contents derived from separate analyses of rice grain fractions (aleurone, embryo and endosperm) closely matched the total content obtained by analysis of the whole rice grain. Conclusion A high-throughput micro-scaled method has been developed which enables digestion of small quantities of plant samples for subsequent elemental profiling by ICP-spectrometry. The method constitutes a valuable tool for screening of mutants and transformants. In addition, the method facilitates studies of the distribution of essential trace elements between and within plant organs which is relevant for, e.g., breeding programmes aiming at improvement of the micronutrient density in edible plant parts. Compared to existing vial-in-vial systems, the new method developed here represents a significant methodological advancement in terms of higher capacity, reduced labour consumption, lower material costs, less contamination and, as a consequence, improved analytical accuracy following micro-scaled digestion of plant samples

SGAS 143845.1+145407: A Big, Cool Starburst at Redshift 0.816

We present the discovery and a detailed multi-wavelength study of a strongly-lensed luminous infrared galaxy at z=0.816. Unlike most known lensed galaxies discovered at optical or near-infrared wavelengths this lensed source is red, r-Ks = 3.9 [AB], which the data presented here demonstrate is due to ongoing dusty star formation. The overall lensing magnification (a factor of 17) facilitates observations from the blue optical through to 500micron, fully capturing both the stellar photospheric emission as well as the re-processed thermal dust emission. We also present optical and near-IR spectroscopy. These extensive data show that this lensed galaxy is in many ways typical of IR-detected sources at z~1, with both a total luminosity and size in accordance with other (albeit much less detailed) measurements in samples of galaxies observed in deep fields with the Spitzer telescope. Its far-infrared spectral energy distribution is well-fit by local templates that are an order of magnitude less luminous than the lensed galaxy; local templates of comparable luminosity are too hot to fit. Its size (D~7kpc) is much larger than local luminous infrared galaxies, but in line with sizes observed for such galaxies at z~1. The star formation appears uniform across this spatial scale. In this source, the luminosity of which is typical of sources that dominate the cosmic infrared background, we find that star formation is spatially extended and well organised, quite unlike the compact merger-driven starbursts which are typical for sources of this luminosity at z~0.Comment: 18 pages, 10 figure

Resonant Lifetime of Core-Excited Organic Adsorbates from First Principles

We investigate by first-principles simulations the resonant electron-transfer lifetime from the excited state of an organic adsorbate to a semiconductor surface, namely isonicotinic acid on rutile TiO$_2$(110). The molecule-substrate interaction is described using density functional theory, while the effect of a truly semi-infinite substrate is taken into account by Green's function techniques. Excitonic effects due to the presence of core-excited atoms in the molecule are shown to be instrumental to understand the electron-transfer times measured using the so-called core-hole-clock technique. In particular, for the isonicotinic acid on TiO$_2$(110), we find that the charge injection from the LUMO is quenched since this state lies within the substrate band gap. We compute the resonant charge-transfer times from LUMO+1 and LUMO+2, and systematically investigate the dependence of the elastic lifetimes of these states on the alignment among adsorbate and substrate states.Comment: 24 pages, 6 figures, to appear in Journal of Physical Chemistry