On T-Semisimplicity of Iwasawa Modules and Some Computations with Z3-Extensions

Thesis (Ph.D.)--University of Washington, 2016-08

Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams

We present a complete solution to the problem of coherent-mode decomposition of the most general anisotropic Gaussian Schell-model (AGSM) beams, which constitute a ten-parameter family. Our approach is based on symmetry considerations. Concepts and techniques familiar from the context of quantum mechanics in the two-dimensional plane are used to exploit the Sp(4, R) dynamical symmetry underlying the AGSM problem. We take advantage of the fact that the symplectic group of first-order optical system acts unitarily through the metaplectic operators on the Hilbert space of wave amplitudes over the transverse plane, and, using the Iwasawa decomposition for the metaplectic operator and the classic theorem of Williamson on the normal forms of positive definite symmetric matrices under linear canonical transformations, we demonstrate the unitary equivalence of the AGSM problem to a separable problem earlier studied by Li and Wolf [Opt. Lett. 7, 256 (1982)] and Gori and Guattari [Opt. Commun. 48, 7 (1983)]. This conn ction enables one to write down, almost by inspection, the coherent-mode decomposition of the general AGSM beam. A universal feature of the eigenvalue spectrum of the AGSM family is noted.

Strong electron correlation of Re 5d electrons in Ca2FeReO6

We have investigated the electronic structure of a double perovskite Ca2FeReO6 using photoemission spectroscopy and LDA+U bandstructure calculations. Small spectral weight at the Fermi level observed above the metal–insulator transition temperature, gradually disappears with decreasing T, forming a small (≤50 meV) energy gap. To reproduce this small energy gap, we require a very large effective U (Ueff) for Re (4 eV) in addition to Ueff of 4 eV for Fe. From simple calculations in terms of the ionic radii, we demonstrate that the Fe–Re bandwidth is smaller than that of Fe–Mo in Ca2FeMoO6, which should yield a strong electron correlation in the Re 5d bands.

On the local Tamagawa number conjecture for Tate motives

There is a wonderful conjecture of Bloch and Kato that generalizes both the analytic Class Number Formula and the Birch and Swinnerton-Dyer conjecture. The conjecture itself was generalized by Fukaya and Kato to an equivariant formulation. In this thesis, I provide a new proof for the equivariant local Tamagawa number conjecture in the case of Tate motives for unramified fields, using Iwasawa theory and (φ,Γ)-modules, and provide some work towards extending the proof to tamely ramified fields.

Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(φ) of a two-dimensional modular Galois representation φ. We start with the p-adic Galois representation φ0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(φ0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let φ denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting φ0, and the characteristic power series of the Selmer group Sel(ad(φ)) is given by a p-adic L-function interpolating L(1, ad(φk)) for weight k + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(φ) ⊗ ν−1), where ν is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

The structure of Selmer groups

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper Λ-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper Λ-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed.

On the coefficients of the characteristic series of the U-operator

A conceptual proof is given of the fact that the coefficients of the characteristic series of the U-operator acting on families of overconvegent modular forms lie in the Iwasawa algebra.

On Selmer groups and factoring p-adic L-functions

Thesis (Ph.D.)--University of Washington, 2016-06

IRAS 00183-7111: ALMA and X-ray view of an ultra luminous infrared galaxy

The aim of this Thesis work is to study the multi-frequency properties of the Ultra Luminous Infrared Galaxy (ULIRG) IRAS 00183-7111 (I00183) at z = 0.327, connecting ALMA sub-mm/mm observations with those at high energies in order to place constraints on the properties of its central power source and verify whether the gas traced by the CO may be responsible for the obscuration observed in X-rays. I00183 was selected from the so-called Spoon diagnostic diagram (Spoon et al. 2007) for mid-infrared spectra of infrared galaxies based on the equivalent width of the 6.2 μm Polycyclic Aromatic Hydrocarbon (PAH) emission feature versus the 9.7 μm silicate strength. Such features are a powerful tool to investigate the contribution of star formation and AGN activity in this class of objects. I00183 was selected from the top-left region of the plot where the most obscured sources, characterized by a strong Si absorption feature, are located. To link the sub-mm/mm to the X-ray properties of I00183, ALMA archival Cycle 0 data in Band 3 (87 GHz) and Band 6 (270 GHz) have been calibrated and analyzed, using CASA software. ALMA Cycle 0 was the Early Science program for which data reprocessing is strongly suggested. The main work of this Thesis consisted in reprocessing raw data to provide an improvement with respect to the available archival products and results, which were obtained using standard procedures. The high-energy data consists of Chandra, XMM-Newton and NuSTAR observations which provide a broad coverage of the spectrum in the energy range 0.5 − 30 keV. Chandra and XMM archival data were used, with an exposure time of 22 and 22.2 ks, respectively; their reduction was carried out using CIAO and SAS software. The 100 ks NuSTAR are still private and the spectra were obtained by courtesy of the PI (K. Iwasawa). A detailed spectral analysis was done using XSPEC software; the spectral shape was reproduced starting from simple phenomenological models, and then more physical models were introduced to account for the complex mechanisms that involve this source. In Chapter 1, an overview of the scientific background is discussed, with a focus on the target, I00183, and the Spoon diagnostic diagram, from which it was originally selected. In Chapter 2, the basic principles of interferometry are briefly introduced, with a description of the calibration theory applied to interferometric observations. In Chapter 3, ALMA and its capabilities, both current and future, are shown, explaining also the complex structure of the ALMA archive. In Chapter 4, the calibration of ALMA data is presented and discussed, showing also the obtained imaging products. In Chapter 5, the analysis and discussion of the main results obtained from ALMA data is presented. In Chapter 6, the X-ray observations, data reduction and spectral analysis are reported, with a brief introduction to the basic principle of X-ray astronomy and the instruments from which the observations were carried out. Finally, the overall work is summarized, with particular emphasis on the main obtained results and the possible future perspectives.