The Kronecker limit formulas via the distribution relation

In this paper, we give a proof of the classical Kronecker limit formulas using the distribution relation of the Eisenstein-Kronecker series. Using a similar idea, we then prove $p$-adic analogues of the Kronecker limit formulas for the $p$-adic Eisenstein-Kronecker functions defined in our previous paper

Algebraic theta functions and p-adic interpolation of Eisenstein-Kronecker numbers

We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke $L$-function of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincare bundle of an elliptic curve. We introduce general methods to study the algebraic and $p$-adic properties of reduced theta functions for CM abelian varieties. As a corollary, when the prime $p$ is ordinary, we give a new construction of the two-variable $p$-adic measure interpolating special values of Hecke $L$-functions of imaginary quadratic fields, originally constructed by Manin-Vishik and Katz. Our method via theta functions also gives insight for the case when $p$ is supersingular. The method of this paper will be used in subsequent papers to study the precise $p$-divisibility of critical values of Hecke $L$-functions associated to Hecke characters of quadratic imaginary fields for supersingular $p$, as well as explicit calculation in two-variables of the $p$-adic elliptic polylogarithm for CM elliptic curves.Comment: 55 pages, 2 figures. Minor misprints and errors were correcte

On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves

In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic analogue of the result of Beilinson and Levin expressing the complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. Our result is valid even if the elliptic curve has supersingular reduction at p.Comment: 61 pages, v2. Sections concerning the Hodge realization was moved to the appendi

Quantized vortices in superfluid helium and atomic Bose-Einstein condensates

This article reviews recent developments in the physics of quantized vortices in superfluid helium and atomic Bose-Einstein condensates. Quantized vortices appear in low-temperature quantum condensed systems as the direct product of Bose-Einstein condensation. Quantized vortices were first discovered in superfluid 4He in the 1950s, and have since been studied with a primary focus on the quantum hydrodynamics of this system. Since the discovery of superfluid 3He in 1972, quantized vortices characteristic of the anisotropic superfluid have been studied theoretically and observed experimentally using rotating cryostats. The realization of atomic Bose-Einstein condensation in 1995 has opened new possibilities, because it became possible to control and directly visualize condensates and quantized vortices. Historically, many ideas developed in superfluid 4He and 3He have been imported to the field of cold atoms and utilized effectively. Here, we review and summarize our current understanding of quantized vortices, bridging superfluid helium and atomic Bose-Einstein condensates. This review article begins with a basic introduction, which is followed by discussion of modern topics such as quantum turbulence and vortices in unusual cold atom condensates.Comment: 99 pages, 20 figures, Review articl

Observation of Hysteretic Transport Due to Dynamic Nuclear Spin Polarization in a GaAs Lateral Double Quantum Dot

We report a new transport feature in a GaAs lateral double quantum dot that emerges only for magnetic field sweeps and shows hysteresis due to dynamic nuclear spin polarization (DNP). This DNP signal appears in the Coulomb blockade regime by virtue of the finite inter-dot tunnel coupling and originates from the crossing between ground levels of the spin triplet and singlet extensively used for nuclear spin manipulations in pulsed gate experiments. The unexpectedly large signal intensity is suggestive of unbalanced DNP between the two dots, which opens up the possibility of controlling electron and nuclear spin states via DC transport.Comment: 5 pages, 4 figure