Singular values of some modular functions

We study the properties of special values of the modular functions obtained from Weierstrass P-function at imaginary quadratic points.Comment: 19 pages,corrected typo

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

Precise variational tunneling rates for anharmonic oscillator with g<0

We systematically improve the recent variational calculation of the imaginary part of the ground state energy of the quartic anharmonic oscillator. The results are extremely accurate as demonstrated by deriving, from the calculated imaginary part, all perturbation coefficients via a dispersion relation and reproducing the exact values with a relative error of less than $10^{-5}$. A comparison is also made with results of a Schr\"{o}dinger calculation based on the complex rotation method.Comment: PostScrip