1,002 research outputs found
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 -adic analogues of the Kronecker limit formulas
for the -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 -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
-adic properties of reduced theta functions for CM abelian varieties. As a
corollary, when the prime is ordinary, we give a new construction of the
two-variable -adic measure interpolating special values of Hecke
-functions of imaginary quadratic fields, originally constructed by
Manin-Vishik and Katz. Our method via theta functions also gives insight for
the case when is supersingular. The method of this paper will be used in
subsequent papers to study the precise -divisibility of critical values of
Hecke -functions associated to Hecke characters of quadratic imaginary
fields for supersingular , as well as explicit calculation in two-variables
of the -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
Opening Remarks 2
書名: Higher Education Research: Challenges and Prospects. Report of RIHE’s 50th Anniversary International Symposium, 202
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