ON THE REAL HODGE AND p-ADIC REALIZATIONS OF THE ELLIPTIC POLYLOGARITHM FOR CM ELLIPTIC CURVES
نویسندگان
چکیده
In this paper, we give an explicit description of the complex 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. 0. Introduction 0.
منابع مشابه
Realizations of the Elliptic Polylogarithm for CM elliptic curves
In these notes, we give an overview of our paper [BKT] which gives an explicit description of the de Rham and p-adic realizations of the elliptic polylogarithm, for a general elliptic curve defined over a subfield of C in the de Rham case and for a CM elliptic curve defined over its field of complex multiplication and with good reduction at the primes above p ≥ 5 in the p-adic case. As explaine...
متن کاملA Survey of the Hodge-Arakelov Theory of Elliptic Curves I
The purpose of the present manuscript is to give a survey of the Hodge-Arakelov theory of elliptic curves (cf. [Mzk1,2]) — i.e., a sort of “Hodge theory of elliptic curves” analogous to the classical complex and p-adic Hodge theories, but which exists in the global arithmetic framework of Arakelov theory — as this theory existed at the time of the workshop on “Galois Actions and Geometry” held ...
متن کاملConnections and Related Integral Structures on the Universal Extension of an Elliptic Curve
§0. Introduction §1. The Étale Integral Structure on the Universal Extension §2. The Étale Integral Structure for an Ordinary Elliptic Curve §2.1. Some p-adic Function Theory §2.2. The Verschiebung Morphism §3. Compactified Hodge Torsors §4. The Étale Integral Structure on the Hodge Torsors §4.1. Notation and Set-Up §4.2. Degenerating Elliptic Curves §4.3. Ordinary Elliptic Curves §4.4. The Gen...
متن کاملAnabelian Geometry in the Hodge-Arakelov Theory of Elliptic Curves
The purpose of the present manuscript is to survey some of the main ideas that appear in recent research of the author on the topic of applying anabelian geometry to construct a “global multiplicative subspace”— i.e., an analogue of the well-known (local) multiplicative subspace of the Tate module of a degenerating elliptic curve. Such a global multiplicative subspace is necessary to apply the ...
متن کاملGalois Representations and Elliptic Curves
An elliptic curve over a field K is a projective nonsingular genus 1 curve E over K along with a chosen K-rational point O of E, which automatically becomes an algebraic group with identity O. If K has characteristic 0, the n-torsion of E, denoted E[n], is isomorphic to (Z/nZ) over K. The absolute Galois group GK acts on these points as a group automorphism, hence it acts on the inverse limit l...
متن کامل