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$p$-adic Beilinson conjecture for ordinary Hecke motives associated to imaginary quadratic fields

joint work with Guido Kings

RIMS Kôkyûroku Bessatsu B25: Algebraic Number Theory and Related Topics 2009, eds. T. Ichikawa, M. Kida, T. Yamazaki, June (2011), 9-30.

Abstract. The purpose of this article is to show that the result of [BK] may be used to prove the $p$-adic Beilinson conjecture at non-critical points of motives associated to Hecke characters of an imaginary quadratic field $K$, for a prime $p$ which splits in $K$. For simplicity, we assume in this article that the imaginary quadratic field $K$ has class number one and that the Hecke character $\psi$ we consider corresponds to an elliptic curve with complex multiplication defined over $\mathbb{Q}$.

p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure

joint work with Guido Kings

American J. Math. 132, no. 6 (2010), 1609-1654;
Available by subscription at Project MUSE.

Abstract. The specializations of the motivic elliptic polylog are called motivic Eisenstein classes. For applications to special values of L-functions, it is important to compute the realizations of these classes. In this paper, we prove that the syntomic realization of the motivic Eisenstein classes, restricted to the ordinary locus of the modular curve, may be expressed using p-adic Eisenstein-Kronecker series. These p-adic modular forms are defined using the two-variable p-adic measure with values in p-adic modular forms constructed by Katz.

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

joint work with Shinichi Kobayashi

Duke Math. J. 153 no. 2 (2010), 229-295; Available by subscription at Project Euclid.

Abstract. 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.

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

joint work with Shinichi Kobayashi and Takeshi Tsuji

Annales scientifiques de l'ENS 43, fascicule 2 (2010), 185-234;

Available by subscription at the Société Mathématique de France.

Abstract. 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.

Realizations of the elliptic polylogarithm for CM elliptic curves

joint work with Shinichi Kobayashi and Takeshi Tsuji

RIMS Kôkyûroku Bessatsu B12: Algebraic Number Theory and Related Topics 2007, eds. M. Asada, H. Nakamura and H. Takahashi, August (2009), 33--50.

Abstract. In these notes, we give an overview of our paper [BKT] which gives an explicit description of the real Hodge and p-adic realizations of the elliptic polylogarithm for CM elliptic curves.

Algebraic theta functions and Eisenstein-Kronecker numbers

joint work with Shinichi Kobayashi

RIMS Kôkyûroku Bessatsu B4: Proceedings of the Symposium on Algebraic Number theory and Related Topics, eds. K. Hashimoto, Y. Nakajima and H. Tsunogai, December (2007), 63--78; Available at RIMS.

Abstract. In this paper, we give an overview of our previous paper concerning the investigation of the algebraic and p-adic properties of Eisenstein-Kronecker numbers using Mumford's theory of algebraic theta functions.

Specialization of the p-adic polylogarithm to p-th power roots of unity

Doc. Math., Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 73-97; Available at Documenta Mathematica.

Abstract. The purpose of this paper is to calculate the restriction of the p-adic polylogarithm sheaf to p-th power torsion points.

On the p-adic realization of elliptic polylogarithms for CM-elliptic curves

Duke Math. J. 113 (2002), 193-236; Available by subscription at Project Euclid.

Abstract. Let E be a CM-elliptic curve over Q with good ordinary reduction at a prime p greater than or equal to 5. The purpose of this paper is to construct the p-adic elliptic polylogarithm of E, following the method of A. Beilinson and A. Levin. Our main result is that the specializations of this object at torsion points give the special values of the one-variable p-adic L-function of the Grossencharakter associated to E.

Syntomic cohomology as a p-adic absolute Hodge cohomology

Math. Z. 242/3 (2002), 443-480; Available by subscription at Springer Link.

Abstract. The purpose of this paper is to interpret rigid syntomic cohomology, defined by Amnon Besser, as a p-adic absolute Hodge cohomology. This is a p-adic analogue of a work of Beilinson which interprets Beilinson-Deligne cohomology in terms of absolute Hodge cohomology. In the process, we will define a theory of p-adic absolute Hodge cohomology with coefficients, which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients.

Rigid syntomic cohomology and p-adic polylogarithms

J. Reine Angew. Math. 529 (2000), 205-237; Available by subscription at Atypon Link.

Abstract. The main purpose of this paper is to construct the p-adic realization of the classical polylogarithm following the method of Beilinson and Deligne as explained by Huber and Wildeshaus. A simplicial construction of the p-adic polylogarithm was previously obtained by Somekawa. In this paper, we will give a sheaf theoretic interpretation of this construction. In particular, we will give an interpretation of the p-adic polylogarithm as an object in the p-adic analogue of the category of variation of mixed Hodge structures. We will also calculate the restriction of this object to torsion points, and will prove a result which is compatible with the results of Gros-Kurihara, Gros and Somekawa.