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

Abstract

Abstract. 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.0. Introduction 0.1. Introduction. In the paper [BL], Beilinson and Levin constructed the elliptic polylogarithm, which is an element in absolute Hodge or ℓ-adic cohomology of an elliptic curve minus the identity. This construction is a generalization to the case of elliptic curves of the construction by Beilinson and Deligne of the polylogarithm sheaf on the projective line minus three points. The purpose of this paper is to study the p-adic realization of the elliptic polylogarithm for an elliptic curve with complex multiplication, and to investigate its relation to p-adic L-functions associated to the elliptic curve, even for the case where the elliptic curve has supersingular reduction at the prime p.To achieve our goal, we first describe the de Rham realization of the elliptic polylogarithm for a general elliptic curve defined over a subfield of C. In particular, we explicitly describe the connection of the elliptic polylogarithm using rational functions. This result is of independent interest. Similar results were obtained by Levin and Racinet [LR] Section 5.1.3. A different description for this connection was also given by Besser and Solomon [BS].Using the de Rham realization of the elliptic polylogarithm, we then construct the p-adic realization of the elliptic polylogarithm as a filtered overconvergent F -isocrystal on the elliptic curve minus the identity, when the elliptic curve has complex multiplication and good reduction at a fixed prime p ≥ 5. Our main result Theorem 4.15 is an explicit description of the p-adic Using this description, we calculate the specializations of the p-adic elliptic polylogarithm to torsion points of order prime to p (more precisely, torsion points of order prime to p. See the Overview for details), and prove that the specializations give the p-adic Eisenstein-Kronecker numbers, which are special values of the p-adic distribution interpolating Eisenstein-Kronecker numbers. This result is a generalization of the result of [Ba3], where we have dealt only with the one variable case for an ordinary prime. A similar result concerning the specialization in two-variables was obtained in [BKi], again for ordinary primes, using a very different method. The result of the current paper is valid even when p is supersingular.The p-adic Eisenstein-Kronecker numbers are related to special values of p-adic L-functions which p-adically interpolate special values of Hecke L-f...

Overview on Elliptic Multiple Zeta Values

“…The terminology for (21) varies in the literature. In [44], it is called "a meromorphic Jacobi form", while in [1], it is referred to as "the Kronecker theta function". Since F τ is meromorphic, it has a Laurent expansion in α.…”

“…The second goal is to collect various results on the structure of the algebra of A-elliptic multiple zeta values, which appeared in work of the author [36], as well as in joint work [5,6]. 1…”

p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

“…which is equal to df (t) on ]0[, hence the function f (t)−E Furthermore, since the power series exp(Ω −1 p λ(t)) gives a homomorphism of formal groups E and G m isomorphically mapping E[π] to the group of p-th root of unity [BKT2] §2.2, we have for each y ∈ Z × p the equality…”

“…The elliptic analogue of the classical polylogarithm sheaf was first constructed by Belinson and Levin [BL]. In previous research, we studied the p-adic realization of the elliptic polylogarithm sheaf for CM elliptic curves ([Ba2], [BKT2], [BKT1]). As in the classical case, the p-adic Eisenstein-Kronecker function defined in this paper should be related to specializations at p-power torsion points of the p-adic realization of the elliptic polylogarithm sheaf.…”

The Hodge realization of the polylogarithm on the product of multiplicative groups

“…In fact, the construction of the polylogarithm was extended to elliptic curves by Beilinson-Levin [BL94], to abelian schemes by Wildeshaus [Wil97] and Kings [Kin09], and more recently to general commutative group schemes by Huber and Kings [HK18]. The explicit description of the Hodge realization of the polylogarithm was given for elliptic curves by Beilinson-Levin [BL94] (see also [BKT10,Appendix]), and the description of the topological sheaf underlying the Hodge realization of the polylogarithm for general abelian varieties was given by Levin [Lev97] and Blottière [Blo09]. The purpose of this article is to describe directly the Hodge realization of the polylogarithm as an explicit class in absolute Hodge cohomology for the product of multiplicative groups.…”