# zbMATH — the first resource for mathematics

On the de Rham and $$p$$-adic realizations of the elliptic polylogarithm for CM elliptic curves. (English) Zbl 1197.11073
The present paper aims at studying the $$p$$-adic realization of the elliptic polylogarithm for a CM elliptic curve $$E$$, even in case $$E$$ has supersingular reduction at $$p$$: First of all, the de Rham realization of the elliptic polylog for a general elliptic curve defined over a subfield of $$\mathbb C$$ along with its connection is described. This realization gives a coherent module with a connection underlying the polylog sheaf in the Hodge and $$p$$-adic cases. The $$p$$-adic realization of the elliptic polylog is constructed as a filtered, overconvergent $$F$$-isocrystal on $$E$$ minus the identity in case some requirements on the curve are fulfilled.
The main results of this paper are the following: The Frobenius structure on the $$p$$-adic realization of the elliptic polylog sheaf is explicitly described in terms of overconvergent functions, which are solutions of the $$p$$-adic differential equations arising from the compatibility of the Frobenius with the connection on the elliptic polylog. This explicit description allows the authors to compute the specializations of the $$p$$-adic elliptic polylog to torsion points of $$E$$ of order prime to $$p$$. As a generalization of [K. Bannai, “On the $$p$$-adic realization of elliptic polylogarithms for CM-elliptic curves”, Duke Math. J. 113, No. 2, 193–236 (2002; Zbl 1019.11018)], the authors prove that these specializations are given by $$p$$-adic Eisenstein-Kronecker numbers, i.e. special values of the $$p$$-adic distribution interpolating “ordinary” Eisenstein-Kronecker numbers. Note that a similar result was obtained in [K. Bannai and G. Kings, “$$p$$-adic elliptic polylogarithms, $$p$$-adic Eisenstein series and Katz measure, arXiv 0707:3747, to appear in Am. J. Math. 132, No. 6, 1609–1654 (2010; Zbl 1225.11075)] using different methods. The authors also show that in case the elliptic curve in question has good reduction a the primes above $$p$$, the $$p$$-adic Kronecker-Eisenstein numbers are related to special values of $$p$$-adic $$L$$-functions which $$p$$-adically interpolate special values of certain Hecke $$L$$-functions associated to imaginary quadratic fields.
In an appendix, an alternative description of the real Hodge realization in terms of multi-valued meromorphic functions, which solve similar differential equations as in the $$p$$-adic case, is given. This description differs from the ones of A. Beilinson and A. Levin in [Proc. Symp. Pure Math. 55, Pt. 2, 123–190 (1994; Zbl 0817.14014)] and of J. Wildeshaus in [“Realizations of polylogarithms”, Lecture Notes in Mathematics. 1650. Berlin: Springer (1997; Zbl 0877.11001)].

##### MSC:
 11G55 Polylogarithms and relations with $$K$$-theory 11G05 Elliptic curves over global fields 11G15 Complex multiplication and moduli of abelian varieties 14F30 $$p$$-adic cohomology, crystalline cohomology 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Full Text: