On the de Rham and \(p\)-adic realizations of the elliptic polylogarithm for CM elliptic curves.

*(English)*Zbl 1197.11073The 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)].

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)].

Reviewer: Oliver Petras (Bonn)

##### 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 |