Eisenstein Classes, Elliptic Soulé Elements and the ℓ-Adic Elliptic Polylogarithm

Abstract

In this paper we study systematically the ℓ-adic realization of the elliptic polylogarithm in the context of sheaves of Iwasawa modules. This leads to a description of the elliptic polylogarithm in terms of elliptic units. As an application we prove a precise relation between ℓ-adic Eisenstein classes and elliptic Soulé elements. This allows to give a new proof of the formula for the residue of the ℓ-adic Eisenstein classes at the cusps and the formula for the cup-product construction in [HK99], which relies only on the explicit description of elliptic units. This computation is the main input in the proof of Bloch-Kato's compatibility conjecture 6.2. needed in the proof of Tamagawa number conjecture for the Riemann zeta function.

Rankin–Eisenstein classes and explicit reciprocity laws

“…Here Λ(H Zp t N ) is a sheaf of Iwasawa modules whose definition we recall below. These classes appeared (although not under this name) in an earlier paper of the first author [Kin15], and we recall below one of the main results of that paper, which asserts that the image of c EI b,N under the k-th moment map, for any k ≥ 0, coincides with Beilinson's weight kétale Eisenstein class. We also prove two distribution relations describing how the classes c EI b,N behave under pushforward maps, which will be used in the construction of the Euler system in the following sections.…”

“…which are the sheaves defined and studied in [Kin15]. These sheaves can and should be viewed as sheaves of modules under the sheaf of Iwasawa algebras Λ(H Zp ) := Λ(H Zp 0 ).…”

“…to the k-th symmetric power (we recall the definition in §2.2 below). The interpolation of these classes is carried out using the formalism of Lambda-adic sheaves introduced in [Kin15]. We consider the sheaf of Iwasawa modules Λ(H Zp t N ) associated to H Zp and its canonical order N section t N , which is equipped with moment maps mom k : Λ(H Zp t N ) → TSym k H Zp for all integers k ≥ 0.…”

“…We consider the sheaf of Iwasawa modules Λ(H Zp t N ) associated to H Zp and its canonical order N section t N , which is equipped with moment maps mom k : Λ(H Zp t N ) → TSym k H Zp for all integers k ≥ 0. One of the main results of [Kin15] is that there is a class, the Eisenstein-Iwasawa class,…”

Polylogarithm for families of commutative group schemes

“…j D * j * D Log(d)). This class has the advantage of having very good norm compatibility properties, which are useful in Iwasawa theoretic applications (see [Ki15]).…”