zbMATH — the first resource for mathematics

Duality theorems for abelian varieties over \(\mathbb{Z}_ p\)-extensions. (English) Zbl 0746.14011
Algebraic number theory - in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values \(L\)-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 471-492 (1989).
[For the entire collection see Zbl 0721.00006.]
The purpose of this paper is the study of \(p\)-adic heights in a \(\mathbb{Z}_ p\)-extension \(k_ \infty/k\) of a number field \(k\) for an abelian variety \(A\). The author looks at the Selmer group \(H^ i({\mathfrak O}_ \infty,{\mathcal A}(p))\) (\(fppf\) cohomology of the \(p\)-divisible group deduced from the Néron model) and its structure as module over the complete group algebra \(\Lambda\) of the Galois group \(\text{Gal}(k_ \infty/k)\). He takes the \(\Lambda\)-torsion part, \(T({\mathcal A})\), of its Pontryagin dual. He reviews the definition of the adjoint module [with the definition given by B. Perrin-Riou in here thesis, “Arithmétique des courbes elliptiques et theorie d’Iwasawa,” Mém. Soc. Math. Fr., Nouv. Sér. 17 (1984; Zbl 0599.14020)] and constructs a map from \(T({\mathcal A}')\) to the adjoint of \(T({\mathcal A^ 0})\). Here \(T({\mathcal A}')\) is obtained by taking the dual of \({\mathcal A}\) instead of it and \(T({\mathcal A}^ 0)\) by replacing \({\mathcal A}\) by its connected component \({\mathcal A}^ 0\).
Theorem 3.1 states that the map is an isomorphism (up to finite groups) in the ordinary case. From this the author deduces a functional equation for the (algebraic) \(p\)-adic \(L\)-function as well as consequences on the Tate-Shafarevich group. — The method uses (as in Schneider’s papers) the Artin-Mazur duality theory and a descent diagram. The paper gives also some results in the case of supersingular reduction.

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11R23 Iwasawa theory
14K05 Algebraic theory of abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
11R42 Zeta functions and \(L\)-functions of number fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)