# 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.

##### MSC:
 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)