\(p\)-adic \(L\)-functions, Iwasawa theory and Heegner points. (Fonctions \(L\) \(p\)-adiques, théorie d’Iwasawa et points de Heegner.) (French) Zbl 0664.12010

Perrin-Riou, Bernadette

Bull. Soc. Math. Fr. 115, 399-456 (1987).

Let \(E\) be a modular elliptic curve over \({\mathbb Q}\) of conductor \(N\). Let \(k\) be an imaginary quadratic field such that each prime factor of \(N\) splits in \(k\). Denote by \(k_{\infty}/k\) the unique \({\mathbb Z}^ 2_ p\)-extension of \(k\); it is the composition of the cyclotomic extension \(C_{\infty}\) and the dihedral extension \(D_{\infty}.\)
The author studies an Iwasawa \(\Lambda_{D_{\infty}/k}\)-module \(H_{\infty}\) constructed by means of the Heegner points belonging to \(E(D_ n)\), for every finite subextension \(D_ n\) of \(D_{\infty}/k\). \(H_{\infty}\) turns out to be free of dimension \(\leq 1.\)
The characteristic series \({\mathcal L}_ p(E| k_{\infty})\) of the Pontryagin dual of the Selmer group \(S_ p(k_{\infty})\) should play a role in the study of the \(p\)-adic functions attached to \(E\). The multiplicity of its zeros is studied in the paper and, moreover, some conjectures in the Gross-Zagier style, relating its principal values to a height pairing defined over \(H_{\infty}\) are formulated.
In an appendix the author studies also the variation by isogeny of the algebraic \(p\)-adic \(L\)-function \({\mathcal L}_ p(E| k_{\infty})\).

Reviewer: P.Bayer

MSC:

11R23	Iwasawa theory
11R42	Zeta functions and \(L\)-functions of number fields
11S40	Zeta functions and \(L\)-functions
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25	Arithmetic ground fields for curves

Keywords:

Iwasawa module; modular elliptic curve; Heegner points; height pairing; algebraic \(p\)-adic \(L\)-function

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