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Variation de la fonction \(L\) \(p\)-adique par isogĂ©nie. (Variation of the \(p\)-adic \(L\)-function by isogeny). (French) Zbl 0747.11056
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, 347-358 (1989).
[For the entire collection see Zbl 0721.00006.]
Let \(p\) be an odd prime number, \(W\) a \(p\)-adic representation of \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\). Let \(\mathbb{Q}_ \infty\) be the cyclotomic \(\mathbb{Z}_ p\) extension of \(\mathbb{Q}\), with Galois group \(\Gamma\) over \(\mathbb{Q}\). Choose a lattice \(L\) of \(W\), stable for \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\). Greenberg constructed an Iwasawa module on the algebra \(\mathbb{Z}_ p[[\Gamma]]\). Suppose it is of \(\mathbb{Z}_ p[[\Gamma]]\)-torsion, an element \(f_ L\) is defined, which is called its “characteristic series”. Suppose \(\mu(L)\) the \(\mu\)-invariant of \(f_ L\).
The aim of the paper is to investigate the variation of \(\mu(L)\) with \(L\). The results can be compared to the variation of the \(p\)-adic periods. It permits to conjecture on a link between the characteristic series and the \(p\)-adic \(L\)-functions.

11S40 Zeta functions and \(L\)-functions
11R23 Iwasawa theory
14K02 Isogeny
11F85 \(p\)-adic theory, local fields