A family of cubic fields and zeros of 3-adic \(L\)-functions
The author studies cyclic cubic fields \(K_n\) defined by the polynomials \[ X^3 - (n^3-2n^2+3n-3)X^2 - n^2X - 1, \] where \(n\in \mathbb{Z},\;n\neq 1\). He is able to explicitly determine the fundamental units of \(K_n\). As an application he shows that the 3-adic \(L\)-function \(L_3(s,\chi)\) for \(K_n\) has a zero arbitrarily close to \(s=1\) if \(n\) is sufficiently close to 1 (both distances in the 3-adic metric) and \(n\) satisfies certain extra conditions. There exist such \(n\) if Schinzel's hypothesis H is true; for this hypothesis, see \textit{A. Schinzel} and \textit{W. Sierpiński} [Acta Arith. 4, 185-208 (1958; Zbl 0082.25802), Erratum 5, 259 (1958)]. The proof relies on the 3-adic class number formula and makes use of the fact that the \(\lambda^-\)-invariant of \(K_n\) is bounded (in fact, constant). In a previous work [in: Dilcher, Karl (ed.), Number theory. Fourth conference of the Canadian Number Theory Association, July 2-8, 1994; CMS Conf. Proc. 15, 393-396 (1995; Zbl 0840.11050)] the author found 2-adic \(L\)-functions with zeros close to \(s=1\).
scientific article; zbMATH DE number 1011064
scientific article