\(q\)-adic valuation and additive distribution relation for certain \(q\)-periodic functions
The author gives a local analogue for Tate elliptic curves \(\overline{K}^\times / q^{\mathbb{Z}}\) over a local field \(K\) of the additive distribution law for Siegel functions on classical complex elliptic curves \(\mathbb{C}/\Omega\), that he obtained with \textit{G. Robert} in a previous article [``Une relation de distribution additive satisfaite par une famille de fonctions elliptiques'', Bull. Soc. Math. Fr. (to appear)]. Given a local field \(K\), an element \(q\) of \(K^\times \) with strictly positive valuation, and a torsion point \(\psi\) in \(\overline{K}^\times / q^{\mathbb{Z}}\), he introduces in a quite natural way a local Siegel function associated to the discrete subgroup \(q^{\mathbb{Z}} \psi^{\mathbb{Z}}\) of \(K^\times \) and an ad hoc local discriminant function. Then he is able to compute the \(q\)-valuations of these functions evaluated at certain torsion points (theorem 3.13) and to prove that they satisfy an additive distribution relation (theorem 4.1). In the appendix, the author uses the results above to give a nice simplification of local computations of \(q\)-adic valuations for products of elliptic resolvents which were used to construct quadratic Stickelberger elements in a previous article with \textit{W. Bley} and \textit{Ph. Cassou-Noguès} [J. Algebra 179, 145-190 (1996; Zbl 0863.11074)].
additive distribution law for Siegel functions
scientific article; zbMATH DE number 1037289
scientific article