On l-adic Galois L-functions

Abstract

Let ${z}\;\in \;\mathbb{Q}$ and let γ be an l-adic path on $\mathbb{P}\frac{1}{\mathbb{Q}}\backslash \{0,1,\infty\}\;\mathrm{from}\;\vec{01}\;\mathrm{to}\;z$. For any $\sigma\in Gal(\bar{\mathbb{Q}}/\mathbb{Q})$, the element $x^{-\kappa(\sigma)}f_{\gamma}(\sigma)\;\in\;\pi_{1}(\mathbb{P}\frac{1}{\mathbb{Q}}\backslash \{0,1,\infty\}, \vec{01})_{pro-l}$. After the embedding of π₁ into $\mathbb{Q}\{\{X,Y\}\}$ we get the formal power series $\Delta_\gamma(\sigma)\;\in\;\mathbb{Q}\{\{X,Y\}\}$. We shall express coefficients of $\Delta_\gamma(\sigma)$as integrals over $(\mathbb{Z}_l)^r$ with respect to some measures $K_r(z)$. The measures $K_r(z)$ are constructed using the tower $(\mathbb{P}\frac{1}{\mathbb{Q}}\backslash(\{0,\infty\}\cup\mu_{l^n})_{n\in\mathbb{N}}$ of coverings of $\mathbb{P}\frac{1}{\mathbb{Q}}\backslash\{0,1,\infty\}$. Using the integral formulas we shall show congruence relations between coefficients of the formal power series $\Delta_\gamma(\sigma)$. The measures allow the construction of l-adic functions of non-Archimedean analysis, which however rest mysterious. Only in the special case of the measures $K_1(\vec{10})\;\mathrm{and}\;K_1(-1)$we recover the familiar Kubota–Leopoldt l-adic L-functions. We recover also l-adic analogues of Hurwitz zeta functions. Hence we get also l-adic analogues of L-series for Dirichlet characters.

Mathematics Subject Classification (2010). 11G55, 11G99, 14G32, 14H30