$p$-adic regulator
Let $K$, $n$, ${r}_{1}$, ${r}_{2}$, $\{{\epsilon}_{n},\mathrm{\dots},{\epsilon}_{r-1}\}$, and $||\cdot |{|}_{i}$ be as in the entry regulator^{}, but with $K$ taken to be a CM field.
Define the $p$-adic logarithm ${\mathrm{log}}_{p}:{\u2102}_{p}^{\times}\to {\u2102}_{p}$ by
${\mathrm{log}}_{p}(x)=-{\displaystyle \sum _{k=1}^{\mathrm{\infty}}}{\displaystyle \frac{{(1-x)}^{k}}{k}}$ |
Let ${A}_{K,p}$ be the $(r-1)\times (r-1)$ matrix with general entry given by ${a}_{i,j}={\mathrm{log}}_{p}{||{\epsilon}_{j}||}_{i}$. The absolute value^{} of the determinant^{} of this matrix is again independent of your choice of basis for the units and of the ordering of the embeddings^{}. This value is called the $p$-adic regulator of $K$, and is denoted by ${R}_{p,K}$, or ${R}_{p}(K)$.
References
- 1 L. C. Washington, Introduction to Cyclotomic Fields^{}, Springer-Verlag, New York.
Title | $p$-adic regulator |
---|---|
Canonical name | PadicRegulator |
Date of creation | 2013-03-22 14:14:13 |
Last modified on | 2013-03-22 14:14:13 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 5 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 11R27 |
Related topic | PAdicExponentialAndPAdicLogarithm |
Defines | $p$-adic logarithm |