PlanetMath: $p$-adic regulator

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-adic regulator

(Definition)

Let , , , , $\{\varepsilon_n,\ldots,\varepsilon_{r-1}\}$ , and $\vert\vert\cdot\vert\vert _i$ be as in the entry regulator, but with taken to be a CM field.

Define the -adic logarithm $\log_p: \mathbb{C}_p^\times\rightarrow \mathbb{C}_p$ by

$\displaystyle \log_p(x)=-\sum_{k=1}^\infty \frac{(1-x)^k}{k}$

Let $A_{K,p}$ be the $(r-1)\times (r-1)$ matrix with general entry given by $a_{i,j}=\log_p \vert\vert\varepsilon_j\vert\vert _i$ . 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 -adic regulator of , and is denoted by $R_{p,K}$ , or .

Bibliography

1: L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.

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-adic logarithm

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Cross-references: embeddings, ordering, units, basis, independent, determinant, absolute value, matrix, logarithm, field, regulator
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This is version 2 of $p$

-adic regulator, born on 2004-03-10, modified 2006-08-17.
Object id is 5679, canonical name is PAdicRegulator.
Accessed 2186 times total.

Classification:

AMS MSC:

11R27 (Number theory :: Algebraic number theory: global fields :: Units and factorization)

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