P-adic exponential function: Difference between revisions

Revision as of 15:49, 12 May 2017

Revision as of 10:49, 8 May 2019

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The power series

:<math>\log(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n},</math>

converges for ''x'' in '''C'''_''p'' satisfying |''x''|_''p''&nbsp;&lt;&nbsp;1 and so defines the '''''p''-adic logarithm function''' log_''p''(''z'') for |''z''&nbsp;&minus;&nbsp;1|_''p''&nbsp;&lt;&nbsp;1 satisfying the usual property log_''p''(''zw'')&nbsp;=&nbsp;log_''p''''z''&nbsp;+&nbsp;log_''p''''w''. The function log_''p'' can be extended to all of {{SubSup|'''C'''|''p''|×}} (the set of nonzero elements of '''C'''_''p'') by imposing that it continues to satisfy this last property and setting log_''p''(''p'')&nbsp;=&nbsp;0. Specifically, every element ''w'' of {{SubSup|'''C'''|''p''|×}} can be written as ''w''&nbsp;=&nbsp;''p^r''''·ζ·''z'' with ''r'' a rational number, ζ a root of unity, and |''z''&nbsp;&minus;&nbsp;1|_''p''&nbsp;&lt;&nbsp;1,<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.44}}</ref> in which case log_''p''(''w'')&nbsp;=&nbsp;log_''p''(''z'').<ref>In factoring ''w'' as above, there is a choice of a root involved in writing ''p^r'' since ''r'' is rational; however, different choices differ only by multiplication by a root of unity, which gets absorbed into the factor ζ.</ref> This function on {{SubSup|'''C'''|''p''|×}} is sometimes called the '''Iwasawa logarithm''' to emphasize the choice of log_''p''(''p'')&nbsp;=&nbsp;0. In fact, there is an extension of the logarithm from |''z''&nbsp;&minus;&nbsp;1|_''p''&nbsp;&lt;&nbsp;1 to all of {{SubSup|'''C'''|''p''|×}} for each choice of log_''p''(''p'') in '''C'''_''p''.<ref>{{harvnb|Cohen|2007|loc=§4.4.11}}</ref>

==Properties==

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For ''z'' in the ___domain of exp_''p'', we have exp_''p''(log_''p''(1+''z''))&nbsp;=&nbsp;1+''z'' and log_''p''(exp_''p''(''z''))&nbsp;=&nbsp;''z''.

The roots of the Iwasawa logarithm log_''p''(''z'') are exactly the elements of '''C'''_''p'' of the form ''p^r''''·ζ where ''r'' is a rational number and ζ is a root of unity.<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.45}}</ref>

Note that there is no analogue in '''C'''_''p'' of [[Euler's identity]], ''e''^2''πi''&nbsp;=&nbsp;1. This is a corollary of [[Strassmann's theorem]].