University of Illinois Urbana-Champaign

Generalized Kummer congruences and Iwasawa invariants

Gunaratne, Haputantirige Sunil

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https://hdl.handle.net/2142/20942

Description

Title
Generalized Kummer congruences and Iwasawa invariants
Author(s)
Gunaratne, Haputantirige Sunil
Issue Date
1991
Doctoral Committee Chair(s)
Ullom, Stephen V.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
  • We obtain the following generalization of the Kummer congruence: $$G\sb{c}(j,\chi,n) = -\left\lbrack{p\sp{-1}\Delta\sb{\rm c}\atop j}\right\rbrack {1\over n}(1 - \chi\omega\sp{-n}(p)\ p\sp{n-1}) B\sb{n,\chi\omega\sp{-n}}\in\doubz\sb{p}\lbrack\chi\rbrack ,$$where $B\sb{n,\chi}$ is the generalized Bernoulli number associated to the Dirichlet character $\chi,\ \Delta\sb{\rm c}$ is the difference operator$$\Delta\sb{\rm c} x\sb{n} = x\sb{n+c} - x\sb{n}\ {\rm and}\ \left\lbrack {p\sp{-1}\Delta\sb{\rm c}\atop j}\right\rbrack$$is a binomial coefficient operator.
  • The classical generalization of the Kummer congruence is$$K\sb{c}(j,\chi,n) = -p\sp{-j}\Delta\sbsp{\rm c}{j}{1\over n}(1 - \chi\omega\sp{-n}(p)\ p\sp{n-1})\ B\sb{n,\chi\omega\sp{-n}}\in\doubz\sb{p}\lbrack \chi\rbrack .$$We show that this is periodic (mod p) in the sense that$$K\sb{c}(j,\omega\sp{m},n)\equiv K\sb{c} (j\sp\prime,\omega\sp{m},n\sp\prime) (mod\ p\doubz\sb{p})$$if $j\equiv j\sp\prime$ $(mod\ p-1),\ j,\ j\sp\prime > 0,$ and $n\equiv n\sp\prime$ $(mod\ p-1).$
  • As a special case of a more general result on the $\mu$ and $\lambda$ invariants of a p-adic measure, we characterize the Iwasawa invariants $\mu(\chi)$ and $\lambda(\chi)$ as $\mu(\chi)$ = $min\{ord\sb\pi(G\sb{c}(j,\chi,n))\mid j\geq0\}$ and $\lambda(\chi) = min\{j\mid ord\sb\pi(G\sb{c}(j,\chi,n)) = \mu(\chi)\}$ provided that $(c,p) = 1,$ where $\pi$ is a local parameter of $\doubq\sb{\rm p}\lbrack\chi\rbrack.$
  • The Iwasawa characterization of $\mu$ = 0 and a theorem of Kida on p-adic measures are obtained as by products of the method used.
Type of Resource
text
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http://hdl.handle.net/2142/20942
Copyright and License Information
Copyright 1991 Gunaratne, Haputantirige Sunil

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