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Generalized Kummer congruences and Iwasawa invariants

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Title: Generalized Kummer congruences and Iwasawa invariants
Author(s): Gunaratne, Haputantirige Sunil
Doctoral Committee Chair(s): Ullom, Stephen V.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
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.
Issue Date: 1991
Type: Text
Language: English
Rights Information: Copyright 1991 Gunaratne, Haputantirige Sunil
Date Available in IDEALS: 2011-05-07
Identifier in Online Catalog: AAI9210821
OCLC Identifier: (UMI)AAI9210821

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