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.

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