{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20942"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20942","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Generalized Kummer congruences and Iwasawa invariants","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.","abstract_html":"We obtain the following generalization of the Kummer congruence: $<span class=\"etd-inline-math\">G\\sb{c}(j,\\chi,n) = -\\left\\lbrack{p\\sp{-1}\\Delta\\sb{\\rm c}\\atop j}\\right\\rbrack {1\\over n}(1 - \\chi&omega;\\sp{-n}(p) p\\sp{n-1}) B\\sb{n,\\chi&omega;\\sp{-n}}\\in\\doubz\\sb{p}\\lbrack\\chi\\rbrack ,</span>$where $B\\sb{n,\\chi}$ is the generalized Bernoulli number associated to the Dirichlet character $\\chi,\\ \\Delta\\sb{\\rm c}$ is the difference operator$<span class=\"etd-inline-math\">\\Delta\\sb{\\rm c} x\\sb{n} = x\\sb{n+c} - x\\sb{n} {\\rm and} \\left\\lbrack {p\\sp{-1}\\Delta\\sb{\\rm c}\\atop j}\\right\\rbrack</span>$is a binomial coefficient operator.","abstract_has_math":true,"creators":["Gunaratne, Haputantirige Sunil"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ullom, Stephen V."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:53:45Z","date_published":"2011-05-07T12:53:45Z","updated_at":"2026-07-22T22:25:17Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1991 Gunaratne, Haputantirige Sunil"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9210821","AAI9210821"],"render_values":[{"text":"(UMI)AAI9210821","href":null,"code":true},{"text":"AAI9210821","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20942","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ullom, Stephen V."]},{"key":"dc:creator","label":"Author","values":["Gunaratne, Haputantirige Sunil"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:53:45Z","10000-01-01","1991"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1991 Gunaratne, Haputantirige Sunil"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9210821","AAI9210821","http://hdl.handle.net/2142/20942"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Made available in DSpace on 2011-05-07T12:53:45Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9210821.pdf: 1150288 bytes, checksum: 805205f4f0d443eced9b73510be6668f (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:47:25Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:21:23-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Generalized Kummer congruences and Iwasawa invariants"]}]}],"canonical_facts":{"dc:contributor":["Ullom, Stephen V."],"dc:creator":["Gunaratne, Haputantirige Sunil"],"dc:date":["2011-05-07T12:53:45Z","10000-01-01","1991"],"dc:description":["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.","Made available in DSpace on 2011-05-07T12:53:45Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9210821.pdf: 1150288 bytes, checksum: 805205f4f0d443eced9b73510be6668f (MD5) Previous issue date: 1991","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:47:25Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:21:23-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["(UMI)AAI9210821","AAI9210821","http://hdl.handle.net/2142/20942"],"dc:language":["eng"],"dc:rights":["Copyright 1991 Gunaratne, Haputantirige Sunil"],"dc:subject":["Mathematics"],"dc:title":["Generalized Kummer congruences and Iwasawa invariants"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:17Z"}