{"id":{"repo_id":"buffalo","oai_identifier":"oai:ubir.buffalo.edu:10477/79898"},"canonical_url":"https://search.dev.ndltd.org/etd/buffalo/oai:ubir.buffalo.edu:10477/79898","repository":{"repo_id":"buffalo","name":"Buffalo","base_url":"https://ubir.buffalo.edu/oai/request"},"display":{"title":"Exponential Sums and Ramification Data of Artin-Schreier-Witt Extensions","abstract":"Ph.D.","abstract_html":"Ph.D.","abstract_has_math":false,"creators":["Schmidt, Matthew"],"institution":"State University of New York at Buffalo","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Zhu, Hui June","Mathematics"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-07-30T15:10:51Z","date_published":"2019-07-30T15:10:51Z","updated_at":"2026-07-27T19:05:19Z","subjects":["mathematics"],"languages":["eng"],"rights":["Users of works found in University at Buffalo Institutional Repository (UBIR) are responsible for identifying and contacting the copyright owner for permission to reuse. University at Buffalo Libraries do not manage rights for copyright-protected works and cannot assist with permissions.","Copyright retained by author."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10477/79898","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Zhu, Hui June","Mathematics"]},{"key":"dc:creator","label":"Author","values":["Schmidt, Matthew"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-07-30T15:10:51Z","2019","2019-05-07 12:32:31"]},{"key":"dc:publisher","label":"Institution","values":["State University of New York at Buffalo"]},{"key":"dc:type","label":"Dc Type","values":["Text","Dissertation"]}]},{"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":["Users of works found in University at Buffalo Institutional Repository (UBIR) are responsible for identifying and contacting the copyright owner for permission to reuse. University at Buffalo Libraries do not manage rights for copyright-protected works and cannot assist with permissions.","Copyright retained by author."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/10477/79898"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Ph.D.","This thesis contains two parts:In part 1, we introduce and develop T-adic Dwork theory for L-functions of exponential sums associated to one-variable rational functions, interpolating p-power order exponential sums over affinoids. Namely, we prove a generalization of the Dwork-Monsky-Reich trace formula and apply it to establish an analytic continuation of the C-function. We compute the lower (T,p)-adic bound, the Hodge polygon, for this C-function. Along the way, we also show why a strictly T-adic theory will not work in this case. We then use this T-adic theory to study an explicit Dwork cohomology allowing us to compute the degree of the L-function of exponential sums over affinoids and compute their corresponding Hodge polygon. In part 2, we study the Schmid-Witt symbol for higher local fields. For a local field of characteristic p>0, K, the combination of local class field theory and Artin-Schreier-Witt theory yield what is known as the Schmid-Witt symbol. The symbol encodes interesting data about the ramification theory of p-extensions of K and we can, for example, use it to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula for the Schmid-Witt symbol of Artin-Schreier extensions of local fields. Later, his formula was generalized to Artin-Schreier-Witt extensions, but still over a local field. In this thesis we generalize Schmid's formula to compute the Artin-Schreier-Witt-Parshin symbol for Artin-Schreier-Witt extensions of two-dimensional local fields. Similar methods can be used to show analogue results in the n-dimensional case."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Exponential Sums and Ramification Data of Artin-Schreier-Witt Extensions"]}]}],"canonical_facts":{"dc:contributor":["Zhu, Hui June","Mathematics"],"dc:creator":["Schmidt, Matthew"],"dc:date":["2019-07-30T15:10:51Z","2019","2019-05-07 12:32:31"],"dc:description":["Ph.D.","This thesis contains two parts:In part 1, we introduce and develop T-adic Dwork theory for L-functions of exponential sums associated to one-variable rational functions, interpolating p-power order exponential sums over affinoids. Namely, we prove a generalization of the Dwork-Monsky-Reich trace formula and apply it to establish an analytic continuation of the C-function. We compute the lower (T,p)-adic bound, the Hodge polygon, for this C-function. Along the way, we also show why a strictly T-adic theory will not work in this case. We then use this T-adic theory to study an explicit Dwork cohomology allowing us to compute the degree of the L-function of exponential sums over affinoids and compute their corresponding Hodge polygon. In part 2, we study the Schmid-Witt symbol for higher local fields. For a local field of characteristic p>0, K, the combination of local class field theory and Artin-Schreier-Witt theory yield what is known as the Schmid-Witt symbol. The symbol encodes interesting data about the ramification theory of p-extensions of K and we can, for example, use it to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula for the Schmid-Witt symbol of Artin-Schreier extensions of local fields. Later, his formula was generalized to Artin-Schreier-Witt extensions, but still over a local field. In this thesis we generalize Schmid's formula to compute the Artin-Schreier-Witt-Parshin symbol for Artin-Schreier-Witt extensions of two-dimensional local fields. Similar methods can be used to show analogue results in the n-dimensional case."],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/10477/79898"],"dc:language":["eng"],"dc:publisher":["State University of New York at Buffalo"],"dc:rights":["Users of works found in University at Buffalo Institutional Repository (UBIR) are responsible for identifying and contacting the copyright owner for permission to reuse. University at Buffalo Libraries do not manage rights for copyright-protected works and cannot assist with permissions.","Copyright retained by author."],"dc:subject":["mathematics"],"dc:title":["Exponential Sums and Ramification Data of Artin-Schreier-Witt Extensions"],"dc:type":["Text","Dissertation"]},"updated_at":"2026-07-27T19:05:19Z"}