{"id":{"repo_id":"penn","oai_identifier":"oai:repository.upenn.edu:20.500.14332/28565"},"canonical_url":"https://search.dev.ndltd.org/etd/penn/oai:repository.upenn.edu:20.500.14332/28565","repository":{"repo_id":"penn","name":"University of Pennsylvania","base_url":"https://repository.upenn.edu/server/oai/request"},"display":{"title":"Representations of Fundamental Groups of Abelian Varieties in Characteristic P","abstract":"Let $A_g$ be an abelian variety of dimension $g$ and $p$-rank $\\lambda \\leq 1$ over an algebraically closed field of characteristic $p>0$. We compute the number of homomorphisms from $\\pi_1^{\\text{\\'et}}(A_g)$ to $GL_n(\\mathbb F_q)$, where $q$ is any power of $p$. We show that for fixed $g$, $\\lambda$, and $n$, the number of such representations is polynomial in $q$. We show that the set of such homomorphisms forms a constructable set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial. In the last chapter we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups. As a corollary, we deduce that when $\\lambda=0$, \\[\\frac{\\#\\Hom(\\pi_1^{\\text{\\'et}}(A_g),GL_n(\\mathbb F_q))}{\\#GL_n(\\mathbb F_q)}\\] is a Laurent polynomial in $q$.","abstract_html":"Let <span class=\"etd-inline-math\">A<sub>g</sub></span> be an abelian variety of dimension $g$ and $p$-rank $\\lambda \\leq 1$ over an algebraically closed field of characteristic $p&gt;0$. We compute the number of homomorphisms from <span class=\"etd-inline-math\">&pi;<sub>1</sub><sup>\\text{\\&#x27;et}</sup>(A<sub>g</sub>)</span> to <span class=\"etd-inline-math\">GL<sub>n</sub>(\\mathbb F<sub>q</sub>)</span>, where $q$ is any power of $p$. We show that for fixed $g$, $\\lambda$, and $n$, the number of such representations is polynomial in $q$. We show that the set of such homomorphisms forms a constructable set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial. In the last chapter we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups. As a corollary, we deduce that when $\\lambda=0$, \\[\\frac{\\#\\Hom(\\pi_1^{\\text{\\&#x27;et}}(A_g),GL_n(\\mathbb F_q))}{\\#GL_n(\\mathbb F_q)}\\] is a Laurent polynomial in $q$.","abstract_has_math":true,"creators":["Frankel, Brett"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Ted Chinburg"],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-01-01","date_published":"2016-01-01","updated_at":"2026-07-24T03:46:00Z","subjects":[],"languages":["en"],"rights":["Brett Frankel"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://repository.upenn.edu/handle/20.500.14332/28565","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Ted Chinburg"]},{"key":"dc:creator","label":"Author","values":["Frankel, Brett"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-05-17T15:56:23.000"]},{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2023-05-22T16:45:44Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2016-04-21T00:00:00Z"]},{"key":"dc:date.issued","label":"Date","values":["2016-01-01"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation/Thesis"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Brett Frankel"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://repository.upenn.edu/handle/20.500.14332/28565"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let $A_g$ be an abelian variety of dimension $g$ and $p$-rank $\\lambda \\leq 1$ over an algebraically closed field of characteristic $p>0$. We compute the number of homomorphisms from $\\pi_1^{\\text{\\'et}}(A_g)$ to $GL_n(\\mathbb F_q)$, where $q$ is any power of $p$. We show that for fixed $g$, $\\lambda$, and $n$, the number of such representations is polynomial in $q$. We show that the set of such homomorphisms forms a constructable set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial. In the last chapter we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups. As a corollary, we deduce that when $\\lambda=0$, \\[\\frac{\\#\\Hom(\\pi_1^{\\text{\\'et}}(A_g),GL_n(\\mathbb F_q))}{\\#GL_n(\\mathbb F_q)}\\] is a Laurent polynomial in $q$."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Doctor of Philosophy (PhD)"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Representations of Fundamental Groups of Abelian Varieties in Characteristic P"]}]}],"canonical_facts":{"dc:contributor.advisor":["Ted Chinburg"],"dc:creator":["Frankel, Brett"],"dc:date":["2023-05-17T15:56:23.000"],"dc:date.accessioned":["2023-05-22T16:45:44Z"],"dc:date.available":["2016-04-21T00:00:00Z"],"dc:date.issued":["2016-01-01"],"dc:description.abstract":["Let $A_g$ be an abelian variety of dimension $g$ and $p$-rank $\\lambda \\leq 1$ over an algebraically closed field of characteristic $p>0$. We compute the number of homomorphisms from $\\pi_1^{\\text{\\'et}}(A_g)$ to $GL_n(\\mathbb F_q)$, where $q$ is any power of $p$. We show that for fixed $g$, $\\lambda$, and $n$, the number of such representations is polynomial in $q$. We show that the set of such homomorphisms forms a constructable set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial. In the last chapter we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups. As a corollary, we deduce that when $\\lambda=0$, \\[\\frac{\\#\\Hom(\\pi_1^{\\text{\\'et}}(A_g),GL_n(\\mathbb F_q))}{\\#GL_n(\\mathbb F_q)}\\] is a Laurent polynomial in $q$."],"dc:description.degree":["Doctor of Philosophy (PhD)"],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://repository.upenn.edu/handle/20.500.14332/28565"],"dc:language":["en"],"dc:rights":["Brett Frankel"],"dc:title":["Representations of Fundamental Groups of Abelian Varieties in Characteristic P"],"dc:type":["Dissertation/Thesis"]},"updated_at":"2026-07-24T03:46:00Z"}