{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/34193"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/34193","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On congruence function fields with many rational places","abstract":"In this thesis, we study congruence function fields, in particular those with many rational places. This thesis consists of three parts, the first two parts present our results in two different aspects of function fields with many rational places, namely maximal function fields and asymptotically good towers of function fields. The third part concerns Selmer groups of elliptic curves over the rational function field. Let $\\mathcal{H}$ be the Hermitian function field, and $\\mathcal{C}$ be a maximal function field, both over the same finite field. In the first part of this thesis, we analyze the Artin representation of $\\mathcal{H}$ and improve the lower bound for the possible degree of the extension $\\mathcal{H}/\\mathcal{C}$ when it is Galois. We then apply the lower bound to show that the generalized Giulietti-Korchm{\\'a}ros function field defined over $\\mathbb{F}_{q^{2n}}$ is not a Galois subfield of the Hermitian function field $\\mathcal{H}$ over $\\mathbb{F}_{q^{2n}}$ for $n\\geq 3$ odd and $q\\geq 3$. Combining the lower bound with some group theoretical arguments, we also generalize an example given by Garcia and Stichtenoth by showing that when $q$ is an odd prime, the function field $\\mathcal{X}_3=\\mathbb{F}_{q^6}(x,y)$ with $x^{q^2}-x=y^{(q^n+1)/(q+1)}$ is not a Galois subfield of the Hermitian function field over the same finite field. The second part is about improving lower bounds of the Ihara constant. Let $\\mathcal{X}$ be a curve over $\\mathbb{F}_q$ and let $N(\\mathcal{X})$, $g(\\mathcal{X})$ be its number of rational places and genus respectively. The Ihara constant $A(q)$ is defined by $A(q)=\\limsup_{g(\\mathcal{X})\\rightarrow\\infty}N(\\mathcal{X})/g(\\mathcal{X})$. We use a variant of Serre's class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$. In the last part, we calculate the distribution of Selmer groups arising from a $2$-isogeny for a family of quadratic twists of the elliptic curves with full $2$-torsions over the rational function field $\\mathbb{F}_q(x)$ for odd $q$. In particular, we show that the sizes of these Selmer groups are almost always bounded. The calculation relies heavily on various estimates of twisted character sums over function fields.","abstract_html":"In this thesis, we study congruence function fields, in particular those with many rational places. This thesis consists of three parts, the first two parts present our results in two different aspects of function fields with many rational places, namely maximal function fields and asymptotically good towers of function fields. The third part concerns Selmer groups of elliptic curves over the rational function field. Let $\\mathcal{H}$ be the Hermitian function field, and $\\mathcal{C}$ be a maximal function field, both over the same finite field. In the first part of this thesis, we analyze the Artin representation of $\\mathcal{H}$ and improve the lower bound for the possible degree of the extension $\\mathcal{H}/\\mathcal{C}$ when it is Galois. We then apply the lower bound to show that the generalized Giulietti-Korchm{\\&#x27;a}ros function field defined over <span class=\"etd-inline-math\">\\mathbb{F}<sub>q<sup>2n</sup></sub></span> is not a Galois subfield of the Hermitian function field $\\mathcal{H}$ over <span class=\"etd-inline-math\">\\mathbb{F}<sub>q<sup>2n</sup></sub></span> for $n\\geq 3$ odd and $q\\geq 3$. Combining the lower bound with some group theoretical arguments, we also generalize an example given by Garcia and Stichtenoth by showing that when $q$ is an odd prime, the function field <span class=\"etd-inline-math\">\\mathcal{X}<sub>3</sub>=\\mathbb{F}<sub>q<sup>6</sup></sub>(x,y)</span> with <span class=\"etd-inline-math\">x<sup>q<sup>2</sup></sup>-x=y<sup>(q<sup>n</sup>+1)/(q+1)</sup></span> is not a Galois subfield of the Hermitian function field over the same finite field. The second part is about improving lower bounds of the Ihara constant. Let $\\mathcal{X}$ be a curve over <span class=\"etd-inline-math\">\\mathbb{F}<sub>q</sub></span> and let $N(\\mathcal{X})$, $g(\\mathcal{X})$ be its number of rational places and genus respectively. The Ihara constant $A(q)$ is defined by <span class=\"etd-inline-math\">A(q)=\\limsup<sub>g(\\mathcal{X})\\rightarrow\\infty</sub>N(\\mathcal{X})/g(\\mathcal{X})</span>. We use a variant of Serre&#x27;s class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$. In the last part, we calculate the distribution of Selmer groups arising from a $2$-isogeny for a family of quadratic twists of the elliptic curves with full $2$-torsions over the rational function field <span class=\"etd-inline-math\">\\mathbb{F}<sub>q</sub>(x)</span> for odd $q$. In particular, we show that the sizes of these Selmer groups are almost always bounded. The calculation relies heavily on various estimates of twisted character sums over function fields.","abstract_has_math":true,"creators":["Mak, Kit Ho"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Duursma, Iwan M.","Schenck, Henry K.","Ullom, Stephen V.","Zaharescu, Alexandru"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-09-18T21:05:00Z","date_published":"2012-09-18T21:05:00Z","updated_at":"2026-07-22T22:25:30Z","subjects":["function fields","maximal curves","Ihara constants","asymptotic bounds","subcover problem"],"languages":["en"],"rights":["Copyright 2012 Kit Ho Mak"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/34193","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Duursma, Iwan M.","Schenck, Henry K.","Ullom, Stephen V.","Zaharescu, Alexandru"]},{"key":"dc:creator","label":"Author","values":["Mak, Kit Ho"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-09-18T21:05:00Z","2012-08"]},{"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":["function fields","maximal curves","Ihara constants","asymptotic bounds","subcover problem"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2012 Kit Ho Mak"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/34193"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, we study congruence function fields, in particular those with many rational places. This thesis consists of three parts, the first two parts present our results in two different aspects of function fields with many rational places, namely maximal function fields and asymptotically good towers of function fields. The third part concerns Selmer groups of elliptic curves over the rational function field. Let $\\mathcal{H}$ be the Hermitian function field, and $\\mathcal{C}$ be a maximal function field, both over the same finite field. In the first part of this thesis, we analyze the Artin representation of $\\mathcal{H}$ and improve the lower bound for the possible degree of the extension $\\mathcal{H}/\\mathcal{C}$ when it is Galois. We then apply the lower bound to show that the generalized Giulietti-Korchm{\\'a}ros function field defined over $\\mathbb{F}_{q^{2n}}$ is not a Galois subfield of the Hermitian function field $\\mathcal{H}$ over $\\mathbb{F}_{q^{2n}}$ for $n\\geq 3$ odd and $q\\geq 3$. Combining the lower bound with some group theoretical arguments, we also generalize an example given by Garcia and Stichtenoth by showing that when $q$ is an odd prime, the function field $\\mathcal{X}_3=\\mathbb{F}_{q^6}(x,y)$ with $x^{q^2}-x=y^{(q^n+1)/(q+1)}$ is not a Galois subfield of the Hermitian function field over the same finite field. The second part is about improving lower bounds of the Ihara constant. Let $\\mathcal{X}$ be a curve over $\\mathbb{F}_q$ and let $N(\\mathcal{X})$, $g(\\mathcal{X})$ be its number of rational places and genus respectively. The Ihara constant $A(q)$ is defined by $A(q)=\\limsup_{g(\\mathcal{X})\\rightarrow\\infty}N(\\mathcal{X})/g(\\mathcal{X})$. We use a variant of Serre's class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$. In the last part, we calculate the distribution of Selmer groups arising from a $2$-isogeny for a family of quadratic twists of the elliptic curves with full $2$-torsions over the rational function field $\\mathbb{F}_q(x)$ for odd $q$. In particular, we show that the sizes of these Selmer groups are almost always bounded. The calculation relies heavily on various estimates of twisted character sums over function fields.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-06-06T17:54:15Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Mak_Kit Ho.pdf: 805598 bytes, checksum: d45cce82bb496168657ed8b08745cd20 (MD5)","Made available in DSpace on 2012-09-18T21:05:00Z (GMT). No. of bitstreams: 2 Mak_KitHo.pdf: 805598 bytes, checksum: d45cce82bb496168657ed8b08745cd20 (MD5) license.txt: 4053 bytes, checksum: d7cc01782951859905acb6354e0ece9c (MD5)"]},{"key":"dc:title","label":"Title","values":["On congruence function fields with many rational places"]}]}],"canonical_facts":{"dc:contributor":["Duursma, Iwan M.","Schenck, Henry K.","Ullom, Stephen V.","Zaharescu, Alexandru"],"dc:creator":["Mak, Kit Ho"],"dc:date":["2012-09-18T21:05:00Z","2012-08"],"dc:description":["In this thesis, we study congruence function fields, in particular those with many rational places. This thesis consists of three parts, the first two parts present our results in two different aspects of function fields with many rational places, namely maximal function fields and asymptotically good towers of function fields. The third part concerns Selmer groups of elliptic curves over the rational function field. Let $\\mathcal{H}$ be the Hermitian function field, and $\\mathcal{C}$ be a maximal function field, both over the same finite field. In the first part of this thesis, we analyze the Artin representation of $\\mathcal{H}$ and improve the lower bound for the possible degree of the extension $\\mathcal{H}/\\mathcal{C}$ when it is Galois. We then apply the lower bound to show that the generalized Giulietti-Korchm{\\'a}ros function field defined over $\\mathbb{F}_{q^{2n}}$ is not a Galois subfield of the Hermitian function field $\\mathcal{H}$ over $\\mathbb{F}_{q^{2n}}$ for $n\\geq 3$ odd and $q\\geq 3$. Combining the lower bound with some group theoretical arguments, we also generalize an example given by Garcia and Stichtenoth by showing that when $q$ is an odd prime, the function field $\\mathcal{X}_3=\\mathbb{F}_{q^6}(x,y)$ with $x^{q^2}-x=y^{(q^n+1)/(q+1)}$ is not a Galois subfield of the Hermitian function field over the same finite field. The second part is about improving lower bounds of the Ihara constant. Let $\\mathcal{X}$ be a curve over $\\mathbb{F}_q$ and let $N(\\mathcal{X})$, $g(\\mathcal{X})$ be its number of rational places and genus respectively. The Ihara constant $A(q)$ is defined by $A(q)=\\limsup_{g(\\mathcal{X})\\rightarrow\\infty}N(\\mathcal{X})/g(\\mathcal{X})$. We use a variant of Serre's class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$. In the last part, we calculate the distribution of Selmer groups arising from a $2$-isogeny for a family of quadratic twists of the elliptic curves with full $2$-torsions over the rational function field $\\mathbb{F}_q(x)$ for odd $q$. In particular, we show that the sizes of these Selmer groups are almost always bounded. The calculation relies heavily on various estimates of twisted character sums over function fields.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-06-06T17:54:15Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Mak_Kit Ho.pdf: 805598 bytes, checksum: d45cce82bb496168657ed8b08745cd20 (MD5)","Made available in DSpace on 2012-09-18T21:05:00Z (GMT). No. of bitstreams: 2 Mak_KitHo.pdf: 805598 bytes, checksum: d45cce82bb496168657ed8b08745cd20 (MD5) license.txt: 4053 bytes, checksum: d7cc01782951859905acb6354e0ece9c (MD5)"],"dc:identifier":["http://hdl.handle.net/2142/34193"],"dc:language":["en"],"dc:rights":["Copyright 2012 Kit Ho Mak"],"dc:subject":["function fields","maximal curves","Ihara constants","asymptotic bounds","subcover problem"],"dc:title":["On congruence function fields with many rational places"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:30Z"}