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 Title: On congruence function fields with many rational places Author(s): Mak, Kit Ho Director of Research: Duursma, Iwan M. Doctoral Committee Chair(s): Schenck, Henry K. Doctoral Committee Member(s): Duursma, Iwan M.; Ullom, Stephen V.; Zaharescu, Alexandru Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): function fields maximal curves Ihara constants asymptotic bounds subcover problem 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. Issue Date: 2012-09-18 URI: http://hdl.handle.net/2142/34193 Rights Information: Copyright 2012 Kit Ho Mak Date Available in IDEALS: 2012-09-18 Date Deposited: 2012-08
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