dc.contributor.author Jain, Lalit Kumar dc.date.accessioned 2008-05-01 12:51:40 (GMT) dc.date.available 2008-05-01 12:51:40 (GMT) dc.date.issued 2008-05-01T12:51:40Z dc.date.submitted 2008 dc.identifier.uri http://hdl.handle.net/10012/3626 dc.description.abstract Let $E$ be an elliptic curve over the rationals without complex multiplication such that any elliptic curve $\mathbb{Q}$-isogenous to $E$ has trivial $\mathbb{Q}$-torsion. Koblitz conjectured that the number of primes less than $x$ for which $|E(\mathbb{F}_p)|$ is prime is asymptotic to $$C_E\frac{x}{(\log{x})^2}$$ for $C_E$ some constant dependent on $E.$ Miri and Murty showed that for infinitely many $p,$ $|E(\mathbb{F}_p)|$ has at most 16 prime factors using the lower bound sieve and assuming the Generalized Riemann Hypothesis. This thesis generalizes Koblitz's conjectures to a function field setting through Drinfeld modules. Let $\phi$ be a Drinfeld module of rank 2, and $\mathbb{F}_q$ a finite field with every $\mathbb{F}_q[t]$-isogeny having no $\mathbb{F}_q[t]$-torsion points and with $\text{End}_{\overline{k}}(\phi)=\mathbb{F}_q[t].$ Furthermore assume that for each monic irreducible $l\in \mathbb{F}_q[t],$ the extension generated by adjoining the $l$-torsion points of $\phi$ to $\mathbb{F}_q(t)$ is geometric. Then there exists a positive constant $C_{\phi}$ depending on $\phi$ such that there are more than $$C_{\phi}\frac{q^x}{x^2}$$ monic irreducible polynomials $P$ with degree less then $x$ such that $\chi_{\phi}(P)$ has at most 13 prime factors. To prove this result we develop the theory of Drinfeld modules and a translation of the lower bound sieve to function fields. en dc.language.iso en en dc.publisher University of Waterloo en dc.subject Number Theory en dc.subject Function Fields en dc.subject Koblitz's Conjecture en dc.title Koblitz's Conjecture for the Drinfeld Module en dc.type Master Thesis en dc.pending false en dc.subject.program Pure Mathematics en uws-etd.degree.department Pure Mathematics en uws-etd.degree Master of Mathematics en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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