## Koblitz's Conjecture for the Drinfeld Module

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.comment.hidden | Mr. Clews, I have resubmitted my thesis on UWSpace, and made several of the corrections you mention (copied below). However one correction I did not make was the suppresion of A.1 in the appendix. I did not treat it as a Chapter...indeed I now call the appendix, 'Appendix A' but I did leave the section numbering as A.1. One main reason I did this is to properly deal with equation/section numbering in the appendix. An example of an accepted thesis where this was done is : http://uwspace.uwaterloo.ca/handle/10012/1057 If you still do not approve of this style, please let me know and also let me know how to suppress the section numbering while maintaining numbering in the appendix. Thank You for your help and your meticulous review. Lalit Jain | 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 |