Good towers of function Fields

Nhut Nguyen

Research output: Book/ReportPh.D. thesisResearch

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Abstract

Algebraic curves are used in many different areas, including error-correcting codes. In such applications, it is important that the algebraic curve C meets some requirements. The curve must be defined over a finite field GF(q) with q elements, and then the curve also should have many points over this field. There are limits on how many points N(C) an algebraic curve C defined over a finite field can have.

An invariant of the curve which is important in this context is the curve’s genus g(C). Hasse and Weil proved that N(C)≤q+1+2g(C) √q and this bound can in general not be improved. However if the genus is large compared with q, the bound can be improved. Drinfeld and Vladut showed the asymptotic result:

A(q)≔limsup ( N(C)/ g(C)→∞ g(C)) ≤ √q-1.

The quantity A(q) is called Ihara’s constant. If q is a square, it is known that A(q)=√q-1, while the value of the A(q) is unknown for all other values of q.

In this thesis, we study a construction using Drinfeld modules that produces explicitly defined families of algebraic curves that asymptotically achieve Ihara’s constant. Such families of curves can also be described using towers of function fields. Restated in this language the aim of the project is to find good and optimal towers. Using the theory of Drinfeld modules and computer algebraic techniques, some new examples of good towers are obtained. We analyse towers of Drinfeld modular curves describing certain equivalence classes of rank 2 Drinfeld modules. Using rank 3 Drinfeld modules further examples of good towers are produced.

Original languageEnglish
Place of PublicationKgs. Lyngby
PublisherTechnical University of Denmark
Number of pages125
Publication statusPublished - 2015
SeriesDTU Compute PHD-2015
Number394
ISSN0909-3192

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