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Abstract
Algebraic curves are used in many different areas, including errorcorrecting 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)) ≤ √q1.
The quantity A(q) is called Ihara’s constant. If q is a square, it is known that A(q)=√q1, 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.
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)) ≤ √q1.
The quantity A(q) is called Ihara’s constant. If q is a square, it is known that A(q)=√q1, 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 language  English 

Place of Publication  Kgs. Lyngby 

Publisher  Technical University of Denmark 
Number of pages  125 
Publication status  Published  2015 
Series  DTU Compute PHD2015 

Number  394 
ISSN  09093192 
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 1 Finished

Good towers of function fields
Nguyen, N., Beelen, P., Ritzenthaler, C., Geil, H. O. & Bouw, I. I.
01/08/2012 → 21/01/2016
Project: PhD