A master thesis from Aalborg University
On bounding the number of rational places of function fields
[Om grænser på antallet af rationale places af funktionslegemer]
Author(s)
Term
4. term
Education
Publication year
2022
Submitted on
2022-06-03
Pages
51 pages
Abstract
Det overordnede tema for projektet er algebraiske funktionslegemer. Specifikt undersøger vi flere forskellige grænser på antallet af rationale places af funktionslegemer over endelige legemer. De nødvendige reskaber for at kunne vurdere sådanne grænser introduceres i det første kapitel. Dette indebærer places, valueringer, divisorer og Riemann-Roch rum. We definerer også Weierstraß semigrupper og sammenholder dem med funktionslegemer. Vi behandler dernæst projektets hovedpunkt ved at præsentere og vurdere fem forskellige grænser. Vores udgangspunkt er Hasse-Weil grænsen, som hastigt forbedres af Serre. Vi antager dernæst yderligere viden om funktionslegemet for at kunne undersøge Lewittes og Geil-Matsumoto grænserne, hvor den sidstnævnte generaliseres yderligere af Beelen og Ruano. Endelig anvender vi grænserne på diverse familier af funktionslegemer, for hvilke vi nævner nogle af deres kendte egenskaber. Vi sammenligner resultaterne af grænserne for hvert funktionslegeme med eksempler.
The overall theme of this report is that of algebraic function fields. Specifically, we examine several bounds on the number of rational places of function fields over finite fields. The necessary tools for discussing such bounds are introduced in the first chapter. This includes places, valuations, divisors and Riemann-Roch spaces. We also define Weierstraß semigroups and relate them to function fields. We then undertake the main task of the report by presenting and assessing five different bounds. Our point of departure is the Hasse-Weil bound, which is swiftly improved upon by Serre. We then assume further knowledge of our function field in order to examine the Lewittes and Geil-Matsumoto bound, the latter of which is further generalised by Beelen and Ruano. Lastly, we apply the bounds on several families of function fields, for which we mention some of their known properties. We compare the results of the bounds for each function field by examples.
Keywords
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