Full Text Available
Note: Clicking the button above will open the full text document at the original institutional repository in a new window.
Torsion bounds for Drinfeld modules with complex multiplication
Thesis (PhD)--Stellenbosch University, 2020.
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | en_ZA |
| Published: |
Stellenbosch : Stellenbosch University.
2020
|
| Subjects: | |
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1867613781129953280 |
|---|---|
| access_status_str | Open Access |
| author | Rabenantoandro, Andry Nirina |
| author2 | Breuer, Florian |
| author_browse | Breuer, Florian Rabenantoandro, Andry Nirina |
| author_facet | Breuer, Florian Rabenantoandro, Andry Nirina |
| author_sort | Rabenantoandro, Andry Nirina |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University. |
| description | Thesis (PhD)--Stellenbosch University, 2020. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/108144 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:41:35.993Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2020 |
| publishDateRange | 2020 |
| publishDateSort | 2020 |
| publisher | Stellenbosch : Stellenbosch University. |
| publisherStr | Stellenbosch : Stellenbosch University. |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/108144 Torsion bounds for Drinfeld modules with complex multiplication Rabenantoandro, Andry Nirina Breuer, Florian Wagner, Stephan Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Drinfeld modular varieties Torsion theory (Algebra) Multiplication, Complex Theorem, Clark and Pollack's Function algebras Torsion bounds UCTD Thesis (PhD)--Stellenbosch University, 2020. ENGLISH ABSTRACT: The main objective of the present thesis is to prove an analogue for Drinfeld modules of a theorem due to Clark and Pollack. The cardinality of the group of K-rational torsion points of an elliptic curve EjK with complex multiplication defined over a number field K of degree d is uniformly bounded by Cd log log d for some absolute and effective constant C > 0, i.e. the constant C > 0 depends neither on E nor on K. Let F be a global function field over Fq and A the ring of elements of F regular away from a fixed prime ¥. Let r 1 be an integer. We prove that there exists a positive constant CA,r > 0 depending only on A and r such that for any field extension L of degree d over F and any Drinfeld A-module jjL of rank r with complex multiplication defined over L and such that the endomorphism ring of j is the maximal order in its CM field, the cardinality of the A-module of L-rational torsion points of j is bounded by CA,rd log log d. The constant depends neither on j nor on L. For a given A and r the constant CA,r is effective and we get an explicit formula for it. The above result is not the full analogue of Clark and Pollack’s theorem but rather a weaker version since it requires the endomorphism ring of j to be the maximal order in its CM field. However, when A = Fq[T], F = Fq(T) and r = 2 we obtain the full analogue of Clark and Pollack’s result by proving the analogue of what they called the Isogeny Torsion Theorem in [CP15]. AFRIKAANSE OPSOMMING: Die hoofdoel van hierdie tesis is om ’n analoog vir Drinfeld modules te bewys van ’n stelling te danke aan Clark en Pollack wat die volgende beweer. Die kardinaliteit van die groep K-rasionale torsiepunte van ’n elliptiese kromme EjK met komplekse vermenigvuldiging gedefinieà ´nr oor ’n getalveld K van graad d is eenvormig begrens deur Cd log log d vir ’n absolute en effektiewe konstante C > 0, dit wil sê die konstante C > 0 hang nie van E of van K af nie. Laat F ’n globale funksieveld oor Fq wees en A die ring van elemente van F reëlmatig weg vanaf ’n vaste priem ¥. Laat r 1 ’n heelgetal wees. Ons bewys dat daar ’n positiewe konstante CA,r > 0 is afhangende slegs van A en r sodanig dat vir enige velduitbreiding L van graad d oor F en enige Drinfeld A-module jjL van rang r met ingewikkelde vermenigvuldiging gedefinieer o or L e n s odanig dat die endomorphism ring van j is die maksimale orde in sy CM-veld, die kardinaliteit van die A-module van L-rasionale torsiepunte van j begrens word deur CA,rd log log d. Die konstante hang nie van j of van L af nie. Vir ’n gegewe A en r die konstante CA,r is effektief en ons kry ’n eksplisiete formule daarvoor. Die bogenoemde resultaat is nie die volledige analoog van Clark en Pollack se stelling nie, maar eerder ’n swakker weergawe, aangesien dit vereis dat die endomorfisme van j die maksimale orde in sy CM-veld. Wanneer A = Fq[T], F = Fq(T) en r = 2, verkry ons die volledige analoog van Clark en Pollack se resultaat deur die analoog te bewys van wat hulle die Isogeny Torsion Stelling in [CP15] genoem het. Doctoral 2020-02-25T13:30:20Z 2020-04-28T12:21:50Z 2020-02-25T13:30:20Z 2020-04-28T12:21:50Z 2020-04 Thesis http://hdl.handle.net/10019.1/108144 en_ZA Stellenbosch University. xiv, 75 pages application/pdf Stellenbosch : Stellenbosch University. |
| spellingShingle | Drinfeld modular varieties Torsion theory (Algebra) Multiplication, Complex Theorem, Clark and Pollack's Function algebras Torsion bounds UCTD Rabenantoandro, Andry Nirina Torsion bounds for Drinfeld modules with complex multiplication |
| title | Torsion bounds for Drinfeld modules with complex multiplication |
| title_full | Torsion bounds for Drinfeld modules with complex multiplication |
| title_fullStr | Torsion bounds for Drinfeld modules with complex multiplication |
| title_full_unstemmed | Torsion bounds for Drinfeld modules with complex multiplication |
| title_short | Torsion bounds for Drinfeld modules with complex multiplication |
| title_sort | torsion bounds for drinfeld modules with complex multiplication |
| topic | Drinfeld modular varieties Torsion theory (Algebra) Multiplication, Complex Theorem, Clark and Pollack's Function algebras Torsion bounds UCTD |
| url | http://hdl.handle.net/10019.1/108144 |
| work_keys_str_mv | AT rabenantoandroandrynirina torsionboundsfordrinfeldmoduleswithcomplexmultiplication |