Spelling suggestions: "subject:"drinfeld (modules dde)"" "subject:"drinfeld (modules dee)""
1 |
Special Values of the Goss L-function and Special PolynomialsLutes, Brad Aubrey 2010 August 1900 (has links)
Let K be the function field of an irreducible, smooth projective curve X defined over Fq. Let [lemniscate] be a fixed point on X and let A [a subset of or is equal to] K be the Dedekind domain of functions which are regular away from [lemniscate]. Following the work of Greg Anderson, we define special polynomials and explain how they are used to define an A-module (in the case where the class number of A and the degree of [lemniscate] are both one) known as the module of special points associated to the Drinfeld A-module [rho]. We show that this module is finitely generated and explicitly compute its rank. We also show that if K is a function field such that the degree of [lemniscate] is one, then the Goss L-function, evaluated at 1, is a finite linear combination of logarithms evaluated at algebraic points. We conclude with examples showing how to use special polynomials to compute special values of both the Goss L-function and the Goss zeta function.
|
2 |
On the coefficients of Drinfeld modular forms of higher rankBasson, Dirk Johannes 04 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Rank 2 Drinfeld modular forms have been studied for more than 30 years, and
while it is known that a higher rank theory could be possible, higher rank
Drinfeld modular forms have only recently been de ned. In 1988 Gekeler
published [Ge2] in which he studies the coe cients of rank 2 Drinfeld modular
forms. The goal of this thesis is to perform a similar study of the coe cients
of higher rank Drinfeld modular forms.
The main results are that the coe cients themselves are (weak) Drinfeld
modular forms, a product formula for the discriminant function, the rationality
of certain naturally de ned modular forms, and the computation of
some Hecke eigenforms and their eigenvalues. / AFRIKAANSE OPSOMMING: Drinfeld modulêre vorme van rang 2 word al vir meer as 30 jaar bestudeer
en alhoewel dit lankal bekend is dat daar Drinfeld modulêre vorme van hoër
rang moet bestaan, is die de nisie eers onlangs vasgepen. In 1988 het Gekeler
die artikel [Ge2] gepubliseer waarin hy die koeffisiënte van Fourier reekse van
rang 2 Drinfeld modulêre vorme bestudeer. Die doel van hierdie proefskrif is
om dieselfde studie vir Drinfeld modulêre vorme van hoër rang uit te voer.
Die hoofresultate is dat die koeffi siënte self (swak) Drinfeld modulêre
vorme is, `n produk formule vir die diskriminant funksie, die feit dat sekere
natuurlik gede finiëerde modulêre vorme rasionaal is, en die vasstelling van
Hecke eievorme en hul eiewaardes.
|
3 |
Cyclotomic polynomials (in the parallel worlds of number theory)Bamunoba, Alex Samuel 12 1900 (has links)
Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: It is well known that the ring of integers Z and the ring of polynomials A = Fr[T] over a
finite field Fr have many properties in common. It is due to these properties that almost all
the famous (multiplicative) number theoretic results over Z have analogues over A. In this
thesis, we are devoted to utilising this analogy together with the theory of Carlitz modules.
We do this to survey and compare the analogues of cyclotomic polynomials, the size of their
coefficients and cyclotomic extensions over the rational function field k = Fr(T). / AFRIKAANSE OPSOMMING: Dit is bekend dat Z, die ring van heelgetalle en A = Fr[T], die ring van polinome oor ’n
eindige liggaam baie eienskappe in gemeen het. Dit is as gevolg van hierdie eienskappe dat
feitlik al die bekende multiplikative resultate wat vir Z geld, analoë in A het. In hierdie tesis,
fokus ons op die gebruik van hierdie analogie saam met die teorie van die Carlitz module.
Ons doen dit om ’n oorsig oor die analoë van die siklotomiese polinome, hul koëffisiënte, en
siklotomiese uitbreidings oor die rasionele funksie veld k = Fr(T).
|
4 |
Drinfeld modules and their application to factor polynomialsRandrianarisoa, Tovohery Hajatiana 12 1900 (has links)
Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Major works done in Function Field Arithmetic show a strong analogy between
the ring of integers Z and the ring of polynomials over a nite eld Fq[T]. While
an algorithm has been discovered to factor integers using elliptic curves, the
discovery of Drinfeld modules, which are analogous to elliptic curves, made it
possible to exhibit an algorithm for factorising polynomials in the ring Fq[T].
In this thesis, we introduce the notion of Drinfeld modules, then we demonstrate
the analogy between Drinfeld modules and Elliptic curves. Finally, we
present an algorithm for factoring polynomials over a nite eld using Drinfeld
modules. / AFRIKAANSE OPSOMMING: 'n Groot deel van die werk wat reeds in funksieliggaam rekenkunde voltooi
is toon 'n sterk verband tussen die ring van heelgetalle, Z; en die ring van
polinome oor 'n eindige liggaam, F[T]: Terwyl daar alreeds 'n algoritme, wat
gebruik maak van elliptiese kurwes, ontwerp is om heelgetalle te faktoriseer,
het die ontdekking van Drinfeld modules, wat analoog is aan elliptiese kurwes,
dit moontlik gemaak om 'n algoritme te konstrueer om polinome in die ring
F[T] te faktoriseer.
In hierdie tesis maak ons die konsep van Drinfeld modules bekend deur sekere
aspekte daarvan te bestudeer. Ons gaan voort deur 'n voorbeeld te voorsien
wat die analoog tussen Drinfeld modules en elliptiese kurwes illustreer. Uiteindelik,
deur gebruik te maak van Drinfeld modules, bevestig ons hierdie analoog
deur die algoritme vir die faktorisering van polinome oor eindige liggame te
veskaf.
|
5 |
An analogue of the Andre-Oort conjecture for products of Drinfeld modular surfacesKarumbidza, Archie 03 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: This thesis deals with a function eld analog of the André-Oort conjecture. The (classical) André-Oort conjecture concerns the distribution of special points on Shimura
varieties. In our case we consider the André-Oort conjecture for special points in the
product of Drinfeld modular varieties. We in particular manage to prove the André-
Oort conjecture for subvarieties in a product of two Drinfeld modular surfaces under
a characteristic assumption. / AFRIKAANSE OPSOMMING: Hierdie tesis handel van 'n funksieliggaam analoog van die André-Oort Vermoeding.
Die (Klassieke) André-Oort Vermoeding het betrekking tot die verspreiding van
spesiale punte op Shimura varietiete. Ons geval beskou ons die André-Oort Vermoeding
vir spesiale punte op die produk Drinfeldse modulvarietiete. In die besonders,
bewys ons die André-Oort Vermoeding vir ondervarieteite van 'n produk van twee
Drinfeldse modulvarietiete, onderhewig aan 'n karakteristiek-aanname.
|
6 |
Sur la conjecture d'André-Oort et courbes modulaires de DrinfeldBREUER, Florian 08 November 2002 (has links) (PDF)
Nous démontrons une version pour la caractéristique p d'un cas spécial de la conjecture d'André-Oort. Plus précisement, soit Z le produit de n courbes modulaires de Drinfeld, et soit X une sous-variété algébrique irréductible de Z. Alors nous démontrons que X contient un ensemble Zariski-dense de points CM (c.a.d. points correspondant aux n-uples de A-modules de Drinfeld de rang 2 avec mulitplications complexes, où A=F_q[T], et q est une puissance d'un nombre prémier impair) si et seulement si X est une sous-variété dite modulaire. Notre approche répose sur une approche (en caractéristique 0) due à Edixhoven.
|
7 |
The Lang-Trotter conjecture for Drinfeld modulesTweedle, David January 2011 (has links)
In 1986, Gupta and Murty proved the Lang-Trotter conjecture in the case of elliptic curves having complex multiplication, conditional on the generalized Riemann hypothesis. That is, given a non-torsion point P∈E(ℚ), they showed that P (mod p) generates E(𝔽p) for infinitely many primes p, conditional on the generalized Riemann hypothesis. We demonstrate that Gupta's and Murty's result can be translated into an unconditional result in the language of Drinfeld modules. We follow the example of Hsu and Yu, who proved Artin's conjecture unconditionally in the case of sign normalized rank one Drinfeld modules. Further, we will cover all necessary background information.
|
8 |
Propriété de Bogomolov pour les modules de Drinfeld à multiplications complexesBauchère, Hugues 16 September 2013 (has links) (PDF)
Notons A:=Fq[T] et k:=Fq(T). Soient φ un A-module de Drinfeld défini sur la clôture algébrique de k et h sa hauteur canonique. Soient K/k une extension finie et L/K une extension galoisienne infinie. Par analogie avec la terminologie utilisée par E. Bombieri et U. Zannier, on dit que L a la propriété (B,φ) s'il existe une constante strictement positive qui minore h sur L privé des points de torsion de φ. S. David et A. Pacheco ont montré que pour tout module de Drinfeld φ, la clôture abélienne de K a la propriété (B,φ). Dans cette thèse nous généralisons, dans le cadre des modules de Drinfeld à multiplications complexes, ce résultat.
|
9 |
The Lang-Trotter conjecture for Drinfeld modulesTweedle, David January 2011 (has links)
In 1986, Gupta and Murty proved the Lang-Trotter conjecture in the case of elliptic curves having complex multiplication, conditional on the generalized Riemann hypothesis. That is, given a non-torsion point P∈E(ℚ), they showed that P (mod p) generates E(𝔽p) for infinitely many primes p, conditional on the generalized Riemann hypothesis. We demonstrate that Gupta's and Murty's result can be translated into an unconditional result in the language of Drinfeld modules. We follow the example of Hsu and Yu, who proved Artin's conjecture unconditionally in the case of sign normalized rank one Drinfeld modules. Further, we will cover all necessary background information.
|
10 |
Traces of Hecke operators on Drinfeld modular forms via point countsDe Vries, Sjoerd January 2023 (has links)
In this licentiate thesis, we study the action of Hecke operators on Drinfeld cusp forms via the theory of crystals over function fields. The thesis contains one preliminary chapter, in which we recall some basic theory of Drinfeld modules and Drinfeld modular forms, as well as the Eichler-Shimura theory developed by Böckle. The core of the thesis consists of Chapter II, in which we prove a Lefschetz trace formula for crystals over stacks and deduce a Ramanujan bound for Drinfeld modular forms, and Chapter III, in which we compute traces and slopes of Hecke operators. We formulate several questions and conjectures based on our data. We also include an appendix in which we discuss the relationship between traces of an operator in positive characteristic and its eigenvalues.
|
Page generated in 0.0741 seconds