The nonvanishing of almost-prime twists of modular L-functions

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dc.contributor.advisorRalaivaosaona, Dimbinainaen_ZA
dc.contributor.authorAndrianarisoa, Tolotranirina Gabrielen_ZA
dc.contributor.otherStellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.en_ZA
dc.date.accessioned2023-11-29T10:35:37Zen_ZA
dc.date.accessioned2024-01-08T21:56:01Zen_ZA
dc.date.available2023-11-29T10:35:37Zen_ZA
dc.date.available2024-01-08T21:56:01Zen_ZA
dc.date.issued2023-11en_ZA
dc.descriptionThesis (MSc)--Stellenbosch University, 2023.en_ZA
dc.description.abstractENGLISH ABSTRACT: 𝐿 functions are special types of Dirichlet series which often hold fundamen tal arithmetic information. Hence, they are among the most important objects in analytic number theory. In this thesis, we consider the so called Hecke 𝐿 function 𝐿(𝑠, 𝑓, 𝜒𝑑) associated to a given normalized holomorphic newform 𝑓 twisted by the Kronecker symbol 𝜒𝑑. It is well known that the twisted 𝐿(𝑠, 𝑓, 𝜒𝑑) converges absolutely for Re(𝑠) > 1 and admits a functional equation which extends it analytically to the whole complex plane. The value of 𝐿(𝑠, 𝑓, 𝜒𝑑) at 𝑠 = 1/2 is of special interest. For instance, if the form 𝑓 parametrizes a twisted elliptic curve 𝐸 of given rank 𝑟 ≥ 0, then the Birch Swinnerton Dyer conjecture asserts that 𝑟 is precisely the order of vanishing of 𝐿(𝑠, 𝑓, 𝜒𝑑) at 𝑠 = 1/2. In this work, we ϐix a holomorphic newform 𝑓 of weight at least 2, level 𝑁 with trivial nebentype and consider the family of twisted 𝐿 functions 𝐿(𝑠, 𝑓, 𝜒𝑑) where 𝑑 is any fundamental discriminant with (𝑑, 𝑁) = 1. Using an adapta tion of a method by Iwaniec, we prove that there are inϐinitely many funda mental discriminants 𝑑 such that 𝐿(1/2, 𝑓, 𝜒𝑑) ≠ 0. In addition, following an idea outlined by Hoffstein and Luo, using combinatorial sieve, we prove that the same holds for inϐinitely many almost prime fundamental discriminants 𝑑 with at most 84 prime factors. Further improvement of this result, which relies on properties of some multiple Dirichlet series, is also discussed in this work. Under some assumptions on certain weight factors, it is possible to reduce the number 84 to just 4.en_ZA
dc.description.abstractAFRIKAANSE OPSOMMING: 𝐿 funksies is spesiale tipes Dirichlet reekse wat dikwels fundamentele arit metiese inligting bevat. Daarom is hulle een van die belangrikste objekte in analitiese getalteorie. In hierdie tesis ondersoek ons die sogenaamde Hecke 𝐿 funksie 𝐿(𝑠, 𝑓, 𝜒𝑑) wat geassosieer word met ’n gegewe genormaliseerde ho lomorfe nuwe vorm 𝑓 wat deur die Kronecker simbool 𝜒𝑑 verdraai is. Dit is al gemeen bekend dat die verdraaide 𝐿(𝑠, 𝑓, 𝜒𝑑) absoluut konvergeer vir Re(𝑠) > 1 en ’n funksionele vergelyking het wat dit analities tot die hele komplekse vlak uitbrei. Die waarde van 𝐿(𝑠, 𝑓, 𝜒𝑑) by 𝑠 = 1/2 is van besondere belang. Byvoor beeld, as die vorm 𝑓 ’n verdraaide elliptiese kromme𝐸 van ’n gegewe rang 𝑟 ≥ 0 parametriseer, beweer die Birch Swinnerton Dyer vermoede dat 𝑟 presies die orde van nulstelling van 𝐿(𝑠, 𝑓, 𝜒𝑑) by 𝐿(𝑠, 𝑓, 𝜒𝑑) is. In hierdie werk, bepaal ons ’n holomorfe nuwe vorm 𝑓 van gewig van ten minste 2, met ’n vlak 𝑁 met ’n triviale nebentipe, en ons oorweeg die familie van verdraaide 𝐿 funksies 𝐿(𝑠, 𝑓, 𝜒𝑑) waar 𝑑 enige fundamentele diskriminant met (𝑑, 𝑁) = 1 is. Deur ’n aanpassing van ’n metode deur Iwaniec, bewys ons dat daar oneindig baie fundamentele diskriminante 𝑑 is sodat 𝐿(1/2, 𝑓, 𝑐ℎ𝑖𝑑) ≠ 0. Daarbenewens bewys ons, volgens ’n idee deur Hoffstein en Luo, deur gebruik te maak van ’n kombinatoriese sif, dat dieselfde waar is vir oneindig baie by kans primêre fundamentele diskriminante 𝑑 met hoogstens 84 primêre fak tore. Verdere verbetering van hierdie resultaat, wat berus op eienskappe van sekere multiple Dirichlet reekse, word ook in hierdie werk bespreek. Onder sekere aanannames oor sekere gewigsfaktore, is dit moontlik om die getal 84 tot net 4 te verminder.af_ZA
dc.description.versionMastersen_ZA
dc.format.extentviii, 97 pagesen_ZA
dc.identifier.urihttps://scholar.sun.ac.za/handle/10019.1/129087en_ZA
dc.language.isoen_ZAen_ZA
dc.language.isoen_ZAen_ZA
dc.publisherStellenbosch : Stellenbosch Universityen_ZA
dc.rights.holderStellenbosch Universityen_ZA
dc.subject.lcshForms, Modularen_ZA
dc.subject.lcshL-functionsen_ZA
dc.subject.lcshDirichlet seriesen_ZA
dc.subject.lcshCurves, Ellipticen_ZA
dc.titleThe nonvanishing of almost-prime twists of modular L-functionsen_ZA
dc.typeThesisen_ZA
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