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

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Stellenbosch : Stellenbosch University
ENGLISH ABSTRACT: ๐ฟ 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 ๐ฟ function ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) associated to a given normalized holomorphic newform ๐‘“ twisted by the Kronecker symbol ๐œ’๐‘‘. It is well known that the twisted ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) converges absolutely for Re(๐‘ ) > 1 and admits a functional equation which extends it analytically to the whole complex plane. The value of ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) at ๐‘  = 1/2 is of special interest. For instance, if the form ๐‘“ parametrizes a twisted elliptic curve ๐ธ of given rank ๐‘Ÿ โ‰ฅ 0, then the Birch Swinnerton Dyer conjecture asserts that ๐‘Ÿ is precisely the order of vanishing of ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) at ๐‘  = 1/2. In this work, we ฯix a holomorphic newform ๐‘“ of weight at least 2, level ๐‘ with trivial nebentype and consider the family of twisted ๐ฟ functions ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) where ๐‘‘ is any fundamental discriminant with (๐‘‘, ๐‘) = 1. Using an adapta tion of a method by Iwaniec, we prove that there are inฯinitely many funda mental discriminants ๐‘‘ such that ๐ฟ(1/2, ๐‘“, ๐œ’๐‘‘) โ‰  0. In addition, following an idea outlined by Hoffstein and Luo, using combinatorial sieve, we prove that the same holds for inฯinitely many almost prime fundamental discriminants ๐‘‘ 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.
AFRIKAANSE OPSOMMING: ๐ฟ 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 ๐ฟ funksie ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) wat geassosieer word met โ€™n gegewe genormaliseerde ho lomorfe nuwe vorm ๐‘“ wat deur die Kronecker simbool ๐œ’๐‘‘ verdraai is. Dit is al gemeen bekend dat die verdraaide ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) absoluut konvergeer vir Re(๐‘ ) > 1 en โ€™n funksionele vergelyking het wat dit analities tot die hele komplekse vlak uitbrei. Die waarde van ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) by ๐‘  = 1/2 is van besondere belang. Byvoor beeld, as die vorm ๐‘“ โ€™n verdraaide elliptiese kromme๐ธ van โ€™n gegewe rang ๐‘Ÿ โ‰ฅ 0 parametriseer, beweer die Birch Swinnerton Dyer vermoede dat ๐‘Ÿ presies die orde van nulstelling van ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) by ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) is. In hierdie werk, bepaal ons โ€™n holomorfe nuwe vorm ๐‘“ van gewig van ten minste 2, met โ€™n vlak ๐‘ met โ€™n triviale nebentipe, en ons oorweeg die familie van verdraaide ๐ฟ funksies ๐ฟ(๐‘ , ๐‘“, ๐œ’๐‘‘) waar ๐‘‘ enige fundamentele diskriminant met (๐‘‘, ๐‘) = 1 is. Deur โ€™n aanpassing van โ€™n metode deur Iwaniec, bewys ons dat daar oneindig baie fundamentele diskriminante ๐‘‘ is sodat ๐ฟ(1/2, ๐‘“, ๐‘โ„Ž๐‘–๐‘‘) โ‰  0. Daarbenewens bewys ons, volgens โ€™n idee deur Hoffstein en Luo, deur gebruik te maak van โ€™n kombinatoriese sif, dat dieselfde waar is vir oneindig baie by kans primeฬ‚re fundamentele diskriminante ๐‘‘ met hoogstens 84 primeฬ‚re 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.
Thesis (MSc)--Stellenbosch University, 2023.