Browsing by Author "Andrianarisoa, Tolotranirina Gabriel"

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    The nonvanishing of almost-prime twists of modular L-functions
    (Stellenbosch : Stellenbosch University, 2023-11) Andrianarisoa, Tolotranirina Gabriel; Ralaivaosaona, Dimbinaina; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH 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.