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Vanishing and non-vanishing of L-series of elliptic curves twisted by Dirichlet characters

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Vanishing and non-vanishing of L-series of elliptic curves twisted by Dirichlet characters

Fearnley, Jack (2001) Vanishing and non-vanishing of L-series of elliptic curves twisted by Dirichlet characters. PhD thesis, Concordia University.

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Abstract

We study the behaviour of L -series of elliptic curves twisted by Dirichlet characters. In particular, we study the vanishing and non vanishing of these L -series at the critical point. We present empirical results indicating the vanishing behaviour of cyclic twists of orders 3, 5, 7 and conductors up to 5000 for elliptic curves of conductor less than 100. We prove results for vanishing in the case of cyclic cubic twists and non-vanishing in the case of cyclic twists of arbitrary prime order. Let L ( E, s ) be the L -series of an elliptic curve E : y 2 = x 3 + Ax + B with A, B ✹ [Special characters omitted.] . If there exists a cyclic cubic character { such that L ( E , 1, {) = 0 or if L ( E , 1) = 0 then the L -series vanishes for an infinite number of cyclic cubic characters. With finite exceptions, if L ( E , 1) ✹ 0 there exist an infinite number of cyclic twists [Special characters omitted.] of prime order k such that L ( E , 1, [Special characters omitted.] ) ✹ 0 for every order k

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (PhD)
Authors:Fearnley, Jack
Pagination:vii, 85 leaves ; 29 cm.
Institution:Concordia University
Degree Name:Theses (Ph.D.)
Program:Mathematics and Statistics
Date:2001
Thesis Supervisor(s):Kisilevsky, H
ID Code:1324
Deposited By:Concordia University Libraries
Deposited On:27 Aug 2009 13:18
Last Modified:08 Dec 2010 10:20
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