Abstract: Vatsal has proved recently a result which has consequences for the existence of rational points of odd prime order ℓ on optimal elliptic curves over ℚ. When the conductor N is squarefree, ℓ ∤ N and the local root number w p = -1 for at least one prime p | N, we offer a somewhat different proof, starting from an explicit cuspidal divisor on X 0 (N). We also prove some results linking the vanishing of L(E,1) with the divisibility by ℓ of the modular parametrization degree, fitting well with the Bloch–Kato conjecture for L( Sym 2 E,2), and with an earlier construction of elements in Shafarevich–Tate groups. Finally (following Faltings and Jordan) we prove an analogue of the result on ℓ-torsion for cuspidal Hecke eigenforms of level one (and higher weight), thereby strengthening some existing evidence for another case of the Bloch–Kato conjecture.
Publication Year: 2005
Publication Date: 2005-12-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 10
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