On the arithmetic of the curves y² = xˡ + A and their Jacobians

Titelangaben

Stoll, Michael:
On the arithmetic of the curves y² = xˡ + A and their Jacobians.
In: Journal für die Reine und Angewandte Mathematik. (1998) Heft 501 . - S. 171-189.
ISSN 1435-5345
DOI: https://doi.org/10.1515/crll.1998.076

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Abstract

Let l be an odd prime, and for a non-zero integer A, let C_A be the normalization of the curve given by the affine equation y^2 = x^l + A, and let J_A be its Jacobian, which is a 1/2(l-1)-dimensional abelian variety defined over Q. We use a method invented by Ed Schaefer to compute the (1-zeta_l)-Selmer grou Sel^(1-zeta_l)(K,J_A) of J_A over K=Q(zeta_l) under suitable hypotheses on A. This leads to bouds for the Mordell-Weil ranks of J_A(K) and of J_A(Q).