On the arithmetic of the curves y² = xˡ + A, II

Titelangaben

Stoll, Michael:
On the arithmetic of the curves y² = xˡ + A, II.
In: Journal of Number Theory. Bd. 93 (2002) Heft 2 . - S. 183-206.
ISSN 0022-314X
DOI: https://doi.org/10.1006/jnth.2001.2727

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Abstract

This paper continues the investigation of the arithmetic of the curves C_A: y^2=x^ℓ+A and their Jacobians J_A, where ℓ is an odd prime and A is an integer not divisible by ℓ, which was begun in an earlier paper. In the first part, we sketch how to extend the formula for the dimension of a certain Selmer group of J_A to the case when A is a (non-zero) square mod ℓ. The second part deals with the L-series of J_A. We determine the corresponding Hecke character and find a formula for the root number of the L-series. This formula is then used to show the “Birch and Swinnerton-Dyer conjecture mod 2”
ord_s=1 L(J_A,s) == rank J_A(Q) mod 2
for those A that are covered by the result of the first part, assuming the ℓ-part of Ш(Q, J_A) to be finite.