On the Arithmetic of the Curves y2=xℓ+A, II

AbstractThis paper continues the investigation of the arithmetic of the curves CA:y2=xℓ+A and their Jacobians JA, 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 JA to the case when A is a (non-zero) square modℓ. The second part deals with the L-series of JA. 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 mod2”ords=1L(JA, s)≡rankJA(Q)mod2 for those A that are covered by the result of the first part, assuming the ℓ-part of Ш(Q, JA) to be finite