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

✍ Scribed by Michael Stoll

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
205 KB
Volume
93
Category
Article
ISSN
0022-314X

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✦ Synopsis

This paper continues the investigation of the arithmetic of the curves C A : y 2 =x a +A and their Jacobians J A , where a is an odd prime and A is an integer not divisible by a, 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 a. 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''

for those A that are covered by the result of the first part, assuming the a-part of I(Q, J A ) to be finite.

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