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Constructing Distinct Curves with Isomorphic Jacobians

✍ Scribed by Everett W. Howe

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
322 KB
Volume
56
Category
Article
ISSN
0022-314X

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✦ Synopsis

We show that the hyperelliptic curves y 2 =x 5 +x 3 +x 2 &x&1 and y 2 =x 5 & x 3 +x 2 &x&1 over the field with three elements are not geometrically isomorphic, and yet they have isomorphic Jacobian varieties. Furthermore, their Jacobians are absolutely simple. We present a method for constructing further such examples. We also present two curves of genus three, one hyperelliptic and one a plane quartic, that have isomorphic absolutely simple Jacobians.

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