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Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields

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Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields Synopsis

The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\mathbb F_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb F_q(t^1/d)$.

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ISBN: 9781470442194
Publication date:
Author: Lisa Berger, Chris Hall, René Pannekoek, Jennifer Mun Young Park, Rachel Pries, Shahed Sharif, Alice Silverberg, Douglas Ulmer
Publisher: American Mathematical Society
Format: Paperback
Pagination: 131 pages
Series: Memoirs of the American Mathematical Society
Genres: Calculus and mathematical analysis