Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields [electronic resource] / Lisa Berger, Chris Hall, Rene Pannekoek, Jennifer Park, Rachel Pries, Shahed Sharif, Alice Silverberg, Douglas Ulmer.
"We study the Jacobian J of the smooth projective curve C of genus r-1 with affine model yr = xr-1(x+ 1)(x + t) over the function field Fp(t), when p is prime and r [greater than or equal to] 2 is an integer prime to p. When q is a power of p and d is a positive integer, we compute the L-function of...
Main Authors: | |
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Language: | English |
Published: |
Providence, RI :
American Mathematical Society,
[2020]
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Series: | Memoirs of the American Mathematical Society,
number 1295 |
Subjects: | |
Online Access: | |
Format: | Electronic eBook |
Summary: |
"We study the Jacobian J of the smooth projective curve C of genus r-1 with affine model yr = xr-1(x+ 1)(x + t) over the function field Fp(t), when p is prime and r [greater than or equal to] 2 is an integer prime to p. When q is a power of p and d is a positive integer, we compute the L-function of J over Fq(t1/d) and show that the Birch and Swinnerton-Dyer conjecture holds for J over Fq(t1/d). When d is divisible by r and of the form p[nu] + 1, and Kd := Fp([mu]d, t1/d), we write down explicit points in J(Kd), show that they generate a subgroup V of rank (r-1)(d-2) whose index in J(Kd) is finite and a power of p, and show that the order of the Tate-Shafarevich group of J over Kd is [J(Kd) : V ]2. When r > 2, we prove that the "new" part of J is isogenous over Fp(t) to the square of a simple abelian variety of dimension [phi](r)/2 with endomorphism algebra Z[[mu]r]+. For a prime with pr, we prove that J[](L) = {0} for any abelian extension L of Fp(t)"-- Provided by publish |
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Note: | "Forthcoming, volume 266, number 1295." |
Bibliography Note: | Includes bibliographical references. |
ISBN: | 9781470462536 (online) |
ISSN: | 0065-9266 ; |