Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields /

Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields /

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Bibliographic Details
Author / Creator:	Berger, Lisa, 1969- author.
Imprint:	Providence, RI : American Mathematical Society, [2020]
Description:	v, 131 pages : illustrations ; 26 cm
Language:	English
Series:	Memoirs of the American Mathematical Society, 0065-9266 ; number 1295 Memoirs of the American Mathematical Society ; no. 1295.
Subject:	Curves, Algebraic. Abelian varieties. Jacobians. Birch-Swinnerton-Dyer conjecture. Rational points (Geometry) Legendre's functions. Finite fields (Algebra) Abelian varieties Birch-Swinnerton-Dyer conjecture Curves, Algebraic Finite fields (Algebra) Jacobians Legendre's functions Rational points (Geometry)
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/12617470

Hidden Bibliographic Details
Other authors / contributors:	Hall, Chris, 1975- author. Pannekoek, René, author. Park, Jennifer Mun Young, author. Pries, Rachel, 1972- author. Sharif, Shahed, 1977- author. Silverberg, Alice, author. Ulmer, Douglas, 1960- author.
ISBN:	9781470442194 1470442191 9781470462536
Notes:	"Forthcoming, volume 266, number 1295." Includes bibliographical references.
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)"--

Mansueto

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Call Number:	QA1.A528 no.1295
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