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Author Berger, Lisa, 1969- author.

Title Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields / Lisa Berger, Chris Hall, Rene Pannekoek, Jennifer Park, Rachel Pries, Shahed Sharif, Alice Silverberg, Douglas Ulmer
Published Providence, RI : American Mathematical Society, [2020]
©2020

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Description 1 online resource (v, 144 pages)
Series Memoirs of the American Mathematical Society, 0065-9266 ; number 1295
Contents The curve, explicit divisors, and relations -- Descent calculations -- Minimal regular model, local invariants, and domination by a product of curves -- Heights and the visible subgroup -- The L-function and the BSD conjecture -- Analysis of J[p] and NS(Xd)tor -- Index of the visible subgroup and the Tate-Shafarevich group -- Monodromy of ℓ-torsion and decomposition of the Jacobian
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 publisher
Notes "Forthcoming, volume 266, number 1295."
Bibliography Includes bibliographical references
Notes Description based on print version record
Subject Curves, Algebraic.
Abelian varieties.
Jacobians.
Birch-Swinnerton-Dyer conjecture.
Rational points (Geometry)
Legendre's functions.
Finite fields (Algebra)
Geometría algebraica
Variedades abelianas
Curvas
Jacobianos
Abelian varieties
Birch-Swinnerton-Dyer conjecture
Curves, Algebraic
Finite fields (Algebra)
Jacobians
Legendre's functions
Rational points (Geometry)
Number theory -- Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] -- Abelian varieties of dimension $> 1$ [See also 14Kxx].
Number theory -- Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] -- Curves of arbitrary genus or genus $\ne 1$ over global fields [See also 14H25].
Number theory -- Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] -- $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Algebraic geometry -- Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx] -- Rational points.
Algebraic geometry -- Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx] -- Global ground fields.
Algebraic geometry -- Abelian varieties and schemes -- Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx].
Form Electronic book
Author 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.
LC no. 2020032015
ISBN 1470462532
9781470462536