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.
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Abelian varieties.
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Jacobians.
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Birch-Swinnerton-Dyer conjecture.
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Rational points (Geometry)
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Legendre's functions.
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Finite fields (Algebra)
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Geometría algebraica
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Variedades abelianas
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Curvas
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Jacobianos
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Abelian varieties
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Birch-Swinnerton-Dyer conjecture
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Curves, Algebraic
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Finite fields (Algebra)
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Jacobians
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Legendre's functions
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Rational points (Geometry)
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Number theory -- Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] -- Abelian varieties of dimension $> 1$ [See also 14Kxx].
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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].
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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]
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Algebraic geometry -- Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx] -- Rational points.
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Algebraic geometry -- Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx] -- Global ground fields.
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Algebraic geometry -- Abelian varieties and schemes -- Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx].
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Form |
Electronic book
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Author |
Hall, Chris, 1975- author.
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Pannekoek, René, author.
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Park, Jennifer Mun Young, author.
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Pries, Rachel, 1972- author.
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Sharif, Shahed, 1977- author.
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Silverberg, Alice, author.
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Ulmer, Douglas, 1960- author.
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LC no. |
2020032015 |
ISBN |
1470462532 |
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9781470462536 |
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