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Record Nr. |
UNINA9910813546003321 |
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Autore |
Berger Lisa <1969-> |
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Titolo |
Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields / / Lisa Berger [and seven others] |
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Pubbl/distr/stampa |
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Providence, Rhode Island : , : American Mathematical Society, , [2020] |
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©2020 |
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ISBN |
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Descrizione fisica |
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1 online resource (144 pages) |
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Collana |
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Memoirs of the American Mathematical Society ; ; Number 1295 |
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Classificazione |
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11G1011G3011G4014G0514G2514K15 |
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Disciplina |
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Soggetti |
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Curves, Algebraic |
Legendre's functions |
Rational points (Geometry) |
Birch-Swinnerton-Dyer conjecture |
Jacobians |
Abelian varieties |
Finite fields (Algebra) |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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"Forthcoming, volume 266, number 1295." |
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Nota di bibliografia |
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Includes bibliographical references. |
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Nota di contenuto |
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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. |
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Sommario/riassunto |
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"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, |
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