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Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields /
"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-functi...
| Clasificación: | Libro Electrónico |
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| Autores principales: | , , , , , , , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Providence, RI :
American Mathematical Society,
[2020]
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| Colección: | Memoirs of the American Mathematical Society,
number 1295 |
| Temas: |
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.
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| Acceso en línea: | Texto completo |
| Sumario: | "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)"-- |
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| Notas: | "Forthcoming, volume 266, number 1295." |
| Descripción Física: | 1 online resource (v, 144 pages) |
| Bibliografía: | Includes bibliographical references. |
| ISBN: | 1470462532 9781470462536 |
| ISSN: | 0065-9266 ; |