Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields
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 th...
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| Weitere Verfasser: | , , , , , , |
| Format: | UnknownFormat |
| Sprache: | eng |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
2020
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| Schriftenreihe: | Memoirs of the American Mathematical Society
volume 266, number 1295 (July 2020) |
| Schlagworte: | |
| Online Zugang: | Inhaltsverzeichnis Inhaltsverzeichnis und Leseprobe |
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Tag hinzufügen | Zusammenfassung: | 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. "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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| Beschreibung: | Literaturverzeichnis: Seite 129-131 Includes bibliographical references |
| Beschreibung: | v, 131 Seiten Illustrationen |
| ISBN: | 9781470442194 978-1-4704-4219-4 9781470462536 |