03673nam 2200697 450 991081354600332120201204082343.01-4704-6253-2(CKB)4100000011437433(MiAaPQ)EBC6346635(RPAM)21697713(PPN)250799235(EXLCZ)99410000001143743320201204d2020 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierExplicit arithmetic of Jacobians of generalized Legendre curves over global function fields /Lisa Berger [and seven others]Providence, Rhode Island :American Mathematical Society,[2020]©20201 online resource (144 pages)Memoirs of the American Mathematical Society ;Number 1295"Forthcoming, volume 266, number 1295."1-4704-4219-1 Includes bibliographical references.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)"--Provided by publisher.Memoirs of the American Mathematical Society ;Number 1295.Curves, AlgebraicLegendre's functionsRational points (Geometry)Birch-Swinnerton-Dyer conjectureJacobiansAbelian varietiesFinite fields (Algebra)Curves, Algebraic.Legendre's functions.Rational points (Geometry)Birch-Swinnerton-Dyer conjecture.Jacobians.Abelian varieties.Finite fields (Algebra)516.35211G1011G3011G4014G0514G2514K15mscBerger Lisa1969-1686950Hall Chris1975-Pannekoek RenéPark Jennifer Mun YoungPries Rachel1972-Sharif Shahed1977-Silverberg AliceUlmer Douglas1960-MiAaPQMiAaPQMiAaPQBOOK9910813546003321Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields4060057UNINA