@article{79d39c5941c7429ba5ffa2eaa142538e,

title = "Explicit Arithmetic of Jacobians of Generalized Legendre Curves over Global Function Fields",

abstract = "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 ≥ 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(t 1/d) and show that the Birch and Swinnerton-Dyer conjecture holds for J over Fq(t 1/d). When d is divisible by r and of the form pν + 1, and Kd := Fp(μd, 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 φ(r)/2 with endomorphism algebra Z[μr] +. For a prime ℓ with ℓ {\l} pr, we prove that J[ℓ](L) = {0} for any abelian extension L of Fp(t).",

keywords = "Abelian variety, Birch and swinnerton-dyer conjecture, Curve, Descent, Endomorphism algebra, Finite field, Function field, Height, Jacobian, Kodaira-spencer map, L-function, Monodromy, Mordell-weil group, N{\'e}ron model, Rank, Tamagawa number, Tate-shafarevich group, Torsion",

author = "Lisa Berger and Chris Hall and Rene Pannekoek and Jennifer Park and Rachel Pries and Shahed Sharif and Alice Silverberg and Douglas Ulmer",

note = "Funding Information: This project was initiated at the workshop on Cohomological methods in abelian varieties at the American Institute of Mathematics, March 26–30, 2012. We thank AIM and the workshop organizers for making this paper possible. The first seven authors thank the last for initiating this project at the AIM workshop and for his leadership of the project. Author Hall was partially supported by Simons Foundation award 245619 and IAS NSF grant DMS-1128155. Author Park was partially supported by NSF grant DMS-10-69236 and NSERC PGS-D and PDF grants. Author Pries was partially supported by NSF grants DMS-11-01712 and DMS-15-02227. Author Silver-berg was partially supported by NSF grant CNS-0831004. Author Ulmer was partially supported by Simons Foundation award 359573. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the supporting agencies. We also thank Karl Rubin and Yuri G. Zarhin for helpful conversations and an anonymous referee for a careful reading and productive comments. Publisher Copyright: {\textcopyright} 2020 American Mathematical Society.",

year = "2020",

month = jul,

doi = "10.1090/memo/1295",

language = "English (US)",

volume = "266",

pages = "1--144",

journal = "Memoirs of the American Mathematical Society",

issn = "0065-9266",

publisher = "American Mathematical Society",

number = "1295",

}