## 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 ł pr, we prove that J[ℓ](L) = {0} for any abelian extension L of Fp(t).

Original language | English (US) |
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Pages (from-to) | 1-144 |

Number of pages | 144 |

Journal | Memoirs of the American Mathematical Society |

Volume | 266 |

Issue number | 1295 |

DOIs | |

State | Published - Jul 2020 |

## 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
- Néron model
- Rank
- Tamagawa number
- Tate-shafarevich group
- Torsion

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics