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 $y^r=x^{r-1}(x+1)(x+t)$ over the function field ${\mathbb{F}}_p(t)$, when $p$ is prime and $r\ge 2$ 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 ${\mathbb{F}}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over ${\mathbb{F}}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^{\hat{I}}½+1$, and $K_d:={\mathbb{F}}_p(\hat{I}{¼}_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V{]}^2$. When $r>2$, we prove that the "new" part of $J$ is isogenous over $\overline{{\mathbb{F}}_p(t)}$ to the square of a simple abelian variety of dimension $\ddot{I}•(r)/2$ with endomorphism algebra $\mathbb{Z}[\hat{I}{¼}_r{]}^{+}$. For a prime $\ell$ with $\ell \nmid pr$, we prove that $J[\ell ](L)=\{0\}$ for any abelian extension $L$ of ${\overline{\mathbb{F}}}_p(t)$.

v1: vi+121 pages. v2: numerous improvements following referee's report. vi+131 pages