Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields

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 ${\mathbb{F}}_q(t^{1\mathrm{/}d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over ${\mathbb{F}}_q(t^{1\mathrm{/}d})$. When $d$ is divisible by $r$ and of the form $p^{\nu }+1$, and $K_d := $\mathbb{F}$_p(\mu_d,t^{1/d})$,wewritedownexplicitpointsin$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 $\varphi (r)\mathrm{/}2$ with endomorphism algebra $\mathbb{Z}[{\mu }_r{]}^{+}$. For a prime $\mathrm{\ell }$ with $\mathrm{\ell }\nmid pr$, we prove that $J[\mathrm{\ell }](L)=\{0\}$ for any abelian extension $L$ of ${\bar{\mathbb{F}}}_p(t)$.