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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 andSwinnerton-Dyer conjecture holds for $J$ over $\mathbb{F}_q(t^{1/d})$. When $d$is divisible by $r$ and of the form $p^\nu +1$, and $K_d :=\mathbb{F}_p(\mu_d,t^{1/d})$, we write down explicit points in $J(K_d)$, showthat 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-Shafarevichgroup 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 asimple abelian variety of dimension $\phi(r)/2$ with endomorphism algebra$\mathbb{Z}[\mu_r]^+$. For a prime $\ell$ with $\ell \nmid pr$, we prove that$J[\ell](L)=\{0\}$ for any abelian extension $L$ of$\overline{\mathbb{F}}_p(t)$.

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