 Main
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 $r1$ with affine model $y^r = x^{r1}(x + 1)(x + t)$ 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 SwinnertonDyer 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 $J(K_d)$, show that they generate a subgroup $V$ of rank $(r1)(d2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the TateShafarevich 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 $\phi(r)/2$ 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 $\overline{\mathbb{F}}_p(t)$.
Many UCauthored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.
Main Content
Enter the password to open this PDF file:













