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Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields

About this Title

Lisa Berger, Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA, Chris Hall, Department of Mathematics, Western University, London, Ontario, N6A 5B7, Canada, René Pannekoek, Department of Mathematics, Imperial College London SW7 2AZ, United Kingdom, Jennifer Park, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, Rachel Pries, Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA, Shahed Sharif, Department of Mathematics, CSU San Marcos, San Marcos, CA 92096, USA, Alice Silverberg, Department of Mathematics, UC Irvine, Irvine, CA 92697, USA and Douglas Ulmer, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Publication: Memoirs of the American Mathematical Society
Publication Year: 2020; Volume 266, Number 1295
ISBNs: 978-1-4704-4219-4 (print); 978-1-4704-6253-6 (online)
Published electronically: July 22, 2022
Keywords: Curve, function field, Jacobian, abelian variety, finite field, Mordell-Weil group, torsion, rank, $L$-function, Birch and Swinnerton-Dyer conjecture, Tate-Shafarevich group, Tamagawa number, endomorphism algebra, descent, height, NĂ©ron model, Kodaira-Spencer map, monodromy
MSC: Primary 11G10, 11G30; Secondary 11G40, 14G05, 14G25, 14K15

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Table of Contents


  • Introduction
  • 1. The curve, explicit divisors, and relations
  • 2. Descent calculations
  • 3. Minimal regular model, local invariants, and domination by a product of curves
  • 4. Heights and the visible subgroup
  • 5. The $L$-function and the BSD conjecture
  • 6. Analysis of $J[p]$ and $\operatorname {NS}(\mathcal {X}_d)_{\mathrm {tor}}$
  • 7. Index of the visible subgroup and the Tate-Shafarevich group
  • 8. Monodromy of $\ell$-torsion and decomposition of the Jacobian
  • A. An additional hyperelliptic family


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 $p^\nu +1$, 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 $(r-1)(d-2)$ 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 $\bar {\mathbb {F}_p(t)}$ 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 $\bar {\mathbb {F}}_p(t)$.

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