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Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
 
Lisa Berger Stony Brook University, Stony Brook, NY
Chris Hall Western University, London, Ontario, Canada
Rene Pannekoek Imperial College, London, UK
Rachel Pries Colorado State University, Fort Collins, CO
Shahed Sharif California State University San Marcos, San Marcos, CA
Alice Silverberg University of California at Irvine, Irvine, CA
Douglas Ulmer Georgia Institute of Technology, Atlanta, GA
Jennifer Park University of Michigan, Ann Arbor, MI
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
Softcover ISBN:  978-1-4704-4219-4
Product Code:  MEMO/266/1295
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6253-6
Product Code:  MEMO/266/1295.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-4219-4
eBook: ISBN:  978-1-4704-6253-6
Product Code:  MEMO/266/1295.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
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Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
Lisa Berger Stony Brook University, Stony Brook, NY
Chris Hall Western University, London, Ontario, Canada
Rene Pannekoek Imperial College, London, UK
Rachel Pries Colorado State University, Fort Collins, CO
Shahed Sharif California State University San Marcos, San Marcos, CA
Alice Silverberg University of California at Irvine, Irvine, CA
Douglas Ulmer Georgia Institute of Technology, Atlanta, GA
Jennifer Park University of Michigan, Ann Arbor, MI
Softcover ISBN:  978-1-4704-4219-4
Product Code:  MEMO/266/1295
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6253-6
Product Code:  MEMO/266/1295.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-4219-4
eBook ISBN:  978-1-4704-6253-6
Product Code:  MEMO/266/1295.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2662020; 131 pp
    MSC: Primary 11; Secondary 14;

    The authors 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, the authors 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 \(\mathbb F_q(t^1/d)\).

  • Table of Contents
     
     
    • Chapters
    • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2662020; 131 pp
MSC: Primary 11; Secondary 14;

The authors 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, the authors 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 \(\mathbb F_q(t^1/d)\).

  • Chapters
  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.