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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
Add to cart
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
Add to cart
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 266; 2020; 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
Memoirs of the American Mathematical Society
Volume: 266; 2020; 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
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