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Softcover ISBN:  9781470442194 
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Softcover ISBN:  9781470442194 
Product Code:  MEMO/266/1295 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470462536 
Product Code:  MEMO/266/1295.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470442194 
eBook ISBN:  9781470462536 
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 DetailsMemoirs of the American Mathematical SocietyVolume: 266; 2020; 131 ppMSC: Primary 11; Secondary 14
The authors 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, the authors 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 \(\mathbb F_q(t^1/d)\).

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 TateShafarevich group

8. Monodromy of $\ell $torsion and decomposition of the Jacobian

A. An additional hyperelliptic family


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

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 TateShafarevich group

8. Monodromy of $\ell $torsion and decomposition of the Jacobian

A. An additional hyperelliptic family