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 
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


Additional Material

RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Requests
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