On the Shafarevich-Tate group of the jacobian of a quotient of the Fermat curve

McCallum, William G.

doi:10.1007/bf01410203

Cited by 22 publications

(24 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One can prove this in a quite similar way to [10,Theorem 1.4]. See also [2] and [7], where the case n = 5 is treated.…”

Section: The Cassels-tate Pairingsupporting

confidence: 55%

See 1 more Smart Citation

On the Tate–Shafarevich group of semistable elliptic curves with a rational 3-torsion

Aoki

2004

Acta Arith.

View full text Add to dashboard Cite

“…One can prove this in a quite similar way to [10,Theorem 1.4]. See also [2] and [7], where the case n = 5 is treated.…”

Section: The Cassels-tate Pairingsupporting

confidence: 55%

“…We begin with a theorem proved by McCallum [10], which is fundamental in our calculation. It enables us to describe the Cassels-Tate pairing , ψ defined in (2) in terms of the Hilbert norm residue symbol…”

Section: The Cassels-tate Pairingmentioning

confidence: 99%

On the Tate–Shafarevich group of semistable elliptic curves with a rational 3-torsion

Aoki

2004

Acta Arith.

View full text Add to dashboard Cite

“…For that reason, these tend to be somewhat unwieldy to implement. Other algorithms use functions on the curve to compute a Selmer group ( [BK,Ca,CF,Fd,FPS,KS,Mc,PS,Sc1,Tp]). These tend to be far easier.…”

Section: Introductionmentioning

confidence: 99%

Computing a Selmer group of a Jacobian using functions on the curve

Schaefer¹

1998

Mathematische Annalen

View full text Add to dashboard Cite

1In general, algorithms for computing the Selmer group of the Jacobian of a curve have relied on either homogeneous spaces or functions on the curve. We present a theoretical analysis of algorithms which use functions on the curve, and show how to exploit special properties of curves to generate new Selmer group computation algorithms. The success of such an algorithm will be based on two criteria that we discuss. To illustrate the types of properties which can be exploited, we develop a (1 − ζ p )-Selmer group computation algorithm for the Jacobian of a curve of the form y p = f (x) where p is a prime not dividing the degree of f .We compute Mordell-Weil ranks of the Jacobians of three curves of this form. We also compute a 2-Selmer group for the Jacobian of a smooth plane quartic curve using bitangents of that curve, and use it to compute a Mordell-Weil rank.

show abstract

“…It remains an open problem whether the inequality rk(J p,r (Q)) ≤ 2 holds in general. One possible approach is to perform a (ζ − 1)-descent on J p,r using its isogeny decomposition over K combined with results of Faddeev ([6]) and McCallum ( [15]). For some time, the author was under the impression that the information on the corresponding Selmer and Shafarevich-Tate groups obtained by this approach produces examples where the inequality rk(J p,r (Q)) ≤ 2 fails.…”

Section: Proposition 21 Let M Be Any Number Field Thenmentioning

confidence: 99%