Explicit isogeny descent on elliptic curves

@article{Miller2013ExplicitID,
  title={Explicit isogeny descent on elliptic curves},
  author={Robert L. Miller and Michael Stoll},
  journal={Math. Comput.},
  year={2013},
  volume={82},
  pages={513-529}
}
In this note, we consider an `-isogeny descent on a pair of elliptic curves over Q. We assume that ` > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finitedimensional F`-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the `-part of the Birch and Swinnerton-Dyer conjectural formula for certain curves of small conductor. 
The class group pairing and p‐descent on elliptic curves
We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows
Second p-descents on elliptic curves
TLDR
An algorithm is described which computes the set of D in the Shafarevich-Tate group such that pD = C and obtains explicit models for these D as curves in projective space, which leads to a practical algorithm for performing explicit 9-descents on elliptic curves over Q.
Explicit 5-descent on elliptic curves
We compute equations for genus one curves representing non-trivial elements of order 5 in the Tate-Shafarevich group of an elliptic curve. We explain how to write the equations in terms of Pfaffians
Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic norm equations
TLDR
A new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations, is proposed and it is shown that the elliptic curves with torsion subgroup of order 3 and rank at least 13, found by Eroshkin, have rank exactly 13.
Vanishing of Some Galois Cohomology Groups for Elliptic Curves
Let \(E/\mathbb {Q}\) be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of \(\mathbb {Q}\) obtained by adjoining the coordinates of the p-torsion points on
Rational points on Jacobians of hyperelliptic curves
  • J. Müller
  • Mathematics
    Advances on Superelliptic Curves and their Applications
  • 2015
TLDR
The Mordell-Weil theorem for Jacobians of hyperelliptic curves over Q is described and how to compute the rank and generators for the Mordell -Weil group is described.
Computing the Cassels-Tate pairing
For an elliptic curve E admitting a p-isogeny φ : E→ E we calculate the Cassels-Tate pairing on S(φ)(E/K)× S(φ)(E/K) using a pushout form. We calculate examples in the p = 3 case of type μ3-nonsplit,
Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces
Let X be one of the 28 Atkin–Lehner quotients of a curve X0(N ) such that X has genus 2 and its Jacobian variety J is absolutely simple. We show that the Shafarevich–Tate group X(J/Q) is trivial.
...
...

References

SHOWING 1-10 OF 82 REFERENCES
Explicit Descent via 4-Isogeny on an Elliptic Curve
TLDR
In the process the 4-isogeny and the isogenous curve are exhibited, the principal homogeneous spaces are explicitly presented, and examples by computing the rank are discussed.
Explicit n-descent on elliptic curves III. Algorithms
This is the third in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The methods we
Descent by 3-isogeny and 3-rank of quadratic fields
In this paper families of elliptic curves admitting a rational isogeny of degree 3 are studied. It is known that the 3-torsion in the class group of the field defined by the points in the kernel of
Some examples of 5 and 7 descent for elliptic curves over Q
Abstract.We perform descent calculations for the families of elliptic curves over Q with a rational point of order n = 5 or 7. These calculations give an estimate for the Mordell-Weil rank which we
Explicit 5-descent on elliptic curves
We compute equations for genus one curves representing non-trivial elements of order 5 in the Tate-Shafarevich group of an elliptic curve. We explain how to write the equations in terms of Pfaffians
Explicit n-descent on elliptic curves, I. Algebra
Abstract This is the first in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The
5-Torsion in the Shafarevich–Tate Group of a Family of Elliptic Curves
Abstract We compute the φ -Selmer group for a family of elliptic curves, where φ is an isogeny of degree 5, then find a practical formula for the Cassels–Tate pairing on the φ -Selmer groups and use
Explicit n-descent on elliptic curves, II. Geometry
TLDR
This paper shows how to realize elements of the n-Selmer group explicitly as curves of degree n embedded in ℙ n–1 through a comparison between an easily obtained embedding into �’ n 2–1 and another map into ™ n 2-1 that factors through the Segre embedding.
...
...