Displaying similar documents to “Heuristics on Tate-Shafarevitch groups of elliptic curves defined over ℚ .”

On second 2-descent and non-congruent numbers

Yi Ouyang, Shenxing Zhang (2015)

Acta Arithmetica

Similarity:

We use the so-called second 2-descent method to find several series of non-congruent numbers. We consider three different 2-isogenies of the congruent elliptic curves and their duals, and find a necessary condition to estimate the size of the images of the 2-Selmer groups in the Selmer groups of the isogeny.

Counting elliptic curves of bounded Faltings height

Ruthi Hortsch (2016)

Acta Arithmetica

Similarity:

We give an asymptotic formula for the number of elliptic curves over ℚ with bounded Faltings height. Silverman (1986) showed that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in ℝ².

On families of 9-congruent elliptic curves

Tom Fisher (2015)

Acta Arithmetica

Similarity:

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose 9-torsion subgroups are isomorphic as Galois modules.