Ranks, $2$Selmer groups, and Tamagawa numbers of elliptic curves with $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$torsion
Abstract
In 2016, BalakrishnanHoKaplanSpicerSteinWeigandt produced a database of elliptic curves over $\mathbb{Q}$ ordered by height in which they computed the rank, the size of the $2$Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over $\mathbb{Q}$ whose rational torsion subgroup is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$. Conditional on GRH and BSD, we compute the rank of $92\%$ of the $202461$ curves with parameter height less than $10^3$. We also compute the size of the $2$Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 DOI:
 10.48550/arXiv.1805.10709
 arXiv:
 arXiv:1805.10709
 Bibcode:
 2018arXiv180510709C
 Keywords:

 Mathematics  Number Theory
 EPrint:
 Open Book Series 2 (2019) 173189