45 citations · 246 across the 34 of their papers we have counts for
5 papers · 1 filter
The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
Manjul Bhargava, Arul Shankar
In this article, we prove that the average rank of elliptic curves over , when ordered by height, is less than (in fact, less than ). As a consequence of our…
The average number of elements in the 4-Selmer groups of elliptic curves is 7
Manjul Bhargava, Arul Shankar
We prove that when all elliptic curves over are ordered by height, the average size of their 4-Selmer groups is equal to 7. As a consequence, we show that a positive p…
Counting -fields with a power saving error term
Arul Shankar, Jacob Tsimerman
We show how the Selberg -sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, w…
Abelian surfaces over finite fields with prescribed groups
Chantal David, Derek Garton, Zachary Scherr +3
Let A be an abelian surface over F_q, the field of q elements. The rational points on A/\F_q form an abelian group A(\F_q) \simeq \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_…
Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves
Arul Shankar, Xiaoheng Wang
In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational Weierstrass point. They showed that th…