The rank of 2-Selmer group associate to $\theta$-congruent numbers

Abstract

We study the parity of rank of $2$-$\mathrm{Selmer}$ groups associated to $\pi /3$ and $2\pi /3$-congruent numbers. Our second result gives some positive densities about $\pi /3$ and $2\pi /3$ non-congruent numbers which can support the even part of Goldfeld's conjecture. We give some necessary conditions such that $n$ is non $\pi /3$-congruent number for elliptic curves $E_n$ whose Shafarevich-Tate group is non-trivial. In the last section, we show that for $n=pq\equiv 5(resp.11)(\mathrm{mod}24)$, the density of non $\pi /3$($resp.$ $2\pi /3$)-congruent numbers is at least 75\%, where $p,q$ are primes.