REVIEW 1 cited by
The rank of 2-Selmer group associate to $\theta$-congruent numbers
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
abstract
We study the parity of rank of $2$-${\rm Selmer}$ 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 $n=pq\equiv 5(resp. \ 11)\pmod{24}$, the density of non $\pi/3$($resp.$ $2\pi/3$)-congruent numbers is at least 75\%, where $p,q$ are primes.
Forward citations
Cited by 1 Pith paper
-
The size of $2$-Selmer groups for the $\frac{\pi}{3}$-congruent number problem
For square-free n ≡ 5 or 13 mod 24 with all prime factors ≡ 1 mod 4, the average 2-Selmer size of the pi/3-congruent curve is 9, yielding positive densities of Selmer rank 0/2 and 1/3.
Discussion (0). Sign in to comment.