REVIEW 1 cited by
Genus Periods, Genus Points and Congruent Number Problem
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
read the original abstract
In this paper, based on an ideal of Tian, we establish a new sufficient condition for a positive integer to be a congruent number in terms of Legendre symbols of prime factors of the positive integer. Our criterion generalizes previous criterions of Heegner, and Birch--Stephens, Monsky, and Tian, and conjecturally provides a list of positive density of congruent numbers. Our method of proving our criterion is to give formulae for the analytic Tate--Shafarevich number in terms of the so-called genus periods and genus points. These formulae are derived from the Waldspurger formula and the generalized Gross--Zagier formula of Yuan-Zhang-Zhang.
Forward citations
Cited by 1 Pith paper
-
Unveiling Arithmetic Statistics of Congruent Number Elliptic Curves via Data Science and Machine Learning
A large empirical study of congruent number elliptic curves up to D < 3×10^6 confirms several Selmer-rank heuristics but does not support the paper's headline claim that Goldfeld's conjecture is rigorously verified.
Discussion (0). Sign in to comment.