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.
Genus Periods, Genus Points and Congruent Number Problem
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
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.
citation-role summary
background 1
citation-polarity summary
fields
math.NT 1years
2025 1verdicts
CONDITIONAL 1roles
background 1polarities
unclear 1representative citing papers
citing papers explorer
-
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.