  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.
