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|Abstract:||In this paper, based on an idea of Tian we establish a new sufficient condition for a positive integer n to be a congruent number in terms of the Legendre symbols for the prime factors of n. Our criterion generalizes previous results of Heegner, Birch-Stephens, Monsky, and Tian, and conjecturally provides a list of positive density of congruent numbers. Our method of proving the criterion is to give formulae for the analytic Tate-Shafarevich number L(n) 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.|
|Electronic Publication Date:||25-Aug-2017|
|Citation:||Tian, Ye, Yuan, Xinyi, Zhang, Shou-Wu. (2017). GENUS PERIODS, GENUS POINTS AND CONGRUENT NUMBER PROBLEM. ASIAN JOURNAL OF MATHEMATICS, 21 (721 - 774). doi:10.4310/AJM.2017.v21.n4.a5|
|Pages:||721 - 774|
|Type of Material:||Journal Article|
|Journal/Proceeding Title:||ASIAN JOURNAL OF MATHEMATICS|
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