Congruent numbers with many prime factors

Abstract

Mohammed Ben Alhocain, in an Arab manuscript of the 10th century, stated that the principal object of the theory of rational right triangles is to find a square that when increased or diminished by a certain number, m becomes a square [Dickson LE (1971) History of the Theory of Numbers (Chelsea, New York), Vol 2, Chap 16]. In modern language, this object is to find a rational point of infinite order on the elliptic curve . Heegner constructed such rational points in the case that m are primes congruent to 5,7 modulo 8 or twice primes congruent to 3 modulo 8 [Monsky P (1990) Math Z 204:45–68]. We extend Heegner's result to integers m with many prime divisors and give a sketch in this report. The full details of all the proofs will be given in ref. 1 [Tian Y (2012) Congruent Numbers and Heegner Points, arXiv:1210.8231].

Lectures on the Birch-Swinnerton-Dyer Conjecture

“…Until recently, this problem seemed largely inaccessible. But Tian [12,13] has introduced a beautiful new idea which should enable us to eventually prove this for many elliptic curves.…”

“…We will begin by briefly explaining what was known about it until recently. By beautifully generalizing some old work of Heegner [8], Ye Tian [12,13] has very recently made important progress on this conjecture for one family of elliptic curves (the quadratic twists of the elliptic curve y 2 = x 3 − x), and it is now an important question to generalize his method to the quadratic twists of all elliptic curves defined over Q. In the latter part of the lectures, I will discuss joint ongoing work (see [2]) with Yongxiong Li, Ye Tian and Shuai Zhai, which makes a first step in this direction by establishing analogous results for the elliptic curve E = X 0 (49) with equation y 2 + xy = x 3 − x 2 − 2x − 1.…”

The Birch-Swinnerton-Dyer conjecture and Heegner points: A survey

“…It seems difficult to obtain the same result for small primes p, especially p = 2. However, Tian ([48], [49]) and Tian-Yuan-S. Zhang ([50]) have proved the 2-part of the B-SD formula in the case of analytic rank zero or one, for "many" (expected to be of a high percentage) quadratic twists of the congruent number elliptic curve:…”

Congruent numbers

“…For the congruent number elliptic curve with such N, the L-function L(E, s) does not usually vanish at s = 1, and it is known that the quantity M(E) = L(E, 1)/Ω is a rational number, where Ω denotes the smallest positive real period of E. The conjecture of Birch and Swinnerton-Dyer (1) predicts that, for these N, the rational number M(E) must always be divisible by 4 k , and Zhao (14,15) found an ingenious proof of this fact. This divisibility plays a central role in Tian's work (2). Both Tian (2) and Zhao (14,15) use induction on the number k of odd prime divisors of N in their work, and it is intriguing to note that there are some striking parallels between the averaging of Heegner points attached to divisors of N used by Tian (2), and the averaging of the values M(E) over divisors of N used by Zhao (14,15), in their separate proofs.…”

“…This divisibility plays a central role in Tian's work (2). Both Tian (2) and Zhao (14,15) use induction on the number k of odd prime divisors of N in their work, and it is intriguing to note that there are some striking parallels between the averaging of Heegner points attached to divisors of N used by Tian (2), and the averaging of the values M(E) over divisors of N used by Zhao (14,15), in their separate proofs. In conclusion, Tian's work (2) is an important milestone in the history of this ancient problem, and, as has always happened in the past, it seems only a matter of time until its generalization to all elliptic curves is established.…”