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Congruent numbers and Heegner points. (English) Zbl 1303.11067

Let us recall that an integer \(n>1\) is congruent if the rank of the elliptic curve \(E^{(n)}:\;ny^2=x^3-x\) is positive. The author proves that for any given integer \(k\geq 0\) there are infnitely many square-free congruent numbers with exactly \(k+1\) odd prime divisors in each residue class of 5, 6, and 7 modulo 8. This result generalizes an earlier results of K. Heegner [Math. Z. 56, 227–253 (1952; Zbl 0049.16202)], B. J. Birch [in: Sympos. Math., Roma 4, Teoria Numeri, Dic. 1968, e Algebra, Marzo 1969, 27–32 (1970; Zbl 0225.14016)], N. M. Stephens [Bull. Lond. Math. Soc. 7, 182–184 (1975; Zbl 0304.10011)] and P. Monsky [Math. Z. 204, No. 1, 45–67 (1990; Zbl 0705.14023)] in case \(k=0\), and Monsky and Gross in case of \(k=1\). The method of the proof is based on the method of Heegner.
Using a combination of induction, Kolyyvagin’s Euler system and a clever generalization of the Gross-Zagier formula, the author deduces the following (partial) result towards the conjecture of Birch and Swinnerton-Dyer: Let \(m\equiv 5,6,7\pmod{8}\) be a square-free integer such that its odd part \(n=p_{0}p_{1}\cdots p_{k},k\geq 0\), has prime factors \(p_{i}\equiv 1\pmod{8}\) for \(1\leq i\leq k\), and satisfies the condition that the field \(\mathbb{Q}(\sqrt{-n})\) has no ideal classes of exact order 4. Then, for the elliptic curve \(E^{(m)}\) over \(\mathbb{Q}\), one has \[ \mathrm{rank}_{\mathbb{Z}}E^{(m)}(\mathbb{Q})=1=\mathrm{ord}_{s=1}L(E^{(m)},s). \] Moreover, in this case the Shafarevitch-Tate group of \(E^{(m)}\) is finite and has odd cardinality.

MSC:

11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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