``Congruent number curves'' ny²=x³-x with |n| congruent to 7 mod 8 and less than a million: Positivity of rank, and list of curves of analytic rank 3

For which numbers n is there a rational Pythagorean triangle of area n? This problem is known to be equivalent to: for which n does the elliptic curve

have a rational point other than the point at infinity and the 2-torsion points with y=0? Such n are said to be ``congruent'' because they are also known to be the numbers that occur as the common difference (``congruum'' in Latin) of a three-term arithmetic progression of squares.

We may, and henceforth do, assume that n is a positive squarefree integer. Fermat proved in effect that the curve E(1) has no rational points of infinite order, and deduced the exponent-4 case of his ``last theorem''. Thus 1 is not a congruent number. For general n, the curve E(n) is a ``quadratic twist'' of E(1), and its group of rational torsion points consists of the point at infinity and the 2-torsion points (0,0), (n,0), (-n,0). Thus n is congruent if and only if E(n) has positive rank. For more information about elliptic curves in general and the family of curves E(n) in particular see N.Koblitz's text Introduction to elliptic curves and modular forms, New York: Springer 1984.

It is known that the first congruent numbers are 5, 6, and 7; for instance, 6 is the area of the 3-4-5 triangle, and the common difference of {(1/2)², (5/2)², (7/2)²}, while 5 is the area of the triangle of sides 9/6, 40/6, 41/6 and the common difference of {(31/12)², (41/12)², (49/12)²}. In each of these three cases, E(n) has rank 1. In fact it is known that the sign of the functional equation of L(E(n),s) is +1 for n congruent to 1, 2, or 3 mod 8, and -1 for n congruent to 5, 6, or 7 mod 8. Thus the conjecture of Birch and Swinnerton-Dyer implies that the rank of E(n) is even if n is of the form 8k+1, 8k+2, or 8k+3, and odd if n is of the form 8k+5, 8k+6, or 8k+7. In particular, if n=8k+5, 8k+6, or 8k+7 then n is conjectured to be congruent.

Theorem. If n=8k+7 and n<10⁶ then n is a congruent number.

The proof combines computation and theory. In my paper ``Heegner point computations'' [Lecture Notes in Computer Science 877 (proceedings of ANTS-1, 5/1994), 122-133], I announced this result with for n<200000; the extension to 10⁶ combines the same theoretical tools with new software (Cremona's MWRANK) and faster hardware.

The main tool is the use of Heegner points of discriminant (-n) on the modular curve X₀(32) to construct a rational point P(n) on E(n). The only possible stumbling block is that P(n) may be a torsion point. But this can be detected in time essentially Cn^1/2 by calculating a real approximation to P(n) with low precision -- we used 19 digits in gp. We found only 339 cases, tabulated below, where P(n) is a torsion point.

Now by the celebrated formula of B.H.Gross and D.Zagier [see their paper ``Heegner points and derivatives of L-series'', Invent. Math. 84 (1986), 225-320], the canonical height of P(n) is proportional to the value at s=1 of the first derivative of the L-series attached to E(n). [Perhaps unexpectedly, testing whether P(n) is a torsion point is faster than directly computing L'(E(n),1), which takes time >>n.] Thus P(n) is a torsion point if and only if that derivative vanishes, that is, if and only if E(n) has analytic rank at least 3. So, it is precisely when the Heegner construction fails that we expect E(n) to have many rational points.

Now the usual conjectures and heuristics suggest that the regulator of E(n) can grow roughly as n^1/2. Thus the minimal nonzero height of a rational point on E(n) can grow roughly as n^1/2 when E(n) has rank 1, but only as n^1/6 when E(n) has rank 3. Still, as n increases, even a point of height roughly n^1/6 can be hard to find, let alone three independent points. Even with MWRANK, we were not able to find a rank-3 subgroup of each of our 339 curves E(n). Nevertheless, in every case MWRANK found at least one nontorsion point, and this sufficed to complete the proof that every positive integer n=8k+7<10⁶ is congruent. This computation took several days on a 500MHz machine.

Below we list the 339 values of n<10⁶ for which the curve has analytic rank at least 3, together with their prime factorizations. (For the conjectural extension of this table to 10⁷, go here.) We also give the numerator and denominator of the x-coordinate of a nontrivial rational solution of ny²=x³-x. In all 339 cases, 2-descent on E(n) and/or on a curve 2-isogenous with E(n) shows that the arithmetic rank is at most 3. In all but 7 cases, the 2-part of the Tate-Shafarevich group SHA of E(n) is trivial if E(n) has arithmetic rank 3 as expected. The 7 exceptional cases are:
n = 68839 = 23 * 41 * 73
n = 450079 = 7 * 113 * 569
n = 473263 = 7 * 17 * 41 * 97
n = 488047 = 7 * 113 * 617
n = 606599 = 7 * 193 * 449
n = 900271 = 31 * 113 * 257
n = 990647 = 7 * 137 * 1033
In each of these cases, MWRANK confirms arithmetic rank 3 (by 2-descent on an isogenous curve), but finds a subgroup of SHA isomorphic with (Z/2Z)².