Congruence Properties of Congruent Numbers

Stephens, N. M.

doi:10.1112/blms/7.2.182

Cited by 35 publications

(22 citation statements)

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“…Numerical evidence from such calculations suggested to the authors of [1] that all positive integers congruent to 5, 6, or 7 modulo 8 should be congruent numbers. Stephens [21] observed that this would follow from a weak form of the conjecture of Birch and Swinnerton-Dyer and asserted that the method of Heegner points could be applied to prove that primes congruent to 5 or 7 modulo 8 or twice primes congruent to 3 modulo 8 are in fact the areas of rational right triangles. B.…”

Section: (N)+o Then 2n Is Not the Area Of Any Right Triangle With Ramentioning

confidence: 96%

A classical Diophantine problem and modular forms of weight 3/2

Tunnell

1983

Invent Math

178

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Section: (N)+o Then 2n Is Not the Area Of Any Right Triangle With Ramentioning

confidence: 96%

A classical Diophantine problem and modular forms of weight 3/2

Tunnell

1983

Invent Math

178

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“…Let r = l/m with l,meZ, (I, m) = 1. Then X = m 4 x, Y = m 6 y satisfy We have now only to invoke the result of Stephens [8] that, subject to the Selmer conjecture, the rank of (3.2) is odd whenever n = 5,6,7 (mod 8) is a positive integer.…”

Section: £Nec{st) = C(uv)mentioning

confidence: 99%

Selmer's Conjecture and Families of Elliptic Curves

Cassels

Schinzel

1982

Bulletin of the London Mathematical Society

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1. In this note we consider the elliptic curve d:y 2 = x { x 2 -( l + f 4 ) 2 } (1.1) over the field C(t) of rational functions with complex coefficients. It turns out that all C(0-r ational points on si satisfy x , y e QU/2, i, t). As a consequence, the curvehas only points of order 2 over Q(t). On the other hand, the famous conjecture of Selmer [7] is shown to imply that for every rational r the curve 28 r \y 2 = x{x 2 -(7 + 7r 4 ) 2 } (1.3) has infinitely many rational points.To place this phenomenon in a more general setting, let R = Q[t] and let F E R[x, y] be irreducible over the algebraic closure of Q{t). If x, y are regarded as coordinates of affine space, the equation F = 0 defines an affine curve over Q(t) and also a family of affine curves F(x, y, r) = 0 over Q, as r runs over the elements of Q such that F(x, y, r) is absolutely irreducible (the remaining values of r form a finite set £).Let # be the projective curve defined over Q(t) obtained from normalization of the completion of the curve F = 0 and let ( € r be the normalization of the completion of the affine curve F(x, y, r) = 0 for r e Q \ £ .If ( ti is a curve of genus 1, then for all but finitely many values r e Q \ £ the curve # r is also of genus 1. Without further comment we ignore the exceptional values of r. Let ^(Q(0) denote the set of Q(t)-rational points on # , and # r (Q) the set of Q-rational points on # r .If %>(Q(t)) and ^r(Q) are nonempty, they have natural structure as groups when one point is selected as origin. They are finitely generated by virtue, respectively, of the theorems of Neron ([6, Theoreme 3]) and of Mordell. Let g{^) (respectively gi^r)) be the reduced rank of #(Q(t)) (respectively # r (Q)). Neron has shown (ibid., Theoreme 6) that there are infinitely many r e Q such that g(^r) $s g{^). Our example shows that there can be strict inequality for every rational r.An analogous result where r is restricted to values in Z has been obtained by D.J. Lewis and A. Schinzel [4]. The curve # given by y 2 = x 4 -( 8 t 2 + 5) 2 has g(<%) = 0, but Selmer's Conjecture implies that g{%> r ) ^ 1 for every integer r. In this example, however, g{^r) = 0 for infinitely many rational r.

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“…Remark 1: The above result when k = 0 is due to Heegner (5), and completed by Birch (6), Stephens (7), and Monsky (8); and that when k = 1 is due to Monsky (8) and Gross (9). Actually Heegner is the first mathematician who found a method to construct fairly general solutions to cubic Diophantine equations (5).…”

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confidence: 99%

Congruent numbers with many prime factors

Tian

2012

Proc. Natl. Acad. Sci. U.S.A.

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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].

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Congruence Properties of Congruent Numbers

Cited by 35 publications

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A classical Diophantine problem and modular forms of weight 3/2

A classical Diophantine problem and modular forms of weight 3/2

Selmer's Conjecture and Families of Elliptic Curves

Congruent numbers with many prime factors

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