On θ-Congruent Numbers, Rational Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves

Abstract

Abstract:The correspondence between right triangles with rational sides, triplets of rational squares in arithmetic succession and integral solutions of certain quadratic forms is well-known. We show how this correspondence can be extended to the generalized notions of rational θ-triangles, rational squares occurring in arithmetic progressions and concordant forms. In our approach we establish one-to-one mappings to rational points on certain elliptic curves and examine in detail the role of solutions of the θ-congruent number problem and the concordant form problem associated with nontrivial torsion points on the corresponding elliptic curves. This approach allows us to combine and extend some disjoint results obtained by a number of authors, to clarify some statements in the literature and to answer some hitherto open questions.

Four rational squares in arithmetic progressions and a family of elliptic curves with positive Mordell-Weil rank

“…Any of the curves E r;s is isomorphic to the intersection of the two quadrics in projective three-space given by X 2 0 C rX 2 1 D X 2 2 and X 2 0 C sX 2 1 D X 2 3 (see [37]) and hence is intimately related to Euler's concordant form problem described before. Our results give additional information for the special cases which correspond to the curves E k;`;m .…”

“…As was pointed out (and remedied) in [37], in both cases the mappings chosen are not isomorphisms between algebraic varieties, but only mappings of degree 4, which causes a loss of information on effects corresponding to torsion points on the elliptic curve in question. In [23], we gave an explicit (and lavishly illustrated) general construction of a rationally defined isomorphism between a rationally defined smooth intersection of two quadrics in projective threespace and an elliptic curve in Weierstraß form which maps a distinguished rational point in the intersection of the two quadrics to the point at infinity of the elliptic curve.…”

“…Clearly, a number n is congruent if and only if n and n are concordant; hence Euler's problem generalizes the congruent number problem. One can readily check (see [37]) that there is a 1-1 correspondence between the solutions of the Â-congruent number problem and Euler's concordant form problem, but since the geometric formulation of the problem is somewhat contrived we do not go into any details in this direction, but rather point out how Euler's problem gives rise to a more general problem in a rather natural way.…”

“…To carry out this line of reasoning, we need to identify the torsion points of an elliptic curve of the type considered, which can be easily done. Proof See [32] and also [37] where, however, a different sign convention was used.…”

“…The curve E 2;3;5 D E10,40 with the equation y 2 D x.x C 10/.x C 40/ is isomorphic to the curve E 10,30 with the equation y 2 D x.x 10/.x C 30/ via the translation x 7 ! x 10, and this latter curve is isomorphic to the intersection of the two quadrics given by X via the isomorphism presented in[37]. The 2-torsion points are mapped to the K trivial solutions .1; 0;˙1;˙1/, the point P D .10; 100/ is mapped to the solution .…”