A classical Diophantine problem and modular forms of weight \(3/2\)
From the introduction: ``It is a classical Diophantine problem to determine which integers are the area of some right triangle with rational sides. The main result of this paper is the following. Theorem. Let formal power series in the variable \(q\) be given by \(g=q\prod_1^\infty (1-q^{8n})(1-q^{16n})\) and, for each positive integer \(t\), \(\theta_t=\sum_{-\infty}^{\infty} q^{tn^2}\). Set \(g\theta_2=\sum_1^{\infty} a(n)q^n\) and \(g\theta_4=\sum_1^{\infty} b(n)q^n\). (a) If \(a(n)\neq 0\), then \(n\) is not the area of any right triangle with rational sides. (b) If \(b(n)\neq 0\), then \(2n\) is not the area of any right triangle with rational sides.'' The integers \(D\) contained in the set \(\mathcal C\) of areas of right triangles with rational sides are the classical congruent numbers. From the Pythagorean theorem it follows that \(D\in\mathcal C\) iff the group of rational points \(E^D(\mathbb Q)\) on the elliptic curve with Weierstrass model \(E^D: y^2=x^3-D^2x\) is infinite. This establishes a link with the famous conjecture of Birch and Swinnerton-Dyer on the equivalence of the nonvanishing of the \(L\)-series \(L_E(s)\) at \(s=1\) of the elliptic curve \(E\) over \(\mathbb Q\) and the finiteness of the group \(E(\mathbb Q)\) (see \textit{B. J. Birch} and \textit{H. P. F. Swinnerton-Dyer} [J. Reine Angew. Math. 218, 79--108 (1965; Zbl 0147.02506)]). Several recent results (cf. \textit{J. Coates} and \textit{A. Wiles} [Invent. Math. 39, 223--251 (1977; Zbl 0359.14009)] and \textit{J.-L. Waldspurger} [J. Math. Pures Appl. (9) 60, 375--484 (1981; Zbl 0431.10015)]) are used by the authors to show that the existence of modular forms of weight \(3/2\) (the power series \(g\theta_2\) and \(g\theta_4\) of the theorem are the \(q\)-expansions of such modular forms) with nonvanishing \(D\)th Fourier coefficient implies the finiteness of \(E^D(\mathbb Q)\). A table of square-free non-congruent numbers less than 1000 is added and it is conjectured to be complete. Finally, classical criteria for non-congruent numbersare proved with the same technique.
area of right triangle with rational sides
table of square-free non-congruent numbers
scientific article; zbMATH DE number 3813716
scientific article