A classical Diophantine problem and modular forms of weight \(3/2\)
type: Item
instance of: scholarly article
title: A classical Diophantine problem and modular forms of weight \(3/2\)
zbMATH Open document ID: 0515.10013
DOI: 10.1007/BF01389327
published in: Inventiones Mathematicae
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143024
review text: 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. F…
reviewed by: Roelof J. Stroeker
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 3813716
zbMATH Keywords: area of right triangle with rational sides
zbMATH Keywords: congruent numbers
zbMATH Keywords: elliptic curve
zbMATH Keywords: finiteness of group of rational points
zbMATH Keywords: modular forms of weight 3/2
zbMATH Keywords: table of square-free non-congruent numbers
author: Jerrold B. Tunnell
MaRDI profile type: Publication
Wikidata QID: Q29398849
cites work: Q5675781
cites work: The Congruent Number Problem
cites work: Notes on elliptic curves. II.
cites work: Note on primes of type x2 + 32y2, class number, and residuacity.
cites work: The Class Number of Q(√- p), for p ≡1 (mod 8) A Prime
cites work: On the conjecture of Birch and Swinnerton-Dyer
cites work: Automorphic forms on covering groups of GL(2)
cites work: Q5801410
cites work: The higher reciprocity laws: An example
cites work: On the 2-part of the class number of imaginary quadratic number fields
cites work: The Non-Vanishing of L(1) for Certain Elliptic Curves with no First Descents
cites work: A Relation Between the Two-Component of the Tate-Safarevic Group and L(1) for Certain Elliptic Curves
cites work: On modular forms of half integral weight
cites work: Q5625265
cites work: Congruence Properties of Congruent Numbers
cites work: The arithmetic of elliptic curves
cites work: Algorithm for determining the type of a singular fiber in an elliptic pencil
cites work: Sur les coefficients de Fourier des formes modulaires de poids demi-entier
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description: scientific article; zbMATH DE number 3813716