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A classical Diophantine problem and modular forms of weight $3/2$. (English) Zbl 0515.10013
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 $a(n)\ne 0$, then $n$ is not the area of any right triangle with rational sides. (b) If $b(n)\ne 0$, then $2n$ is not the area of any right triangle with rational sides.” The integers $D$ contained in the set $\cal C$ of areas of right triangles with rational sides are the classical congruent numbers. From the Pythagorean theorem it follows that $D\in\cal C$ iff the group of rational points $E^D(\Bbb Q)$ 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 $\Bbb Q$ and the finiteness of the group $E(\Bbb Q)$ (see {\it B. J. Birch} and {\it H. P. F. Swinnerton-Dyer} [J. Reine Angew. Math. 218, 79--108 (1965; Zbl 0147.02506)]). Several recent results (cf. {\it J. Coates} and {\it A. Wiles} [Invent. Math. 39, 223--251 (1977; Zbl 0359.14009)] and {\it 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 $E^D(\Bbb Q)$. 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.

##### MSC:
 11D25 Cubic and quartic diophantine equations 11F37 Forms of half-integer weight, etc. 11G40 $L$-functions of varieties over global fields 14G10 Zeta-functions and related questions
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##### References:
 [1] Alter, R., Curtz, T.B., Kubota, K.K.: Remarks and results on congruent numbers. Proc. Third Southeastern Conf. on Combinatorics, Graph Theory and Computing 1972, pp. 27-35 · Zbl 0259.10010 [2] Alter, R.: The congruent number problem. Amer. Math. Monthly87, 43-45 (1980) · Zbl 0422.10009 · doi:10.2307/2320381 [3] Birch, B.J., Swinnerton-Dyer, H.P.F.: Notes on elliptic curves II. J. reine angewandte Math.218, 79-108 (1965) · Zbl 0147.02506 · doi:10.1515/crll.1965.218.79 [4] Birch, B.J., Kuyk, W.: Tables on elliptic curves. In: Modular functions of one variable IV. Lecture Notes in Mathematics, vol. 476, pp. 81-144. Berlin-Heidelberg-New York: Springer 1979 [5] Barrucand, P., Cohn, H.: Note on primes of typex 2+32y 2, class number, and residuacity. J. reine angewandte Math.238, 67-70 (1969) · Zbl 0207.36202 · doi:10.1515/crll.1969.238.67 [6] Brown, E.: The class number of $Q(\sqrt { - p} )$ , forp?1 (mod 8) a prime. Proc. Amer. Math. Soc.31, 381-383 (1972) · Zbl 0207.05401 [7] Coates, J., Wiles, A.: On the conjecture of Birch and Swinnerton-Dyer. Invent. Math.39, 223-251 (1977) · Zbl 0359.14009 · doi:10.1007/BF01402975 [8] Cohen, H., Oesterlé, J.: Dimension des espaces de formes modulaires. In: Modular functions of one variable VI. Lecture Notes in Mathematics, vol. 627, pp. 69-78. Berlin-Heidelberg-New York: Springer 1977 [9] Dickson, L.E.: History of the theory of numbers II. Carnegie Institution, Washington, DC (1920) (reprinted by Chelsea, 1966) · Zbl 47.0888.08 [10] Flicker, Y.: Automorphic forms on covering groups ofGL(2). Invent. Math.57, 119-182 (1980) · Zbl 0431.10014 · doi:10.1007/BF01390092 [11] Jones, B.W.: The arithmetic theory of quadratic forms. Math. Assoc. of Amer., Baltimore, MD 1950 · Zbl 0041.17505 [12] Lagrange, J.: Thèse d’Etat de l’Université de Reims, 1976 [13] Moreno, C.J.: The higher reciprocity laws: an example. J. Number Theory12, 57-70 (1980) · Zbl 0426.10024 · doi:10.1016/0022-314X(80)90073-6 [14] Pizer, A.: On the 2-part of the class number of imaginary quadratic number fields. J. Number Theory8, 184-192 (1976) · Zbl 0329.12003 · doi:10.1016/0022-314X(76)90100-1 [15] Razar, M.: The nonvanishing ofL(1) for certain elliptic curves with no first descents. Amer. J. Math.96, 104-126 (1974) · Zbl 0296.14015 · doi:10.2307/2373583 [16] Razar, M.: A relation between the two-component of the Tate-Shafarevitch group andL(1) for certain elliptic curves. Amer. J. Math.96, 127-144 (1974) · Zbl 0296.14016 · doi:10.2307/2373584 [17] Serre, J-P., Stark, H.M.: Modular forms of weight 1/2. In: Modular functions of one variable VI. Lecture Notes in Mathematics, vol. 627, pp. 27-68. Berlin-Heidelberg-New York: Springer 1977 [18] Shimura, G.: On modular forms of half-integral weight. Ann. of Math.97, 440-481 (1973) · Zbl 0266.10022 · doi:10.2307/1970831 [19] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Iwanami Shoten and Princeton University Press 1971 · Zbl 0221.10029 [20] Smith, H.J.: Collected Mathematical Papers, Volume 1, Oxford (1894). (reprinted by Chelsea, 1965) · Zbl 25.0029.02 [21] Stephens, N.M.: Congruence Properties of Congruent numbers. Bull. London Math. Soc. pp. 182-184 (1975) · Zbl 0304.10011 [22] Tate, J.: The arithmetic of elliptic curves. Invent. Math.23, 179-206 (1974) · Zbl 0296.14018 · doi:10.1007/BF01389745 [23] Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In: Modular functions of one variable IV. Lecture Notes in Mathematics, vol. 476, pp. 33-52. Berlin-Heidelberg-New York: Springer 1975 · Zbl 1214.14020 [24] Tate, J.: Number theoretic background. In: Automorphic forms, representations, andL-functions. Proc. Symp. in Pure Math. XXXIII, Part 2, pp. 3-26 (1979) [25] Waldspurger, J.-L.: Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. de Math. pures et appliquées60, (4) 375-484 (1981) · Zbl 0431.10015 [26] Guy, R.K.: Unsolved problems. Amer. Math. Monthly88, 758-761 (1981) · doi:10.2307/2321478