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A classical Diophantine problem and modular forms of weight 3/2. (English) Zbl 0515.10013

MSC:
11B39Fibonacci and Lucas numbers, etc.
11F27Theta series; Weil representation; theta correspondences
14H52Elliptic curves
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
[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(-p) , forp?1 (mod 8) a prime. Proc. Amer. Math. Soc.31, 381-383 (1972)
[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)
[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
[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
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[19]Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Iwanami Shoten and Princeton University Press 1971
[20]Smith, H.J.: Collected Mathematical Papers, Volume 1, Oxford (1894). (reprinted by Chelsea, 1965)
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[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
[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)
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