Euler's concordant forms

• Autorzy: Ken Ono
• Języki publikacji: EN

Słowa kluczowe

EN

Euler's concordant forms problem

Kategorie tematyczne

• 11P83: Partitions; congruences and congruential restrictions
• 05A17: Partitions of integers

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1996-1997
• Tom: 78
• Numer: 2
• Strony: 101-123

Daty

wydano

1996

otrzymano

1995-10-16

poprawiono

1996-03-11

Twórcy

autor

Ken Ono

• School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, U.S.A.
• Department of Mathematics, Penn State University, University Park, Pennsylvania 16802, U.S.A.

Bibliografia

• [1] E. T. Bell, The problems of congruent numbers and concordant forms, Proc. Amer. Acad. Sci. 33 (1947), 326-328.
• [2] M. Bennett, private communication.
• [3] D. Bump, S. Friedberg and J. Hoffstein, Nonvanishing theorems for L-functions of modular forms and their derivatives, Invent. Math. 102 (1990), 543-618.
• [4] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), 223-251.
• [5] F. Diamond and K. Kramer, Modularity of a family of elliptic curves, Math. Res. Lett. 2 (3) (1995), 299-304.
• [6] L. Euler, De binis formulis speciei xx+myy et xx+nyy inter se concordibus et disconcordibus, Opera Omnia Series 1 vol. 5 (1780), 48-60, Leipzig-Berlin-Zürich, 1944.
• [7] D. Husemöller, Elliptic Curves, Springer, New York, 1987.
• [8] A. Knapp, Elliptic Curves, Princeton Univ. Press, 1992.
• [9] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer, 1984.
• [10] V. A. Kolyvagin, Finiteness of E(ℚ) and the Tate-Shafarevich group of E(ℚ) for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 522-540 (in Russian).
• [11] J. Lehman, Levels of positive definite ternary quadratic forms, Math. Comp. 58, 197 (1992), 399-417.
• [12] L. Mai and M. R. Murty, A note on quadratric twists of an elliptic curve, in: Elliptic Curves and Related Topics, CRM Proc. Lecture Notes, Amer. Math. Soc., 1994, 121-124.
• [13] D. Masser and J. Rickert, Simultaneous Pell equations, J. Number Theory, to appear.
• [14] M. R. Murty and V. K. Murty, Mean values of derivatives of modular L-series, Ann. of Math. 133 (1991), 447-475.
• [15] K. Ono, Rank zero quadratic twists of modular elliptic curves, Compositio Math., to appear.
• [16] K. Ono, Twists of elliptic curves, Compositio Math., to appear.
• [17] T. Ono, Variations on a Theme of Euler, Plenum, New York, 1994.
• [18] J. Rickert, Simultaneous rational approximations and related Diophantine equations, Math. Proc. Cambridge Philos. Soc. 113 (1993), 461-472.
• [19] H. P. Schlickewei, The number of subspaces occurring in the p-adic subspace theorem in Diophantine approximation, J. Reine Angew. Math. 406 (1990), 44-108.
• [20] W. M. Schmidt, Norm form equations, Ann. of Math. 96 (1972), 526-551.
• [21] W. M. Schmidt, Diophantine Approximations, Lecture Notes in Math. 785, Springer, 1980.
• [22] B. Schoeneberg, Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen, Math. Ann. 116 (1939), 511-523.
• [23] J.-P. Serre, Divisibilité de certaines fonctions arithmétiques, Enseign. Math. 22 (1976), 227-260.
• [24] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971.
• [25] G. Shimura, On modular forms of half-integral weight, Ann. of Math. 97 (1973), 440-481.
• [26] C. Siegel, Über die analytische Theorie der quadratischen formen, in: Gesammelte Abhandlungen, Bd. 3, Springer, 1966, 326-405.
• [27] J. Silverman, The Arithmetic of Elliptic Curves, Springer, New York, 1986.
• [28] J. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer, New York, 1992.
• [29] J. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math. 72 (1983), 323-334.
• [30] J. L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. 60 (1981), 375-484.
• [31] A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. 141 (1995), 443-551.