Euler's concordant forms
Acta Arithmetica (1996)
- Volume: 78, Issue: 2, page 101-123
- ISSN: 0065-1036
Access Full Article
topHow to cite
topKen Ono. "Euler's concordant forms." Acta Arithmetica 78.2 (1996): 101-123. <http://eudml.org/doc/206936>.
@article{KenOno1996,
author = {Ken Ono},
journal = {Acta Arithmetica},
keywords = {Euler's concordant forms problem; representations of integers; ternary quadratic forms; Euler's concordant forms; congruent number problem; elliptic curve; ranks of quadratic twists; pair of Pell equations; lacunary modular forms},
language = {eng},
number = {2},
pages = {101-123},
title = {Euler's concordant forms},
url = {http://eudml.org/doc/206936},
volume = {78},
year = {1996},
}
TY - JOUR
AU - Ken Ono
TI - Euler's concordant forms
JO - Acta Arithmetica
PY - 1996
VL - 78
IS - 2
SP - 101
EP - 123
LA - eng
KW - Euler's concordant forms problem; representations of integers; ternary quadratic forms; Euler's concordant forms; congruent number problem; elliptic curve; ranks of quadratic twists; pair of Pell equations; lacunary modular forms
UR - http://eudml.org/doc/206936
ER -
References
top- [1] E. T. Bell, The problems of congruent numbers and concordant forms, Proc. Amer. Acad. Sci. 33 (1947), 326-328. Zbl0029.11007
- [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. Zbl0721.11023
- [4] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), 223-251. Zbl0359.14009
- [5] F. Diamond and K. Kramer, Modularity of a family of elliptic curves, Math. Res. Lett. 2 (3) (1995), 299-304. Zbl0867.11041
- [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. Zbl0553.10019
- [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. Zbl0754.11011
- [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. Zbl0806.14025
- [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. Zbl0745.11032
- [15] K. Ono, Rank zero quadratic twists of modular elliptic curves, Compositio Math., to appear. Zbl0876.11025
- [16] K. Ono, Twists of elliptic curves, Compositio Math., to appear. Zbl0887.11025
- [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. Zbl0786.11040
- [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. Zbl0693.10027
- [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. Zbl0432.10029
- [22] B. Schoeneberg, Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen, Math. Ann. 116 (1939), 511-523. Zbl0020.20201
- [23] J.-P. Serre, Divisibilité de certaines fonctions arithmétiques, Enseign. Math. 22 (1976), 227-260. Zbl0355.10021
- [24] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971. Zbl0221.10029
- [25] G. Shimura, On modular forms of half-integral weight, Ann. of Math. 97 (1973), 440-481. Zbl0266.10022
- [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. Zbl0585.14026
- [28] J. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer, New York, 1992. Zbl0752.14034
- [29] J. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math. 72 (1983), 323-334. Zbl0515.10013
- [30] J. L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. 60 (1981), 375-484. Zbl0431.10015
- [31] A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. 141 (1995), 443-551. Zbl0823.11029
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.