A nonvanishing theorem for derivatives of automorphic $L$-functions with applications to elliptic curves
HTML articles powered by AMS MathViewer
- by Daniel Bump, Solomon Friedberg and Jeffrey Hoffstein PDF
- Bull. Amer. Math. Soc. 21 (1989), 89-93
References
- Don Blasius, Laurent Clozel, and Dinakar Ramakrishnan, Algébricité de l’action des opérateurs de Hecke sur certaines formes de Maass, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 16, 705–708 (French, with English summary). MR 920047
- Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic $L$-functions and their derivatives, Ann. of Math. (2) 131 (1990), no. 1, 53–127. MR 1038358, DOI 10.2307/1971508
- J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), no. 3, 223–251. MR 463176, DOI 10.1007/BF01402975
- Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
- Dorian Goldfeld, Gauss’s class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 1, 23–37. MR 788386, DOI 10.1090/S0273-0979-1985-15352-2
- Dorian Goldfeld and Jeffrey Hoffstein, Eisenstein series of ${1\over 2}$-integral weight and the mean value of real Dirichlet $L$-series, Invent. Math. 80 (1985), no. 2, 185–208. MR 788407, DOI 10.1007/BF01388603
- D. Goldfeld, J. Hoffstein, and S. J. Patterson, On automorphic functions of half-integral weight with applications to elliptic curves, Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981), Progr. Math., vol. 26, Birkhäuser, Boston, Mass., 1982, pp. 153–193. MR 685295
- Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
- R. Howe, $\theta$-series and invariant theory, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 275–285. MR 546602 10. H. Jacquet, J. Shalika and I. Piatetski-Shapiro, in preparation. 11. V. Kolyvagin, On groups of Mordell-Weil and Shafarevich-Tate and Weil elliptic curves, Preprint, (1988) (Russian).
- Mark E. Novodvorsky, Automorphic $L$-functions for symplectic group $\textrm {GSp}(4)$, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 87–95. MR 546610
- David E. Rohrlich, Nonvanishing of $L$-functions for $\textrm {GL}(2)$, Invent. Math. 97 (1989), no. 2, 381–403. MR 1001846, DOI 10.1007/BF01389047
- Goro Shimura, On the periods of modular forms, Math. Ann. 229 (1977), no. 3, 211–221. MR 463119, DOI 10.1007/BF01391466
- J. B. Tunnell, A classical Diophantine problem and modular forms of weight $3/2$, Invent. Math. 72 (1983), no. 2, 323–334. MR 700775, DOI 10.1007/BF01389327
- J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), no. 1, 1–132 (French). MR 577010
- J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (French). MR 646366
Additional Information
- Journal: Bull. Amer. Math. Soc. 21 (1989), 89-93
- MSC (1985): Primary 11G40; Secondary 11F11, 11F46, 11F67
- DOI: https://doi.org/10.1090/S0273-0979-1989-15771-6
- MathSciNet review: 983456