Abstract
Ever since Dirichlet’s introduction of the analytic class number formula, special values of L-functions have been the subject of much study and speculation. In this paper we survey some recent results about such values that were presented at this conference. Our attention is essentially restricted to the central values of L-functions associated to certain (holomorphic) newforms. These results have many applications to class numbers of imaginary quadratic fields, Selmer groups of elliptic curves, and K-groups of real quadratic fields, a few of which are included.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
J. H. Bruinier, Non-vanishing modulo pof Fourier coefficients of half-integral weight modular forms,(preprint).
H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields.II, Proc. Roy. Soc. London ser. A 322 (1971), 405–420.
S. Friedberg and J. Hoffstein, Nonvanishing theorems for automophic L-functions on L(2), Ann. Math. 142 (1995), 385–423.
G. Frey, On the Selmer group of twists of elliptic curves with ℚ-rational torsion points, Can. J. Math. XL (1988), 649–665.
D. Goldfeld, Conjectures on elliptic curves over quadratic fields, Number Theory, Carbondale, Springer Lect. Notes 751 (1979), 108–118.
K. Horie, Trace formulae and imaginary quadratic fields, Math. Ann. 288 (1990), 605–612.
K. James, L-series with non-zero central critical value, J. Amer. Math. Soc. 11 (1998), 635–641.
N. Jochnowitz, Congruences between modular forms of half-integral weights and implications for class numbers and elliptic curves,(preprint).
W. Kohnen, On the proportion of quadratic character twists of L-functions attached to cusp forms not vanishing at the central point, J. reine angew. math. (to appear).
W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 173–198.
B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Invent. Math. 76 (1984), 179–330.
J. Nakagawa and K. Horie, Elliptic curves with no rational points, Proc. AMS 104, no.1 (1988), 20–24.
K. Ono and C. Skinner, Nonvanishing of quadratic twists of modular L-functions, Invent. Math., (to appear).
K. Ono and C. Skinner, Fourier coefficients of half-integral weight modular forms modulo 1, Annals of Math. 147 (1998), 451–468.
G. Shimura, On modular forms of half integral weight, Annals of Math. 97, 440–481.
G. Stevens, The cuspidal group and special values of L-functions, Trans. A.M.S. 291, 519–550.
V. Vatsal, Canonical periods and congruence formulae,Duke Math. J. (to appear).
V. Vatsal, Rank-one twists of a cetain elliptic curve,Math. Annalen (to appear).
J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures et Appl. 60 (1981), 375–484.
J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), 219–307.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Kluwer Academic Publishers
About this chapter
Cite this chapter
Bruinier, J.H., James, K., Kohnen, W., Ono, K., Skinner, C., Vatsal, V. (1999). Congruence Properties of Values of L-Functions and Applications. In: Ahlgren, S.D., Andrews, G.E., Ono, K. (eds) Topics in Number Theory. Mathematics and Its Applications, vol 467. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0305-3_6
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0305-3_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7988-1
Online ISBN: 978-1-4613-0305-3
eBook Packages: Springer Book Archive