Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the non-vanishing of cubic twists
of automorphic $L$-series


Author: Xiaotie She
Journal: Trans. Amer. Math. Soc. 351 (1999), 1075-1094
MSC (1991): Primary 11F66; Secondary 11F70, 11M41, 11N75
DOI: https://doi.org/10.1090/S0002-9947-99-02082-6
MathSciNet review: 1451616
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $f$ be a normalised new form of weight $2$ for $\Gamma _{0} (N)$ over ${\mathbb{Q}}$ and $F$, its base change lift to $\mathbb{Q}(\sqrt {-3})$. A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the $L$-function of $F$. There is an algorithm to check the condition for any given form. The new form of level $11$ is used to illustrate our method.


References [Enhancements On Off] (What's this?)

  • 1. Tom M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag (1976). MR 55:7892
  • 2. D. Bump, S. Friedberg and J. Hoffstein, Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives, Annals of Math. 131 53-127 (1990). MR 92e:11053
  • 3. D. Bump, S. Friedberg and J. Hoffstein, Nonvanishing theorems for L-functions of modular forms and their derivatives, Inventiones Math. 102 543-618 (1990). MR 92a:11058
  • 4. D. Bump and J. Hoffstein, Cubic metaplectic forms on GL(3), Inventiones Math. 84 481-505 (1986). MR 87i:11059
  • 5. S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic L-functions on GL(2), Annals of Math. 142 385-423 (1995). MR 96e:11072
  • 6. S. Friedberg, On the imaginary quadratic Doi-Naganuma lifting of modular forms of arbitrary level, Nagoya Math. J. 92 1-20 (1983). MR 85f:10031
  • 7. D. Goldfeld, J. Hoffstein and S. Patterson, On automorphic functions of half-integral weight with applications to elliptic curves, in Number Theory related to Fermat's Last Theorem (N. Koblitz, ed.), Birkhauser, Boston, Basel, Stuttgart, 153-193 (1983). MR 84i:10031
  • 8. I. Gradshteyn and I. Ryzhik, Tables of integral series and products, Fifth edition. MR 94g:00008
  • 9. K. Ireland and M. Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics, Springer-Verlag 84 (1982). MR 83g:12001
  • 10. H. Iwaniec, On the order of vanishing of modular L-functions at the critical point, Seminaire de Theorie des Nombres, Bordeaux 2 365-376 (1990). MR 92h:11040
  • 11. D. Lieman, Nonvanishing of L-series associated to cubic twists of elliptic curves, Annals of Math. 140 81-108 (1994). MR 95g:11044
  • 12. K. Murty and R. Murty, Mean values of derivatives of modular L-series, Annals of Math. 133 447-475 (1991).
  • 13. S.J. Patterson, A cubic analogue of the theta series, J. Reine Agnew Math. 296 125-161 (1977). MR 58:27795b
  • 14. D. Rohrlich, Nonvanishing of L-functions for GL(2), Inventiones Math. 97 381-403 (1989). MR 90g:11062
  • 15. G. Shimura, On the periods of modular forms, Math. Annalen 229 211-221 (1977). MR 57:5911
  • 16. J. Waldspurger, Correspondences de Shimura et quaternions, Forum Math 3 219-307 (1991). MR 92g:11054

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11F66, 11F70, 11M41, 11N75

Retrieve articles in all journals with MSC (1991): 11F66, 11F70, 11M41, 11N75


Additional Information

Xiaotie She
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Address at time of publication: Financial Data Planning Corp., 2140 S. Dixie Hwy., Miami, Florida 33133
Email: xiaoties@fdpcorp.com

DOI: https://doi.org/10.1090/S0002-9947-99-02082-6
Received by editor(s): September 27, 1996
Received by editor(s) in revised form: February 14, 1997
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society