On the non-vanishing of cubic twists

of automorphic -series

Author:
Xiaotie She

Journal:
Trans. Amer. Math. Soc. **351** (1999), 1075-1094

MSC (1991):
Primary 11F66; Secondary 11F70, 11M41, 11N75

MathSciNet review:
1451616

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a normalised new form of weight for over and , its base change lift to . A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the -function of . There is an algorithm to check the condition for any given form. The new form of level is used to illustrate our method.

**1.**Tom M. Apostol,*Introduction to analytic number theory*, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR**0434929****2.**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**, 10.2307/1971508**3.**Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein,*Nonvanishing theorems for 𝐿-functions of modular forms and their derivatives*, Invent. Math.**102**(1990), no. 3, 543–618. MR**1074487**, 10.1007/BF01233440**4.**Daniel Bump and Jeffrey Hoffstein,*Cubic metaplectic forms on 𝐺𝐿(3)*, Invent. Math.**84**(1986), no. 3, 481–505. MR**837524**, 10.1007/BF01388743**5.**Solomon Friedberg and Jeffrey Hoffstein,*Nonvanishing theorems for automorphic 𝐿-functions on 𝐺𝐿(2)*, Ann. of Math. (2)**142**(1995), no. 2, 385–423. MR**1343325**, 10.2307/2118638**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. 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****8.**I. S. Gradshteyn and I. M. Ryzhik,*Table of integrals, series, and products*, 5th ed., Academic Press, Inc., Boston, MA, 1994. Translation edited and with a preface by Alan Jeffrey. MR**1243179****9.**Kenneth F. Ireland and Michael I. Rosen,*A classical introduction to modern number theory*, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, 1982. Revised edition of Elements of number theory. MR**661047****10.**Henryk Iwaniec,*On the order of vanishing of modular 𝐿-functions at the critical point*, Sém. Théor. Nombres Bordeaux (2)**2**(1990), no. 2, 365–376. MR**1081731****11.**Daniel B. Lieman,*Nonvanishing of 𝐿-series associated to cubic twists of elliptic curves*, Ann. of Math. (2)**140**(1994), no. 1, 81–108. MR**1289492**, 10.2307/2118541**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. II*, J. Reine Angew. Math.**296**(1977), 217–220. MR**0563069****14.**David E. Rohrlich,*Nonvanishing of 𝐿-functions for 𝐺𝐿(2)*, Invent. Math.**97**(1989), no. 2, 381–403. MR**1001846**, 10.1007/BF01389047**15.**Goro Shimura,*On the derivatives of theta functions and modular forms*, Duke Math. J.**44**(1977), no. 2, 365–387. MR**0466028****16.**Jean-Loup Waldspurger,*Correspondances de Shimura et quaternions*, Forum Math.**3**(1991), no. 3, 219–307 (French). MR**1103429**, 10.1515/form.1991.3.219

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