## L-series with nonzero central critical value

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- by Kevin James PDF
- J. Amer. Math. Soc.
**11**(1998), 635-641 Request permission

## Abstract:

Given a cusp form $f$ of even integral weight and its associated $L$-function $L(f,s)$, we expect that a positive proportion of the quadratic twists of $L$ will have nonzero central critical value. In this paper we give examples of weight two newforms whose associated $L$-functions have the property that a positive proportion of its quadratic twists have nonzero central critical value.## References

- A. O. L. Atkin and J. Lehner,
*Hecke operators on $\Gamma _{0}(m)$*, Math. Ann.**185**(1970), 134–160. MR**268123**, DOI 10.1007/BF01359701 - 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 - 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**, DOI 10.1007/BF01233440 - H. Davenport and H. Heilbronn,
*On the density of discriminants of cubic fields. II*, Proc. Roy. Soc. London Ser. A**322**(1971), no. 1551, 405–420. MR**491593**, DOI 10.1098/rspa.1971.0075 - G. Frey,
*On the Selmer group of twists of elliptic curves with $\textbf {Q}$-rational torsion points*, Canad. J. Math.**40**(1988), no. 3, 649–665. MR**960600**, DOI 10.4153/CJM-1988-028-9 - Solomon Friedberg and Jeffrey Hoffstein,
*Nonvanishing theorems for automorphic $L$-functions on $\textrm {GL}(2)$*, Ann. of Math. (2)**142**(1995), no. 2, 385–423. MR**1343325**, DOI 10.2307/2118638 - Dorian Goldfeld,
*Conjectures on elliptic curves over quadratic fields*, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 108–118. MR**564926** - D. R. Heath-Brown,
*The size of Selmer groups for the congruent number problem*, Invent. Math.**111**(1993), no. 1, 171–195. MR**1193603**, DOI 10.1007/BF01231285 - D. R. Heath-Brown,
*The size of Selmer groups for the congruent number problem. II*, Invent. Math.**118**(1994), no. 2, 331–370. With an appendix by P. Monsky. MR**1292115**, DOI 10.1007/BF01231536 - 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**, DOI 10.5802/jtnb.33 - K. James,
*An example of an elliptic curve with a positive density of prime quadratic twists which have rank zero*, Proceedings of Topics in Number Theory (G. Andrews and K. Ono, eds.), Kluwer, to appear. - Charles Hopkins,
*Rings with minimal condition for left ideals*, Ann. of Math. (2)**40**(1939), 712–730. MR**12**, DOI 10.2307/1968951 - V. A. Kolyvagin,
*Finiteness of $E(\textbf {Q})$ and SH$(E,\textbf {Q})$ for a subclass of Weil curves*, Izv. Akad. Nauk SSSR Ser. Mat.**52**(1988), no. 3, 522–540, 670–671 (Russian); English transl., Math. USSR-Izv.**32**(1989), no. 3, 523–541. MR**954295**, DOI 10.1070/IM1989v032n03ABEH000779 - J. Larry Lehman,
*Levels of positive definite ternary quadratic forms*, Math. Comp.**58**(1992), no. 197, 399–417, S17–S22. MR**1106974**, DOI 10.1090/S0025-5718-1992-1106974-1 - Wen Ch’ing Winnie Li,
*Newforms and functional equations*, Math. Ann.**212**(1975), 285–315. MR**369263**, DOI 10.1007/BF01344466 - 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**, DOI 10.2307/2118541 - M. Ram Murty and V. Kumar Murty,
*Mean values of derivatives of modular $L$-series*, Ann. of Math. (2)**133**(1991), no. 3, 447–475. MR**1109350**, DOI 10.2307/2944316 - Jin Nakagawa and Kuniaki Horie,
*Elliptic curves with no rational points*, Proc. Amer. Math. Soc.**104**(1988), no. 1, 20–24. MR**958035**, DOI 10.1090/S0002-9939-1988-0958035-0 - Ken Ono,
*Rank zero quadratic twists of modular elliptic curves*, Compositio Math.**104**(1996), no. 3, 293–304. MR**1424558** - —,
*Twists of elliptic curves*, Compositio Math.**106**(1997), 349–360. - K. Ono and C. Skinner,
*Fourier coefficients of half-integral weight modular forms modulo $\ell$*, Ann. of Math. (to appear). - —,
*Non-vanishing of quadratic twists of modular $L$-functions*, Invent. Math. (to appear). - B. Schoeneberg,
*Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen*, Math. Ann.**116**(1939). - Goro Shimura,
*On modular forms of half integral weight*, Ann. of Math. (2)**97**(1973), 440–481. MR**332663**, DOI 10.2307/1970831 - Carl Ludwig Siegel,
*Gesammelte Abhandlungen. Bände I, II, III*, Springer-Verlag, Berlin-New York, 1966 (German). Herausgegeben von K. Chandrasekharan und H. Maass. MR**0197270** - 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** - S. Wong,
*Rank zero twists of elliptic curves*, preprint. - Gang Yu,
*Quadratic twists of a given elliptic curve over ${\mathbb {Q}}$*, preprint.

## Additional Information

**Kevin James**- Affiliation: Department of Mathematics, Pennsylvania State University, 218 McAllister Building, University Park, Pennsylvania 16802-6401
- MR Author ID: 629241
- Email: klj@math.psu.edu
- Received by editor(s): August 13, 1997
- Received by editor(s) in revised form: January 20, 1998
- © Copyright 1998 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**11**(1998), 635-641 - MSC (1991): Primary 11G40
- DOI: https://doi.org/10.1090/S0894-0347-98-00263-X
- MathSciNet review: 1492854