Abstract
For a given normalized newform f of large prime level, we establish a lower bound with respect to the level for the number of normalized newforms g of the same weight and level as of f such that the central L-values of f and g both twisted by a quadratic character do not vanish.
Acknowledgements
The authors would like to thank the anonymous referee for his comments which improved the presentation of the article.
References
[1]
A. Abbes and E. Ullmo,
Comparaison des métriques d’Arakelov et de Poincaré sur
[2] A. Akbary, Non-vanishing of weight k modular L-functions with large level, J. Ramanujan Math. Soc. 14 (1999), no. 1, 37–54. Search in Google Scholar
[3] P. Anamby and S. Das, Jacobi forms, Saito–Kurokawa lifts, their Pullbacks and sup-norms on average, Res. Math. Sci. 10 (2023), no. 1, Paper No. 14. 10.1007/s40687-023-00377-zSearch in Google Scholar
[4]
A. O. L. Atkin and J. Lehner,
Hecke operators on
[5] S. Böcherer, M. Furusawa and R. Schulze-Pillot, On Whittaker coefficients of some metaplectic forms, Duke Math. J. 76 (1994), no. 3, 761–772. 10.1215/S0012-7094-94-07631-XSearch in Google Scholar
[6] W. Duke, The critical order of vanishing of automorphic L-functions with large level, Invent. Math. 119 (1995), no. 1, 165–174. 10.1007/BF01245178Search in Google Scholar
[7] B. Gross, W. Kohnen and D. Zagier, Heegner points and derivatives of L-series. II, Math. Ann. 278 (1987), no. 1–4, 497–562. 10.1007/BF01458081Search in Google Scholar
[8] E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1959. Search in Google Scholar
[9] J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), no. 1, 161–181. 10.2307/2118543Search in Google Scholar
[10] H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math. 17, American Mathematical Society, Providence, 1997. 10.1090/gsm/017Search in Google Scholar
[11] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence, 2004. 10.1090/coll/053Search in Google Scholar
[12] H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families of L-functions, Publ. Math. Inst. Hautes Études Sci. 91 (2000), 55–131. 10.1007/BF02698741Search in Google Scholar
[13] H. Iwaniec and P. Sarnak, The non-vanishing of central values of automorphic L-functions and Landau–Siegel zeros, Israel J. Math. 120 (2000), 155–177. 10.1007/s11856-000-1275-9Search in Google Scholar
[14] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Grad. Texts in Math. 97, Springer, New York, 1984. 10.1007/978-1-4684-0255-1Search in Google Scholar
[15] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982), 32–72. 10.1515/crll.1982.333.32Search in Google Scholar
[16] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268. 10.1007/BF01455989Search in Google Scholar
[17] W. Kohnen, Estimates for Fourier coefficients of Siegel cusp forms of degree two, Compos. Math. 87 (1993), no. 2, 231–240. 10.1007/PL00004730Search in Google Scholar
[18] B. Kumar, Level aspect of simultaneous nonvanishing for the central L-values associated to cusp forms, Proc. Amer. Math. Soc. 148 (2020), no. 3, 979–991. 10.1090/proc/14773Search in Google Scholar
[19] W. Luo, On simultaneous nonvanishing of the central L-values, Proc. Amer. Math. Soc. 145 (2017), no. 10, 4227–4231. 10.1090/proc/13572Search in Google Scholar
[20] A. P. Ogg, On the eigenvalues of Hecke operators, Math. Ann. 179 (1969), 101–108. 10.1007/BF01350121Search in Google Scholar
[21] I. Petrow and M. P. Young, A generalized cubic moment and the Petersson formula for newforms, Math. Ann. 373 (2019), no. 1–2, 287–353. 10.1007/s00208-018-1745-1Search in Google Scholar
[22]
L. Yang,
Average of central L-values for
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
- Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
- Estimates of Picard modular cusp forms
- Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
- Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
- Free groups generated by two unipotent maps
- Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
- Colored multizeta values in positive characteristic
- Simultaneous nonvanishing of central L-values with large level
- Laplace convolutions of weighted averages of arithmetical functions
- Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
- On the geometric trace of a generalized Selberg trace formula
- Elementary properties of free lattices
- Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
- Small generators of abelian number fields
- Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
- Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
Articles in the same Issue
- Frontmatter
- Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
- Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
- Estimates of Picard modular cusp forms
- Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
- Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
- Free groups generated by two unipotent maps
- Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
- Colored multizeta values in positive characteristic
- Simultaneous nonvanishing of central L-values with large level
- Laplace convolutions of weighted averages of arithmetical functions
- Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
- On the geometric trace of a generalized Selberg trace formula
- Elementary properties of free lattices
- Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
- Small generators of abelian number fields
- Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
- Weighted bilinear multiplier theorems in Dunkl setting via singular integrals