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The critical order of vanishing of automorphic \(L\)-functions with large level. (English) Zbl 0838.11035
Let \(f\) be a cuspidal newform of weight 2 for the group \(\Gamma_0 (N)\). Then the twisted \(L\)-functions \(L_f (s, \chi)\) are of interest for many reasons, in particular one studies the order of vanishing at the center \(s=1\) of the critical strip of this \(L\)-function. The two main results of this article are: a) If \(\chi\) is a fixed Dirichlet character of conductor \(q\), then there is a constant \(C_q\) and an absolute constant \(C\) such that for \(N\) prime and \(> C_q\) there are at least \(CN \log^{-2} N\) newforms \(f\) for \(\Gamma_0 (N)\) such that \(L_f (1, \chi) \neq 0\). b) If \(\chi_1, \chi_2\) are distinct Dirichlet characters of conductors \(q_1, q_2\) then there are constants \(C_1\), \(C_2\) depending on \(q_1,q_2\) such that for \(N > C_1\) prime there are at least \(C_2 N \log^{-10} N\) newforms \(f\) for \(\Gamma_0 (N)\) for which \(L_f (s, \chi_1) L_f (s, \chi_2)\) has at most a simple zero at \(s = 1\). The conjectured truth in both cases is a positive proportion of \(N\).

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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[1] S. Böcherer, R. Schulze-Pillot: The Dirichlet series of Koecher and Maass and modular forms of weight 3/2. Math. Z.209 (1992) 273-287 · Zbl 0773.11031 · doi:10.1007/BF02570834
[2] J.-M. Deshouillers, H. Iwaniec: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math.70 (1982) 219-288 · Zbl 0502.10021 · doi:10.1007/BF01390728
[3] W. Duke, J. Friedlander, H. Iwaniec: Bounds for automorphicL-functions. II. Invent. Math.115 (1994) 219-239 · Zbl 0812.11032 · doi:10.1007/BF01231759
[4] D. Goldfeld, J. Hoffstein, D. Lieman: An effective zero free region, Appendix to: Coefficeints of Maass forms and the Siegel zero Ann. Math. (to appear)
[5] F. Gouvêa, B. Mazur: The square-free sieve and the rank of elliptic curves. J. AMS4 (1991) 1-23 · Zbl 0725.11027
[6] B.H. Gross: Heights and the special values ofL-series. In: Number Theory, Proceedings of the 1985 Montreal Conference held June 17-29, 1985, CMS Conference Proceedings, Vol. 7, 1987, 115-187
[7] J. Hoffstein, P. Lockhart: Coefficients of Maass forms and the Siegel zero. Ann. Math. (to appear) · Zbl 0814.11032
[8] H. Iwaniec: On the order of vanishing of modularL-functions at the critical point. In: Sém. Th. des Nombres, Bordeaux2 (1990) 365-376 · Zbl 0719.11029
[9] W. Luo: On the nonvanishing of Rankin SelbergL-functions. Duke Math. J69 (1993) 411-427 · Zbl 0789.11032 · doi:10.1215/S0012-7094-93-06918-9
[10] B. Mazur: Modular curves and the Eisenstein ideal. IHES Publ. Math.47 (1977) 33-186 · Zbl 0394.14008
[11] B. Mazur: On the arithmetic of special values ofL-functions. Invent. Math.55 (1979) 207-240 · Zbl 0426.14009 · doi:10.1007/BF01406841
[12] D.E. Rohrlich: OnL-functions of elliptic curves and cyclotomic towers. Invent. Math.75 (1984) 409-423 · Zbl 0565.14006 · doi:10.1007/BF01388636
[13] D.E. Rohrlich:L-functions and division towers. Math. Ann.281 (1988) 611-632 · Zbl 0656.14013 · doi:10.1007/BF01456842
[14] G. Shimura: Introduction to the arithmetic theory of automorphic functions. Publ. Math. Soc. Japan, Vol. 11. Tokyo-Princeton, 1971 · Zbl 0221.10029
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