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A trace formula for the convolution of Hecke series and its applications. (Russian) Zbl 0893.11019

Kuz’mina, G. V. (ed.) et al., Analytical number theory and function theory. 13. Work collection. Dedicated to the 80th birthday of Yu. V. Linnik. Sankt-Peterburg: Nauka. Zap. Nauchn. Semin. POMI. 226, 14-36 (1996).
Let \(f\) be a newform of weight \(2k\) for the congruence subgroup \(\Gamma_0(N_1)\), \(\chi\) a primitive character \(\pmod{N_2}\) with \((N_1,N_2)=1\), and \(H_f^{(\chi)}(s)\) the twist with \(\chi\) of the Hecke \(L\)-function attached to \(f\). By a suitable normalization, the line \(\sigma=1/2\) plays the role of the critical line for \(H_f^{(\chi)}(s)\), and for \(s=1/2+it\), estimates of the form \(\ll(1+| t|)^{1/2+\varepsilon}N_2^{1/2-\delta+\varepsilon}\) for this function are known. Here \(\delta\geq 0\) is a constant, and the implied constant depends on \(k\), \(N_1\), and \(\varepsilon>0\). The “trivial” estimate with \(\delta=0\) follows from the functional equation by the convexity principle. This was sharpened to \(\delta=1/22\) by W. Duke, J. Friedlander and H. Iwaniec [Invent. Math. 112, 1–8 (1993; Zbl 0765.11038)]. The author obtains a further improvement to \(\delta=1/8\); the result is stated, more generally, for derivatives of \(H_f^{(\chi)}(s)\). The key result in its proof is a trace formula for the sum \[ \sum_f H_f^{(\chi)}(u+1/2-v)\overline H_f^{(\chi)}(u+v-1/2)\lambda_f(d)(f,f)^{-1}, \] where \(f\) runs over a basis of cusp forms for given weight and level, \(\lambda_f(d)\) stands for (suitably normalized) Fourier coefficients of \(f\), and \((\cdot,\cdot)\) denotes the Petersson inner product. The author announces without proof the more uniform estimate \(\ll(1+| t|)^{1/2+\varepsilon}N_1^{1/3+\varepsilon}N_2^{5/12+\varepsilon}\), where the implied constant is now independent of \(N_1\).
For the entire collection see [Zbl 0868.00016].
Reviewer: M.Jutila (Turku)

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11M41 Other Dirichlet series and zeta functions

Citations:

Zbl 0765.11038