Abstract
A trace formula expressing the mean values of the form (k=2,3,...)
via certain arithmetic means on the group Г0(N1) is proved. Here the sum is taken over a normalized orthogonal basis in the space of holomorphic cusp forms of weight 2k with respect to Г0(N1). By H (x)f (s) we denote the Hecke series of the form f, twisted with the primitive character χ (mod N2), and λf(d), (d, N1N2)=1, are the eigenvalues of the Hecke operators
. The trace formula is used for obtaining the estimate
for the newform f for all ε>0, l=0,1,2,.... This improves the known result (Duke-Friedlander-Iwaniec, 1993) with upper bound (1+|t|)2N 1/2−1/22+ε2 on the right-hand side. As a corollary, we obtain the estimate
for the Fourier coefficients of holomorphic cusp forms of weight k+1/2, which improves Iwaniec' result (1987) with exponent 1/4–1/28+ε. Bibliography: 25 titles.
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Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 14–36.
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Bykovskii, V.A. A trace formula for the scalar product of Hecke series and its applications. J Math Sci 89, 915–932 (1998). https://doi.org/10.1007/BF02358528
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DOI: https://doi.org/10.1007/BF02358528