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A trace formula for the scalar product of Hecke series and its applications

Authors:

V A Bykovskii

Abstract:

A trace formula expressing the mean values of the form (k=2,3,...)

$\frac{{\Gamma (2k - 1)}}{{(4\pi )^{2k - 1} }}\sum\limits_f {\frac{{\lambda _f (d)}}{{\left\langle {f,f} \right\rangle }}H_f^{(x)} (s_1 )\frac{{}}{{H_f^{(x)} (\bar s_2 )}}}$

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 _f ^(x) (s) we denote the Hecke series of the form f, twisted with the primitive character χ (mod N₂), and λ_f(d), (d, N₁N₂)=1, are the eigenvalues of the Hecke operators

$T_{2k} (d)f(z) = d^{k - 1/2} \sum\limits_{d_1 d_2 = d} {d_2^{ - 2k} } \cdot \sum\limits_{m(\bmod d_2 )} {f\left( {\frac{{d_1 z + m}}{{d_2 }}} \right)}$

. The trace formula is used for obtaining the estimate

$\frac{{d^l }}{{dt^l }}H_f^{(x)} \left( {\frac{1}{2} + it} \right) \ll _{\varepsilon ,k,l,N_1 } (1 + \left| t \right|)^{1/2 + \varepsilon } N_2^{1/2 - 1/8 + \varepsilon }$

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 ₂ ^{1/2−1/22+ε} on the right-hand side. As a corollary, we obtain the estimate

$c(n) \ll _\varepsilon n^{1/4 - 1/16 + \varepsilon }$

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. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 14–36.

Keywords:

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