A trace formula for convolution of Hecke series and its applications

V. A. Bykovskii

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 13, Tome 226 (1996), pp. 14-36

Résumé

A trace formula expressing the mean values of the form ($k=2,3,\dots$) $$ \frac{\Gamma(2k-1)}{(4\pi)^{2k-1}}\sum_f\frac{\lambda_f(d)}{\langle f,f\rangle}\mathcal H_f^{(\chi)}(s_1)\overline{\mathcal H_f^{(\chi)}(\overline s_2)} $$ via certain arithmetic means on the group $\Gamma_0(N_1)$ 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 $\Gamma_0(N_1)$. By $\mathcal H_f^{(\chi)}(s)$ we denote the Hecke series of the form $f$, twisted with the primitive character $\chi\pmod{N_2}$, and $\lambda_f(d)$, $(d,N_1,N_2)=1$, are the eigenvalues of the Hecke operators $$ T_{2k}(d)f(z)=d^{k-1/2}\sum_{d_1d_2=d}d^{-2k}_2\cdot\sum_{m\,(\operatorname{mod}d_2)}f\Biggl(\frac{d_1z+m}{d_2}\Biggr). $$ The trace formula is used for obtaining the estimate $$ \frac{d^l}{dt^l}\mathcal H_f^{(\chi)}(1/2+it)\ll_{\varepsilon,k,l,N_1}(1+|t|)^{1/2+\varepsilon}N_2^{1/2-1/8+\varepsilon} $$ for the newform $f$ for all $\varepsilon>0$, $l=0,1,2,\dots$. This improves the known result (Duke–Friedlander–Iwaniec, 1993) with upper bound $$ (1+|t|)^2N_2^{1/2-1/22+\varepsilon} $$ on the right-hand side. As a corollary, we obtain the estimate $$ c(n)\ll_\varepsilon h^{1/4-1/16+\varepsilon} $$ for the Fourier coefficients of holomorphic cusp forms of weight $k+1/2$, which improves Iwaniec' result (1987) with exponent $1/4-1/28+\varepsilon$. Bibl. 25 titles.