A note on the cancellations of sums involving Hecke eigenvalues

Abstract

Let f and g be distinct primitive holomorphic cusp forms of even integral weights $k_{1}$ and $k_{2}$ for the full modular group $\Gamma =SL(2,{\mathbb {Z}})$, respectively. Denote by $\lambda _{f}(n)$ and $\lambda _{g}(n)$ the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we consider short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by the multiplicative function $\lambda _{f}(n^{i})\lambda _{f}(n^{j})$ and $\lambda _{f}(n^{i})\lambda _{g}(n^{j})$ for any integers $i,j\ge 1$. We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least $q^{\frac{1}{2}+\varepsilon }$ for arbitrarily small $\varepsilon >0$. In the similar manner, We also establish the nontrivial bounds for short sums of isotypic trace functions twisted by the coefficients $\lambda _{f\times f\times f}(n)$ and $\lambda _{f\times f\times g}(n)$ of triple product L-functions $L(f\times f\times f,s)$ and $L(f\times f\times g,s)$, respectively.