Oscillations and first-ever negative Fourier coefficients of symmetric square L-functions over sparse set
Srinivas Kotyada, Lalit Vaishya
TL;DR
The paper addresses the first-mole moment and sign changes of the Fourier coefficients of the symmetric-square lift $\lambda_{\mathrm{sym}^{2}f}(n)$ on integers represented by reduced binary quadratic forms of a fixed negative discriminant $D$. It develops a Dirichlet-series framework and a decomposition $L(sym^{2}f, D; s)=L(sym^{2}f, s)L(sym^{2}f\otimes \chi_{D}, s)G(s)$, together with smoothing, contour-shifting, and a lower-bound via an auxiliary multiplicative function to derive sharp upper bounds and positive lower bounds for the relevant sums. The main results include an upper bound $S(sym^{2}f, D; X) \ll (N^{2}k^{2}|D|^{3/2})^{1/3+\varepsilon} X^{2/3+\varepsilon}$ and a subconvex-type estimate for the first negative index $n_{sym^{2}f, D}$, namely $n_{sym^{2}f, D} \ll (N^{2}k^{2}|D|^{3/2})^{0.61222+\varepsilon} L(1,\chi_{D})^{-5.51}$ (with refinements depending on the class number $h(D)$). These results contribute to understanding Fourier-coefficient behavior on sparse sets and advance subconvexity-type bounds in the GL(3)×GL(2) setting via symmetric-square $L$-functions. The techniques intertwine mean-value results for representations by discriminant-based forms with analytic properties of twisted and untwisted symmetric-square $L$-functions.
Abstract
Let $sym^{2} f$ denote the symmetric square lift of a Hecke eigenform $f \in S_{k}(Γ_{0}(N))$ with the $n^{\rm th}$-Fourier coefficients $ λ_{sym^{2}f}(n)$. In this article, we prove an estimate for the first moment of the sequence $\{ λ_{sym^{2}f}(\mathcal{Q}(\underline{x}))\}_{\mathcal{Q} \in \mathcal{S}_{D}, \underline{x} \in \mathbb{Z}^{2}}$ where $\mathcal{S}_{D}$ denotes the set of in-equivalent reduced forms of the discriminant $D$. More precisely, we establish an estimate for the following sum: \begin{equation*} \begin{split} S(sym^{2}f, D; X ) &= \sideset{}{^{\flat }}\sum_{\substack{\mathcal{Q}(\underline{x}) \leq X \\ \underline{x} \in \mathbb{Z}^{2} ,~ \mathcal{Q} \in \mathcal{S}_{D} \\ \gcd(\mathcal{Q}(\underline{x}),N) =1 }} λ_{sym^{2}f}(\mathcal{Q}(\underline{x})), \end{split} \end{equation*} Moreover, we consider a question concerning the behavior of signs of the Fourier coefficients $λ_{sym^{2}f}(n),$ supported on the set of integers represented by reduced forms of the discriminant $D$. We determine the size of $n_{sym^{2}f, D}$ (see definition before \thmref{ExtMatKLSW}), in terms of the conductor of the associated $L$-functions.