On a hybrid bound for twisted \(L\)-values
Let \(f \in S_k(M, \psi)\) be a holomorphic newform of weight \(k\), level \(M\) and character \(\psi\). Let \(\chi\) be a primitive character of conductor \(q\). The paper proves the hybrid subconvex bound \[ L(1/2 + it, f \otimes \chi) \ll_{f} (q(1+|t|))^{\frac{1}{2} - \frac{1}{118} + \varepsilon}. \] The result is not new. A stronger result (also valid for general \(\psi\)) has been proved by \textit{G. Harcos} and the reviewer [J. Reine Angew. Math. 621, 53--79 (2008; Zbl 1193.11044); Addendum to appear in Crelle's journal], and the currently strongest and most general result on the market is due to \textit{Han Wu} [``Burgess-like subconvex bounds for \(\mathrm{GL}_2 \times mathrm{GL}_1\)'', \url{arXiv:1209.5950}]. Unlike the two aforementioned papers the present paper does not use families and amplification, but proves directly cancellation in the approximate functional equation. To this end the author writes \(f\) as a linear combination of Poincaré series (hence the proof works only for holomorphic forms), applies Poisson summation and the bilinear estimate for Kloosterman fractions of \textit{W. Duke, J. Friedlander} and \textit{H. Iwaniec} [Invent. Math. 128, No. 1, 23--43 (1997; Zbl 0873.11050)]. The author makes the interesting remark that the best possible bound for these bilinear forms would lead to a hybrid Weyl-type exponent.
scientific article; zbMATH DE number 5886763
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