Bounds for twisted symmetric square \(L\)-functions (Q2856655)

The paper is concerned with proving a bound of the type NEWLINE\[NEWLINE L\left( \frac 12 , Sym^2 f \otimes \chi \right) \ll_{f, \varepsilon} q^{\frac 34 \ell - \delta_\ell +\varepsilon} NEWLINE\]NEWLINE for some \(\delta_\ell > 0\) and any \(\varepsilon >0.\) Here \(f\) is a (fixed) holomorphic newform and \(\chi\) is a primitive character of conductor \(q^\ell,\) with \(q\) prime and \(\ell > 1.\) A generalization to the case when \(f\) is a Maass form is also discussed. Such a bound is subconvex, and is the first instance of a subconvexity bound in the level aspect for a degree three \(L\) function which is not self-dual. Some other relevant results include \textit{V. Blomer} [Math. Z. 260, No. 4, 755--777 (2008; Zbl 1192.11028); Am. J. Math. 134, No. 5, 1385--1421 (2012; Zbl 1297.11046)], \textit{X. Li} [Ann. Math. (2) 173, No. 1, 301--336 (2011; Zbl 1320.11046)], and the author's own subsequent paper [Adv. Math. 235, 74--91 (2013; Zbl 1271.11055)].