The aim of this paper to prove a subconvexity result for \(L\)-functions \(L(1/2, f \times \chi)\), where \(f\) is a (not necessarily self-dual) fixed automorphic form for the group \(\mathrm{SL}(3, \mathbb{Z})\) and \(\chi\) is a Dirichlet character that can be decomposed as \(\chi = \chi_1 \chi_2\) with \(\chi_1\) of modulus \(M_1\) and \(\chi_2\) of modulus \(M_2\). If \(M_1, M_2\) are primes satisfying \(M_2^{1/2}(M_1M_2)^{4\delta} < M_1 < M_2(M_1M_2)^{-3\delta}\) for some \(0 < \delta < 1/28\), then the main result states \[ L(1/2, f \times \chi) \ll (M_1M_2)^{3/4 - \delta + \varepsilon}. \] The factorization condition, although maybe a bit artificial from the point of view of applications, is a very useful requirement in terms of the underlying methods and mimics in a sense the archimedean situation where one is trying to prove subconvexity for \(L(1/2 + it, f)\) in the \(t\)-aspect.
The starting point of the argument is the following seemingly pointless, but ultimately very clever device. An approximate functional equation reduces the problem to bounding sums of the type \[ \sum_{n \asymp N} \lambda(1, n)\chi(n), \] and this is transformed into \[ \sum_{n, m \asymp Nn \equiv m\pmod{M_1}}\lambda(1, n) \chi(m) \delta_{M_1 \mid n - m}. \] The double sum is seemingly a step backwards, but now \(\chi\) and \(\lambda\) are partly separated, which turns out to be a great advantage. The \(\delta\)-symbol is written in terms of Kloosterman’s Farey decomposition, the archetype of the circle method. Then a succession of Poisson summation, Voronoi summation and Cauchy-Schwarz leads to complicated multidimensional character sums whose optimal bounds lead to the desired subconvexity result. It is mainly at this point where the assumption \(M_1, M_2\) prime is used.
For Part I, see [Math. Ann. 358, No. 1–2, 389–401 (2014; Zbl 1312.11037)].