Subconvexity bound for \(\mathrm{GL}(3)\times\mathrm{GL}(2)\) \(L\)-functions: hybrid level aspect (Q6198005)

Let \(P_1\) and \(P_2 \) be two distinct primes. Let \(F\) be a Hecke-Maass cusp form for the congruence subgroup \(\Gamma_0(P_1)\) of \(\mathrm{SL}(3, \mathbb{Z})\) with trivial nebentypus. Let \(f\) be a holomorphic or Maass cusp form for the congruence subgroup \(\Gamma_0(P_2)\) of \(\mathrm{SL}(2, \mathbb{Z})\) with trivial nebentypus. Let \(\mathcal{Q}=P_1^2P_2^3\) be the arithmetic conductor of the Rankin-Selberg convolution of the above two forms. In this paper under review, the authors prove a subconvex bound for the \(\mathrm{GL}(3) \times \mathrm{GL}(2)\) Rankin-Selberg \(L\)-function \(L(s, F \times f )\) in the level aspect: \[L(\frac{1}{2}, F\times f) \ll \mathcal{Q}^{1/4+\varepsilon}\left(\frac{P_1^{1/4}}{P_2^{3/8}}+\frac{P_2^{1/8}}{P_1^{1/4}}\right).\] Since the convexity bound is given by \(\mathcal{Q}^{1/4+\varepsilon}\), the above bound is subconvex in the range \(P_2^{1/2+\varepsilon} < P_1 < P_2^{3/2-\varepsilon}\). This provides a non-trivial subconvex bound in the level aspect for a degree six \(L\)-function which is not a character twist of a fixed \(L\)-function. Note that when \(P_1\approx P_2\), we have \[L(\frac{1}{2},F\times f)\ll \mathcal{Q}^{1/4 -1/40+\varepsilon}.\] The exponent \(\frac{1}{4}-\frac{1}{40}\) appears again in other contexts as well and it seems to be the limit of the delta symbol approach.