Subconvexity for twisted L-functions over number fields via shifted convolution sums

Abstract

Assume that $${\pi}$$ is a cuspidal automorphic $${{\rm GL}_{2}}$$ representation over a number field F. Then for any Hecke character $${\chi}$$ of conductor $${\mathfrak{q}}$$ , the subconvex bound $$L(1/2,\pi \otimes \chi) \ll_{F,\pi,\chi_{\infty},\varepsilon} \mathcal{N}{\mathfrak{q}}^{3/8+\theta/4+\varepsilon}$$ holds for any $${\varepsilon > 0}$$ , where $${\theta}$$ is any constant towards the Ramanujan-Petersson conjecture ( $${\theta = 7/64}$$ is admissible). In these notes, we derive this bound from the spectral decomposition of shifted convolution sums worked out by the author in [21].