Explicit Burgess-like subconvex bounds for $\mathrm{GL}_2 \times \mathrm{GL}_1$

TL;DR

This paper explicitly quantifies how the subconvex bounds for $L(1/2, \, \pi \otimes \chi)$ on $\, \mathrm{GL}_2 \times \mathrm{GL}_1$ depend polynomially on the fixed representation, refining previous results and exploring local test function choices.

Contribution

It provides explicit polynomial bounds in the conductor for the subconvexity problem on $\, \mathrm{GL}_2 \times \mathrm{GL}_1$, and investigates optimal local test functions.

Findings

Explicit polynomial dependence on the conductor $\, \mathbf{C}(\pi_{fin})$.

Analysis of local test function choices at infinite places.

Potential improvements over Michel & Venkatesh's original test function.

Abstract

We make the polynomial dependence on the fixed representation $\pi$ in our previous subconvex bound of $L(1/2,\pi \otimes \chi)$ for $\mathrm{GL}_2 \times \mathrm{GL}_1$ explicit, especially with respect to the usual conductor $\mathbf{C}(\pi_{\mathrm{fin}})$. There is no clue that the original choice, due to Michel & Venkatesh, of the test function at the infinite places should be the optimal one. Hence we also investigate a possible variant of such local choices in some special situations.