Hybrid bounds for $\mathrm{GL}(4)\times \mathrm{GL}(1)$ twisted $L$-functions

Abstract

Let $P,M$ be a two primes such that $(P,M)=1$. Let $\Pi$ be a normalized Hecke-Maaß form on $\mathrm{GL}(4)$ of level $P$, and $\chi$ a primitive Dirichlet character modulo $M$. In this paper, we study the hybrid subconvexity problem for $L(s,\Pi \otimes \chi )$ simultaneously in the level and conductor aspects. Among other things, we prove a hybrid subconvex bound, so long as $M^{1/5}<P<M^{2/5}$.

We added an appendix by showing Lemma 2.1