Explicit Burgess-like subconvex bounds for GL₂ × GL₁

Abstract

We make the polynomial dependence on the fixed representation π in our previous subconvex bound of {L(\frac{1}{2},\pi\otimes\chi)} for {\mathrm{GL}_{2}\times\mathrm{GL}_{1}} explicit, especially in terms of the usual conductor {\mathbf{C}(\pi_{\mathrm{fin}})}. There is no clue that the original choice, due to Michel and 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.

Burgess-like subconvexity for

“…The relevant zeta function has a decomposition as a finite product At an archimedean place , say , [Wu17a, Lemma 3.12(2)] gives the relevant local integral a bound . For the argument is similar, using [Wu17a, Lemma 3.13(2)]. At , [Wu14, Corollary 4.8] is still applicable.…”

“…Its direct generalization, where runs over integral ideals, loses the periodicity for the summand function. Our method can be viewed as a variant of Conrey and Iwaniec’s method (see [Wu17a, § 1.1]). The main common feature is to bring a problem for into the setting for , and to use the available knowledge on the spectral theory of automorphic representations for .…”

On Weyl's Subconvex Bound for Cube-Free Hecke characters: Totally Real Case

“…As mentioned above, the proof relies on the period interpretation of the main formula, which is an intermediate step in one of the two approaches of establishing the formula but can surprisingly not be discarded. In fact, Sarnak's method is the departure point of Michel-Venkatesh's method for GL 2 × GL 1 , which can yield Burgess-type subconvex bounds (see [17,34,35,37]). In §8 we bound the dual side of the Motohashi formula by the fourth moment bound established in §7.…”

Explicit subconvexity for GL 2

“…Remark 1.3. The assumption (1.1) should not be regarded as a condition, since an effective value of 𝐴 is obtained in [22,Theorem 2.1]. In particular, it implies that 𝛿 ′ = (1 − 2𝜃)∕8, 𝐴 = 5∕4 is admissible (note that "𝜋 ′ f in , 𝜒 f in have disjoint ramification" implies 𝐂 f in (𝜋 ′ , 𝜒) = 𝐂 f in [𝜋 ′ , 𝜒] = 1).…”

“…We record the numerical subconvex saving for these values: $$\begin{equation*} \frac{1-2\theta }{40+32A} = \frac{1-2\theta }{80} > \frac{1}{128}, \quad \frac{\delta ^{\prime }}{20+16A} = \frac{1-2\theta }{320} > \frac{1}{1889}. \end{equation*}$$It should even be possible to improve to $$ A=3/4$$ once the relevant sup‐norm result becomes available, see the discussion in [22, §1.3].…”