Addendum: Hybrid bounds for twisted L-functions

Abstract

Abstract. We extend our hybrid bounds for twisted automorphic L-functions L.s; f˝ / on the critical line [J. reine angew. Math. 621 (2008), 53-79] to cusp forms f with arbitrary nebentypus, and correct an error in the proof.

A twisted Motohashi formula and Weyl-subconvexity for $$L$$ L -functions of weight two cusp forms

“…See also the classical work-out as an explicit formula for full level due to Kohnen and Zagier [13], and for general level due to Kohnen [12]. As a consequence of positivity (3), the bounds on harmonic weights (2), and our main result Theorem 1 we derive the following strengthened form of the subconvex bound found in Conrey and Iwaniec as their Corollary 1.2.…”

“…, (2) by [7,4,8]. The weight κ is always considered fixed, and all implicit constants may depend on κ.…”

A generalized cubic moment and the Petersson formula for newforms

“…Corollary 1.2 improves on a hybrid subconvexity result of Blomer and Harcos [BH,Theorem 2'], which holds more generally for f of arbitrary level and nebentype character. In our notation (and assuming (q, r) = 1) the result of Blomer and Harcos takes the form (1.3) L(1/2, f ⊗ χ q ) ≪ (r 1/4 q 3/8 + r 1/2 q 1/4 )(rq) ε .…”

Hybrid level aspect subconvexity for GL(2) × GL(1) Rankin-Selberg L-Functions

“…The works involving estimates of these sums with the divisor functions, known as the additive divisor problems, have a long history (See [DFIw94,Mo94] and the reference given there). Shifted convolution sums with two Fourier coefficients have been extensively investigated in [BlHa14,DFIw93,Ha03,HaMi06,Mi04] for the case when λ g (n) = λ f (n) or λ g (n) = λ f (n) and [HoMu13,HoZh17] for the general case. Thus it is natural to compare the strategies of the existing approaches and ponder about the tools necessary to study the hybrid level aspect problem.…”

“…Using the classical delta method without the conductor lowering trick and following the same process, one would obtain a hybrid range of 0 < η < 2/7. In the meanwhile, Blomer and Harcos in theorem 2 of [BlHa14] establish the following hybrid estimate…”