The Weyl bound for triple product L-functions

A uniform Weyl bound for L-functions of Hilbert modular forms

“…• Blomer-Jana-Nelson [9] for triple products π = π 1 × π 2 × π 3 with π j corresponding to modular forms for SL 2 (Z), π 1 and π 2 fixed and in the archimedean aspect for π 3 .…”

Sup-norm of Hecke–Laplace eigenforms on $$S^3$$

“…Asymptotic notation. In order to simplify the discussion, we shall introduce some asymptotic notation in the spirit of [BJN21,§2.4]. We shall write…”

Analytic twists of GL2×GL2$\rm GL_2\times \rm GL_2$ automorphic forms

“…The best record bound for 𝐿 (1∕2 + 𝑖𝑡, 𝑓 ⊗ 𝑔) is the Weyl type bound 𝐿 (1∕2 + 𝑖𝑡, 𝑓 ⊗ 𝑔) ≪ (1 + |𝑡|) 2∕3+𝜀 due to Blomer, Jana and Nelson [5] by combining in a substantial way representation theory, local harmonic analysis, and analytic number theory. Bernstein and Reznikov showed the bound (1 + |𝑡|) 5∕6+𝜀 in [4] (see Remarks 7.2.2.2).…”

“…So, we only need to consider the case 𝑛 ≍ 𝑇5 ∕𝑋. Now we make use of the fact that 𝜆 1⊞(𝑓×𝑔) (𝑛) = ∑ 𝑙𝑚 2 𝑟=𝑛 𝜆 𝑓 (𝑟)𝜆 𝑔 (𝑟).…”

“…Inserting dyadic partitions to the 𝑙-sum and 𝑚-sum and making a smooth partition of unity into dyadic segments to the 𝑟-sum, we arrive at In order to apply Theorem 1.1, we need to verify that 𝑅 satisfies the condition 𝑅 ≪ 𝑇 12∕5 . Note that 𝑅 ≪ 𝐿𝑀 2 𝑅 ≍ 𝑇5 ∕𝑋 and…”