Level aspect subconvexity for GL(2) × GL(2) 𝐿-functions

Let 𝑓 be a newform of prime level 𝑝 with any central character χ ⁢ ( mod ⁢ p ) χ{\ (mod\ p)}{\ (mod\ p)}{\,(mod\,p)}{\,(mod\,p)} , and let 𝑔 be a fixed cusp form or Eisenstein series for SL 2 ⁢ ( Z ) SL₂(Z) . We prove the uniform subconvexity bound L ⁢ ( 1 / 2 , f ⊗ g ) ≪ p 1 / 2 − 1 / 524 + ε L(1/2,f⊗ g) p1/2-1/524+ε for any ε > 0 ε>0 , where the implied constant depends on 𝑔, 𝜀, and the archimedean parameter of 𝑓. This improves upon the previously best-known result by Harcos and Michel. Our method overcomes existing limitations, which ultimately relies on non-trivial bounds for bilinear forms in Kloosterman fractions pioneered by Duke, Friedlander, and Iwaniec, with later innovations by Bettin and Chandee. In particular, our method avoids the spectral theory of automorphic forms and is independent of bounds towards the Ramanujan conjecture.