Subconvexity Bounds for L-functions of Maass Forms via the Spectral Large Sieve - VERSÃO 2

We establish a subconvexity bound for the central value of the L-function L(1/2, uj) associated with a Hecke-Maass cusp form uj for SL(2, Z) with spectral parameter tj. The convexity bound, derived from the Phragmén-Lindelöf principle, is L(1/2, uj) tj1/2+. By employing the spectral large sieve inequalities of Deshouillers and Iwaniec, combined with the amplification method, we prove that L(1/2, uj) tj1/3+. This result improves upon the convexity bound and matches the Weyl-type exponent in the spectral aspect. The proof relies on the Kuznetsov trace formula to relate the spectral average of L-values to sums of Kloosterman sums, which are then estimated using the spectral large sieve. We provide a rigorous treatment of the off-diagonal terms arising in the amplification process, demonstrating the power of spectral summation formulae in breaking the convexity barrier.