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

We establish a subconvexity bound for the central value of the L-function L(s, uj) associated with a Hecke-Maass cusp form uj for the modular group SL2(Z). By employing the spectral large sieve inequalities developed by Deshouillers and Iwaniec, combined with the amplification method, we derive an upper bound of the form L(1/2, uj) tj1/3+, where tj denotes the spectral parameter of the form. This result improves upon the convexity bound tj1/2+ implied by the Phragmén-Lindelöf principle. The proof relies on the Kuznetsov trace formula to estimate the spectral average of twisted L-functions, effectively exploiting the orthogonality of the Fourier coefficients of the automorphic spectrum. We provide a rigorous treatment of the analytic conductor and the optimization of the amplifier length in the spectral aspect.