in a series of recent works blomer, fouvry, kowalski, milicevic, sawin and myself have been able to solve the vexing problem of evaluating asymptotically the second moment of the central l-values of character twists (of large prime conductor) of a fixed modular form; the solution combines the spectral theory of modular forms, bounds for bilinear sums of kloosterman sums and advanced methods in l-adic cohomology. we will describe the proof and especially the second and third ingredient which is joint work with e. kowalski an w. sawin. references: https://arxiv.org/abs/1411.4467 and https://arxiv.org/abs/1511.01636.