Notes on the Generalized Ramanujan Conjectures

Ramanujan’s original conjecture is concerned with the estimation of Fourier coefficients of the weight 12 holomorphic cusp form ∆ for SL(2,Z) on the upper half plane H. The conjecture may be reformulated in terms of the size of the eigenvalues of the corresponding Hecke operators or equivalently in terms of the local representations which are components of the automorphic representation associated with ∆. This spectral reformulation of the problem of estimation of Fourier coefficients (or more generally periods of automorphic forms) is not a general feature. For example, the Fourier coefficients of Siegel modular forms in several variables carry more information than just the eigenvalues of the Hecke operators. Another example is that of half integral weight cusp forms on H where the issue of the size of the Fourier coefficients is equivalent to special instances of the Lindelof Hypothesis for automorphic L-functions (see [Wal], [I-S]). As such, the general problem of estimation of Fourier coefficients appears to lie deeper (or rather farther out of reach at the present time). In these notes we discuss the spectral or representation theoretic generalizations of the Ramanujan Conjectures (GRC for short).