Estimates for \(p\)-adic valuations of Hecke operator eigenvalues (Q2880698)

The aim of this paper is to establish bounds for the \(p\)-adic valuations of the eigenvalues of Hecke operators acting on spaces of cuspidal automorphic forms. More specifically, if \(\pi\) is a cuspidal automorphic representation of a split reductive group \(G\) over a global field \(F\), and \(v\) is a finite place of \(F\) at which \(\pi\) is unramified, one may attach to \(\pi\) a ''Satake parameter'', a conjugacy class \(c_{\pi, v}\) in the Langlands dual group \(\widehat G(\mathbb{C})\), which is known in many cases to lie in \(\overline{\mathbb{Q}}\). By taking the \(\mathfrak{p}\)-adic valuation, for some prime \(\mathfrak{p}\) of \(\overline{\mathbb{Q}}\) of the same residue characteristic as \(v\), one obtains a map \(N_{\mathfrak{p}}\) from the set of conjugacy classes in \(\widehat{G}(\overline{\mathbb{Q}})\) to the positive Weyl chamber, which is equipped with a natural partial ordering. The main result of the paper is the following theorem:NEWLINENEWLINE- if \(F\) is a function field, then \(N_{\mathfrak{p}}(c_{\pi, v}) \leq \rho\), where \(\rho\) is half the sum of the positive roots;NEWLINENEWLINE- if \(F\) is a number field and \(\pi\) is cohomological at the infinite places of \(F\), then \(N_{\mathfrak{p}}(c_{\pi, v}) \leq \rho + \mu\), where \(\mu\) is given by an explicit formula in terms of the infinitesimal character of \(\pi_w\) for each infinite place \(w\).NEWLINENEWLINEIn the number field case, if \(\pi\) corresponds to a compatible family of representations of the Galois group \(G_F\), then these estimates correspond to the fact that the Newton polygon of any crystalline Galois representation lies on or above the Hodge polygon, a well-known statement in \(p\)-adic Hodge theory. However, the proof given here is wholly automorphic in nature, and does not rely on the existence of Galois representations (which is, of course, presently only known under much more restrictive hypotheses).NEWLINENEWLINEThe article also includes a sketch of some more general but less explicit bounds at non-split and ramified primes when \(F\) is a number field, using the notion of a ``locally integral representation'' introduced in a recent article of \textit{J.-F. Dat} [Am. J. Math. 131, No. 1, 227--255 (2009; Zbl 1165.22015)].