On a conjecture of Ash
The author shows that for certain level-\(p\)-subgroups \(\Gamma\) of GL\((n,{\mathbb Z})\) the Hecke eigenclasses in \(H^*(\Gamma,{\mathbb F}_p)\) which are induced from \(H^*(\Gamma\bmod p,{\mathbb F}_p)\) have Hecke polynomials (for almost every prime \(\ell\)) which coincide with det\((I-\rho(\text{Frob}_\ell) X)\) for some \(n\)-dimensional \(\mathbb{F}_p G\) module, where \(G\) is the absolute Galois group of the rationals and Frob\(_\ell\) some choice of Frobenius element at \(\ell.\) The representation is constructed as a sum of powers of the cyclotomic character. The result confirms a conjecture of \textit{A. Ash} [Duke Math. J. 65, 235-255 (1992; Zbl 0774.11024)] in a particular case.
scientific article; zbMATH DE number 1797468
scientific article