Structure of Hecke algebras of modular forms modulo \(p\)
Let \(p\geq 5\) be a prime number, \(K\) a finite extension field of \(\mathbb Q_p\) with ring of integers \(\mathcal O\) and uniformizing element \(\pi.\) Let \(N\) be a natural number and \(\Gamma\) be one of the congruence groups \(\Gamma_0(N)\) or \(\Gamma_1(N)\) inside \(\text{SL}(2,\mathbb Z).\) Denote by \({\mathcal S}_k\) the module of weight \(k\) cuspidal forms for \(\Gamma\) with \(q\)-expansion in \({\mathcal O}q.\) The author studies local components of the shallow Hecke algebras (i.e., avoiding \(U_\ell\) for primes \(\ell\) dividing \(pN\)) acting on \(\sum_{i\leq k} {\mathcal S}_i\) mod \(\pi.\) As in the case \(N=1\) treated by \textit{J. Bellaïche} and \textit{C. Khare} [Compos. Math. 151, No. 3, 397--415 (2015; Zbl 1328.11046)], it turns out that the local components of the Hecke algebras are fairly often (stated more precisely in the paper) regular complete local algebras of dimension 2. Divided congruence modules and deformation spaces for pseudo-representations play a vital role in the arguments.
scientific article; zbMATH DE number 6679111