Ramanujan-style congruences of local origin
Let \(p\) be a prime and \(k\geq4\) an even integer. Further suppose that \(\ell>3\) is a prime such that \(\mathrm{ord}_{\ell}\left((p^k-1)B_k/2k\right)>0\), where \(B_k\) is the \(k\)-th Bernoulli number. Then the authors show that there exists an eigenform in \(S_k(\Gamma_0(p))\) whose Fourier coefficients satisfy a ``Ramanujan-style'' congruence modulo \(\ell\) similar to the famous 691 congruence for the Ramanujan \(\tau\)-function. The technique used is the construction of explicit Eisenstein series of level \(p\) and weight \(k\) which vanish modulo \(\ell\) at both cusps, together with the Deligne-Serre theory of \(\ell\)-adic Galois representations attached to modular forms. The authors further describe the relation of their work to the \(\ell\)-part of the Bloch-Kato conjecture, and supply new proofs of theorems of \textit{N. Billerey} and \textit{R. Menares} [``On the modularity of reducible mod \(l\) Galois representations'', Preprint, arXiv:1309.3717] concerning the modularity of reducible mod \(\ell\) Galois-representations. The authors then conclude with a detailed description of several illuminating examples.
scientific article; zbMATH DE number 6339007
scientific article