A combinatorial interpretation of Serre's conjecture on modular Galois representations. (Q1415569)

Serre's famous conjecture [cf. \textit{B. Edixhoven}, in: Modular forms and Fermat's last theorem, 209--242 (1997; Zbl 0918.11023)] relates irreducible 2-dimensional modular Galois-representations with certain modular forms of some level \(N\), weight \(k\) and character \(\varepsilon.\) These can be viewed via the Eichler-Shimura-isomorphism as classes in \(H^1(\Gamma_1(N),V)\) for the irreducible SL\(_2\)-module \(V\) of dimension \(k-1.\) Using the Shapiro -Lemma the author turns this into cohomology for SL\(_2({\mathbb Z})\) and makes everything explicit. He thus translates Serre's conjecture into a conjecture relating a continuous absolutely irreducible odd Galois representation \(\rho\) with values in GL\(_2(F)\) (\(F\) a finite field) to an algebraic object: a homomorphism from some group (encoding \(N\) and \(k\)) to \(F\), which is subject to some extra conditions. The proof that this is equivalent to Serre's original conjecture is prepared carefully, and various ways to define modular forms with coefficients in \(F\) are discussed. Not underestimating the merits of this paper, the reviewer fails to see its truly combinatorial nature as well as its claim to give a new interpretation of Serre's conjecture.