Icosahedral Galois Representations

A computation of modular forms of weight one and small level

“…(Custom code existed well before then: examples we know of are due to Cohen-Skoruppa-Zagier, Cremona, Gouvêa and Stein.) By contrast, no such direct method is known in weight one and there were no generally applicable algorithms until more recently-the pioneering work of Buhler [3] and project coordinated by Frey [9] were both focused on computing one or more specific spaces of forms, rather than on a systematic computation.…”

Galois representations of dihedral type over Qp

“…In studying 2-dimensional (irreducible) continuous complex representations of G k the following question is of interest (cf. [2], [7], [5]): Given a continuous projective representation 0 : G k → PGL 2 (C), find the determinant and the Artin conductor of all lifts : G k → GL 2 (C). Now, as any lift has the shape ⊗ φ with any fixed lift and φ a character of G k , and as det( ⊗ φ) = det · φ 2 , one might hope to accomplish this by finding one particular lift for which the following data can be determined:…”

“…again [2], [7], [5]) that the problem of finding a good lift can be reduced to the following local problem: Consider the completion k p of k at a (finite) prime p. Given a projective representation 0,p : G k p → PGL 2 (C) find for some lift p the following:…”

“…Building upon and completing previous investigations [9], [2], [10] of this local problem, in [5] we constructed good lifts under conditions of some generality, and in particular in all cases where k p is a p-adic field Q p . Perhaps somewhat surprisingly the case of dihedral type projective representations, i.e.…”

“…The first statements of (2) are trivial to check, so we have only to verify the entries of the table in (2).…”