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Full-Text Articles in Physical Sciences and Mathematics
Octahedral Extensions And Proofs Of Two Conjectures Of Wong, Kevin Ronald Childers
Octahedral Extensions And Proofs Of Two Conjectures Of Wong, Kevin Ronald Childers
Theses and Dissertations
Consider a non-Galois cubic extension K/Q ramified at a single prime p > 3. We show that if K is a subfield of an S_4-extension L/Q ramified only at p, we can determine the Artin conductor of the projective representation associated to L/Q, which is based on whether or not K/Q is totally real. We also show that the number of S_4-extensions of this type with K as a subfield is of the form 2^n - 1 for some n >= 0. If K/Q is totally real, n > 1. This proves two conjectures of Siman Wong.
Lifting Galois Representations In A Conjecture Of Figueiredo, Wayne Bennett Rosengren
Lifting Galois Representations In A Conjecture Of Figueiredo, Wayne Bennett Rosengren
Theses and Dissertations
In 1987, Jean-Pierre Serre gave a conjecture on the correspondence between degree 2 odd irreducible representations of the absolute Galois group of Q and modular forms. Letting M be an imaginary quadratic field, L.M. Figueiredo gave a related conjecture concerning degree 2 irreducible representations of the absolute Galois group of M and their correspondence to homology classes. He experimentally confirmed his conjecture for three representations arising from PSL(2,3)-polynomials, but only up to a sign because he did not lift them to SL(2,3)-polynomials. In this paper we compute explicit lifts and give further evidence that his conjecture is accurate.
Proven Cases Of A Generalization Of Serre's Conjecture, Jonathan H. Blackhurst
Proven Cases Of A Generalization Of Serre's Conjecture, Jonathan H. Blackhurst
Theses and Dissertations
In the 1970's Serre conjectured a correspondence between modular forms and two-dimensional Galois representations. Ash, Doud, and Pollack have extended this conjecture to a correspondence between Hecke eigenclasses in arithmetic cohomology and n-dimensional Galois representations. We present some of the first examples of proven cases of this generalized conjecture.
Totally Real Galois Representations In Characteristic 2 And Arithmetic Cohomology, Heather Aurora Florence De Melo
Totally Real Galois Representations In Characteristic 2 And Arithmetic Cohomology, Heather Aurora Florence De Melo
Theses and Dissertations
The purpose of this paper is to provide new examples supporting a conjecture of Ash, Doud, and Pollack. This conjecture involves Galois representations taking Gal(Q bar/Q) to the general linear group of 3 x 3 matrices in characterisic 2, and our examples are where complex conjugation is mapped to the identity. Since this case has not yet been examined, the results of this paper are quite significant.
Explicit Computations Supporting A Generalization Of Serre's Conjecture, Brian Francis Hansen
Explicit Computations Supporting A Generalization Of Serre's Conjecture, Brian Francis Hansen
Theses and Dissertations
Serre's conjecture on the modularity of Galois representations makes a connection between two-dimensional Galois representations and modular forms. A conjecture by Ash, Doud, and Pollack generalizes Serre's to higher-dimensional Galois representations. In this paper we discuss an explicit computational example supporting the generalized claim. An ambiguity in a calculation within the example is resolved using a method of complex approximation.