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A character-theoretic approach to Artin's Conjecture
Abstract
This thesis uses character-theoretic methods to gain a deeper understanding of Artin's Conjecture on the holomorphy of L-series. In addition to providing sufficient conditions for Artin's Conjecture to hold, we identify hypothetical minimal counterexamples to this Conjecture. Given a Galois extension of number fields E/F with Galois group G and a character $\psi$ of G, the Artin L-series $L(s,\psi,E/F)$ is defined by an Euler product in the right complex half-plane. These Artin L-functions have been shown to have a meromorphic continuation to the full complex plane. Artin's Conjecture is that these L-series are holomorphic functions except possibly at s = 1. The first result gives a necessary and sufficient condition for when a virtual character can be written as a positive rational linear combination of characters from some specified set of characters of G. This result is used to prove a generalization of the Aramata-Brauer Theorem. This generalization gives a new proof that the L-series for the regular character plus or minus any irreducible character satisfies Artin's Conjecture. Now let $s\sb0$ be a complex point other than 1, while E/F is assumed to be a Galois extension of number fields of minimal degree such that Artin's Conjecture is false at $s\sb0.$ Let G be the Galois group of E/F. It is further assumed that $\zeta\sb{E}(s)$ has a zero of order n at $s\sb0$. The remainder of the thesis studies hypothetical counterexamples to Artin's Conjecture at $s\sb0$ by imposing additional constraints on the structure of G and on the Dedekind zeta function of E. It is shown, in particular, that if G is a solvable group, then a minimal counterexample must have a faithful representation of degree $\le$n. Lastly, we discuss quasisimple minimal counterexamples to Artin's Conjecture in the case where n = 3. All possible minimal counterexamples are explicitly known for the cases where n $<$ 3. The results here work toward a similar knowledge for the case n = 3. In summary, in addition to generalizing the Aramata-Brauer Theorem, this thesis gives new insights into the hypothetical minimal counterexamples to Artin's Conjecture which arise from group-theoretic analysis.
Subject Area
Mathematics
Recommended Citation
Rhoades, Sandra L, "A character-theoretic approach to Artin's Conjecture" (1993). Doctoral Dissertations Available from Proquest. AAI9408333.
https://scholarworks.umass.edu/dissertations/AAI9408333