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Ginsberg, Hy

Mathematics

2010

PhD

Call a virtual character [Greek letter Theta] of a finite group G a Heilbronn character if its inner product with every monomial character of G is nonnegative. If moreover [Greek letterTheta] restricts to a character of every proper subgroup and quotient (where we define the restriction of [Greek letterTheta] to G/N as the sum of the constituents of [Greek letterTheta] whose kernels contain N), but is not a character of G itself, then [Greek letterTheta] is said to be minimal. Heilbronn characters arise naturally in the study of Arlin's Conjecture on the holomorphy of L-series, where a hypothetical minimal counterexample engenders a corresponding minimal Heilbronn character of the associated Galois group. Although motivated originally by this number theoretic application, the study of Heilbronn characters is of independent interest to both group theory and representation theory.
A natural subclass of Heilbronn characters to classify are those that are both minimal and unfaithful, where [Greek letterTheta] is said to be unfaithful if the set {g [Greek letter epsilon] G / [Greek letterTheta](g) = [Greek letterTheta](1)} is nontrivial. Our main result establishes necessary and sufficient conditions for a finite group to possess an unfaithful minimal Heilbronn character. As an application of the main theorem we obtain a corollary bounding the p-rank of the Galois group of a minimal counterexample to Artin's Conjecture by the order of zero of a Dedekind zeta function.