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Numerical verification of the Stark-Chinburg conjecture for some icosahedral representations. (English) Zbl 1082.11082

The authors give a numerical verification of a variant of Stark’s conjecture which they call Stark-Chinburg conjecture. They are the first to show the existence of a certain unit (“Stark unit”) in non-abelian Galois extensions \({\mathcal K}\) which allow two-dimensional representations of icosahedral isomorphism type in \(\text{PGL}_2(\mathbb C)\). In this case the degree \([{\mathcal K}:\mathbb Q]\) is 240. The authors were successful in verifying the conjecture for 14 icosahedral representations with odd quadratic determinant. An illustrative example demonstrating the complexity of the calculations is included. For instance, they present the minimal polynomial of the fourth root of a Stark unit, a polynomial of degree 120 with large coefficients.

MSC:

11Y40 Algebraic number theory computations
11R42 Zeta functions and \(L\)-functions of number fields
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