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On the values of equivariant and Artin L-functions of cyclic extensions of number fields

  • Author(s): Smith, Barry Ried
  • et al.
Abstract

We study the values produced by equivariant Artin L- functions at zero. We begin with three preliminary chapters providing the requisite background. In the fourth chapter, we derive expressions for the norms of the values of Artin L-functions attached to cyclic extensions of degree $2̂}m}p̂}n}$, where p is an odd prime number, m >̲ 1 , and n >̲ 0. We propose hypothetical expressions for the values themselves in terms of the Fitting ideals of two arithmetic modules over the ring of integers in a cyclotomic field, and validate the expressions and their local variants in several cases. In chapter five, the formulas from chapter four are used to study the Brumer- Stark conjecture and its local variants in several new cases. Our methods are similar to those used in the study of degree $2p$ extensions by Greither, Roblot, and Tangedal, excepting the use of the formulas from chapter four that enable proofs in our more general setting. Some results deal only with the annihilation statement of the p -primary part of the conjecture for degree 2p extensions. They hint that something deeper is happening with the p- primary part of the conjecture in the cases where general proofs cannot yet be given. In chapter six, the expressions from chapter four are used to study a new conjecture of Hayes concerning the precise denominators of the values of the equivariant L-functions at zero. A variant of this conjecture is proved for extensions of degree $2̂}m}$ using the formulas from chapter four. Following that, it is shown that the truth of the conjecture is preserved under lowering of the top field. We then prove the conjecture for extensions where both fields are absolutely Abelian of prime conductor. Lastly, a counterexample is provided to a stronger conjecture posed by Hayes in an unpublished manuscript

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