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Computation of the zeros of $ p$-adic $ L$-functions

Authors: R. Ernvall and T. Metsänkylä
Journal: Math. Comp. 58 (1992), 815-830, S37
MSC: Primary 11R23; Secondary 11R42, 11Y70
MathSciNet review: 1122068
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Abstract: The authors have computed the zeros of the Kubota-Leopoldt p-adic L-functions $ {L_p}(s,\chi )$ for some small odd primes p and for a number of Dirichlet characters $ \chi $. The zeros of the corresponding Iwasawa power series $ {f_\theta }(T)$ are also computed. The characters $ \chi $ (associated with quadratic extensions of the pth cyclotomic field) are chosen so as to cover as many different splitting types of $ {f_\theta }(T)$ as possible. The $ \lambda $-invariant of this power series, equal to its number of zeros, assumes values up to 8.

The article is a report on these computations and their results, including the required theoretical background. Much effort is devoted to a study of the accuracy of the computed approximations.

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Keywords: p-adic L-functions, computation of zeros, factorization of polynomials, Newton's tangent method, Abelian fields, Iwasawa theory
Article copyright: © Copyright 1992 American Mathematical Society

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