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Mathematics of Computation

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