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Publication . Part of book or chapter of book . 1982

Zeros of p-Adic L-Functions, II

Samuel S. Wagstaff;
Closed Access
Published: 01 Jan 1982
Publisher: Birkhäuser Boston
Fix an algebraic closure Ωp. of the p-adic numbers ℚp. For s∈ΩpP write ordp(s) for the p-adic ordinal of s. Calculation with p-adic L-functions is facilitated by the existence of two forms for these functions. Kubota and Leopoldt [6] defined a p-adic L-function Lp(s,χ) in terms of the intrinsic variable s. This function is holomorphic in the natural disc ordp (s) > −(p−2)/(p−l). Later, Iwasawa [3,4] defined a formal power series g(T) in terms of the non-intrinsic variable T. The two functions Lp(s,χ) and g(T) are connected by a simple formula ((5) below) under which the natural s-disc corresponds to the open unit T-disc. The zeros of Lp(s,χ) have deeper meaning than those of g(T), but the latter are easier to locate because the Weierstrass preparation theorem applies in the open unit T-disc. Whenever a good approximation is available for a zero in either disc, it can be refined by Newton’s method provided the coefficients are known with enough accuracy.
Subjects by Vocabulary

Microsoft Academic Graph classification: Combinatorics Zero (complex analysis) Function (mathematics) Formal power series Mathematical analysis Legendre symbol symbols.namesake symbols Weierstrass preparation theorem Algebraic closure Holomorphic function Variable (mathematics) Mathematics

arXiv: Mathematics::Number Theory

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Part of book or chapter of book . 1982
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