Zeros of 2-adic 𝐿-functions and congruences for class numbers and fundamental units

Daniel Shanks, Patrick J. Sime, Lawrence C. Washington

Type: Article

Publication Date: 1999-02-10

Citations: 5

DOI: https://doi.org/10.1090/s0025-5718-99-01046-7

Abstract

We study the imaginary quadratic fields such that the Iwasawa <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda 2"> <mml:semantics> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\lambda _{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant equals 1, obtaining information on zeros of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions and relating this to congruences for fundamental units and class numbers.

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Works Cited by This (15)

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+	Introduction to Cyclotomic Fields	1982	Lawrence C. Washington
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+ PDF Chat	On cyclotomic ${\bf Z}_{2}$-extensions of imaginary quadratic fields	1979	Yûji Kida
+	A Family of Cubic Fields and Zeros of 3-adicL-Functions	1997	Lawrence C. Washington
+	On the 2-part of the class number of imaginary quadratic number fields	1976	Arnold Pizer
+ PDF Chat	The quadratic number fields with cyclic 2-classgroups	1983	Richard Morton
+ PDF Chat	Heuristics on class groups of number fields	1984	Henri Cohen H. W. Lenstra
+ PDF Chat	Divisibilit\'e par 8 du nombr des classes des corps quadratiques dont le 2-groupe des classes est cyclique, et r\'eciprocit\'e biquadratique	1973	Par Pierre KAPLAN
+	The Cyclotomic Z 2 -Extension of Imaginary Quadratic Fields	1980	Bruce Ferrero