Congruences for L-Functions

1. Auflage. Softcover version of original hardcover Auflage 2000
ISBN: 978-90-481-5490-6
Verlag: Springer Netherlands

Buch, Englisch, Band 511, 256 Seiten, Previously published in hardcover, Format (B × H): 155 mm x 235 mm, Gewicht: 417 g

Reihe: Mathematics and Its Applications

In [Hardy and Williams, 1986] the authors exploited a very simple idea to obtain a linear congruence involving class numbers of imaginary quadratic fields modulo a certain power of 2. Their congruence provided a unified setting for many congruences proved previously by other authors using various means. The Hardy-Williams idea was as follows. Let d be the discriminant of a quadratic field. Suppose that d is odd and let d = PIP2·. Pn be its unique decomposition into prime discriminants. Then, for any positive integer k coprime with d, the congruence holds trivially as each Legendre-Jacobi-Kronecker symbol (~) has the value + 1 or -1. Expanding this product gives ~ eld e:=l (mod4) where e runs through the positive and negative divisors of d and v (e) denotes the number of distinct prime factors of e. Summing this congruence for o < k < Idl/8, gcd(k, d) = 1, gives ~ (-It(e) ~ (~) =:O(mod2n). eld o

Zielgruppe

Research

Autoren/Hrsg.

Williams, Kenneth S.

Urbanowicz, J.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie

Inhaltsverzeichnis

I. Short Character Sums.- II. Class Number Congruences.- III. Congruences between the Orders of K2-Groups.- IV Congruences among the Values of 2-Adic L-Functions.- V. Applications of Zagier’s Formula (I).- VI. Applications of Zagier’s Formula (II).- Author Index.- List of symbols.