A Kronecker limit formula for totally real fields and arithmetic applications View Full Text


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Article Info

DATE

2015-12

AUTHORS

Sheng-Chi Liu, Riad Masri

ABSTRACT

We establish a Kronecker limit formula for the zeta function ζF(s,A) of a wide ideal class A of a totally real number field F of degree n. This formula relates the constant term in the Laurent expansion of ζF(s,A) at s=1 to a toric integral of a SLn(ℤ)-invariant function logG(Z) along a Heegner cycle in the symmetric space of GLn(ℝ). We give several applications of this formula to algebraic number theory, including a relative class number formula for H/F where H is the Hilbert class field of F, and an analog of Kronecker’s solution of Pell’s equation for totally real multiquadratic fields. We also use a well-known conjecture from transcendence theory on algebraic independence of logarithms of algebraic numbers to study the transcendence of the toric integral of logG(Z). Explicit examples are given for each of these results. More... »

PAGES

8

References to SciGraph publications

  • 2010-01. Genus fields of real biquadratic fields in THE RAMANUJAN JOURNAL
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