Open Access

A Kronecker limit formula for totally real fields and arithmetic applications

Research in Number Theory20151:8

https://doi.org/10.1007/s40993-015-0009-3

Received: 21 April 2015

Accepted: 26 April 2015

Published: 1 September 2015

Abstract

We establish a Kronecker limit formula for the zeta function ζ F (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 ζ F (s,A) at s=1 to a toric integral of a \({SL}_{n}({\mathbb {Z}})\)-invariant function logG(Z) along a Heegner cycle in the symmetric space of \({GL}_{n}({\mathbb {R}})\). 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.

Keywords

Maximal parabolic Eisenstein series Heegner cycle Kronecker limit formula Transcendence