A Kronecker limit formula for totally real fields and arithmetic applications

Sheng-Chi Liu,Riad Masri

doi:10.1007/s40993-015-0009-3

Sheng-Chi Liu, Riad Masri

Open Access

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

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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 \({SL}_{n}({\mathbb {Z}})\)-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.

Highlights

Note that Hecke proved a similar limit formula for real quadratic fields by relating the ideal class zeta function to an integral of E(z, s) over a Heegner cycle in H
Introduction and statement of resultsThe celebrated Kronecker limit formula expresses the constant term in the Laurent expansion at s = 1 of the Dedekind zeta function ζK (s, A) of an ideal class A of an imaginary quadratic field K in terms of the value of log |η(z)| at a Heegner point τA in the complex upper half-plane H where η(z) is the Dedekind eta function ∞η(z) := q1/24 (1 − qn), n=1 q := e2πiz.There are many interesting applications of this formula to algebraic number theory, including relative class number formulae and Kronecker’s “solution” to Pell’s equation
The celebrated Kronecker limit formula expresses the constant term in the Laurent expansion at s = 1 of the Dedekind zeta function ζK (s, A) of an ideal class A of an imaginary quadratic field K in terms of the value of log |η(z)| at a Heegner point τA in the complex upper half-plane H where η(z) is the Dedekind eta function η(z) := q1/24 (1 − qn), n=1 q := e2πiz

Summary

Introduction

Note that Hecke proved a similar limit formula for real quadratic fields by relating the ideal class zeta function to an integral of E(z, s) over a Heegner cycle in H. In this paper we will prove 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 ≥ 2, thereby extending the limit formulae of Hecke and Bump/Goldfeld (see Theorem 1).

Results

Conclusion