A Kronecker limit formula for totally real fields and arithmetic applications

摘要

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 ( ? ) ${SL}_{n}({mathbb {Z}})$ -invariant function log G ( Z ) along a Heegner cycle in the symmetric space of GL n ( ? ) ${GL}_{n}({mathbb {R}})$ . 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 log G ( Z ). Explicit examples are given for each of these results.