The behavior of Hecke L-functions of real quadratic fields at s = 0

Jun, Byungheup; Lee, Jungyun

doi:10.2140/ant.2011.5.1001

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The behavior of Hecke $L$-functions of real quadratic fields at $s=0$

Byungheup Jun and Jungyun Lee

Vol. 5 (2011), No. 8, 1001–1026

DOI: 10.2140/ant.2011.5.1001

Abstract

For a family of real quadratic fields ${K_{n} = ℚ (\sqrt{f (n)})}_{n \in ℕ}$ , a Dirichlet character $χ$ modulo $q$ , and prescribed ideals ${b_{n} \subset K_{n}}$ , we investigate the linear behavior of the special value of the partial Hecke $L$ -function $L_{K_{n}} (s, χ_{n} : = χ \circ N_{K_{n}}, b_{n})$ at $s = 0$ . We show that for $n = q k + r$ , $L_{K_{n}} (0, χ_{n}, b_{n})$ can be written as

\frac{1}{12 q^{2}} (A_{χ} (r) + k B_{χ} (r)),

where $A_{χ} (r), B_{χ} (r) \in ℤ [χ (1), χ (2), \dots, χ (q)]$ if a certain condition on $b_{n}$ in terms of its continued fraction is satisfied. Furthermore, we write $A_{χ} (r)$ and $B_{χ} (r)$ explicitly using values of the Bernoulli polynomials. We describe how the linearity is used in solving the class number one problem for some families and recover the proofs in some cases.

Keywords

special values, Hecke L-functions, real quadratic fields, continued fractions

Mathematical Subject Classification 2000

Primary: 11M06

Milestones

Received: 7 March 2010

Revised: 24 March 2011

Accepted: 8 May 2011

Published: 5 June 2012

Authors

Byungheup Jun
School of Mathematics Korea Institute for Advanced Study Hoegiro 87, Dongdaemun-gu Seoul 130-722 South Korea

Jungyun Lee
School of Mathematics Korea Institute for Advanced Study Hoegiro 87, Dongdaemun-gu Seoul 130-722 South Korea

Vol. 5, No. 8, 2011