On evaluation of $L$-functions over real quadratic fields
Ryotaro Okazaki · Kyoto journal of mathematics · 1991
IntroductionO ur p u rp ose is to give an effective algorithm to com pute special values of L-functions over real quadratic fields and relative class numbers of totally imaginary quadratic extensions of real quadratic fields.In h is p a p e r [3 ], T .Shintani developped a m ethod fo r evaluating L-functions over totally real algebraic num ber fields a t non-positive integers a n d w ro te a class num ber form ula for totally imaginary quadratic extensions of totally real algebraic number fields.H is m ethod is based o n evaluation of certain kind o f p a rtia l zeta functions.H e also described th e detail o f evaluation of such partial zeta functions over real quadratic fields.B ut no m ethod fo r com puting t h e v a lu e s o f c h a ra c te rs a t id e a ls w a s e x p la in d .M o re o v e r h is c la ss n u m b e r fo rm u la involves a group in d e x o f fo rm [E F : -Vic/FEK] (see § 1 fo r th e definition) which has not been determ ined.In this paper, we restrict ourselves to the case of L-functions associated with quadratic characters an d g iv e a com plete algorithm fo r com puting special values of L -functions associated w ith arbitrary quadratic extensions over real quadratic fields by filling those tw o m issin g d e ta ils.W e d e sc rib e a n algorithm fo r com puting th e v a lu e s o f quadratic characters at ideals and the unit indicesThe computation of characters a t id e a ls a re re d u c e d to th a t a t q u a d ra tic in te g e rs.T h e v a l u e o f t h e characters at q u a d ra tic in te g e rs a re w ritte n b y the L egendre symbols a n d th e H ilb e rt symbols over real quadratic fields and the H ilbert symbols a re d e te rm in e d b y p ro p is itio n 4 in § 2. T h e u n it in d ices are w ritte n b y th e H asse's unit indices w hich a re determined by proposition 14 in § 3.A fte r th e above tw o jo b s a re done, w e give som e tables of special values and relative class num bers.F o r this purpose, w e give a n algorithm for enum urating quadratic extensions of real quadratic fields in § 4 .T h e ta b le s a re g iv e n in § 5.T h e au th o r w ishes to express h is sin c e re th a n k s to P ro f.H . S a ito f o r his helpful advices.§ 1 .Preliminaries W e review som e results of [3] in the case of real quadratic fields w ith modifications.T h e following notations are used throughout th e paper except in the argum ent on the