The behavior of Hecke's L-function of real quadratic fields at s=0
aa r X i v : . [ m a t h . N T ] N ov THE BEHAVIOR OF HECKE’S L-FUNCTION OF REALQUADRATIC FIELDS AT s = 0 BYUNGHEUP JUN AND JUNGYUN LEE
Abstract.
For a family of real quadratic fields { K n = Q ( p f ( n )) } n ∈ N ,a Dirichlet character χ modulo q and prescribed ideals { b n ⊂ K n } ,we investigate the linear behaviour of the special value of partialHecke’s L-function L K n ( s, χ n := χ ◦ N K n , b n ) at s = 0. We showthat for n = qk + r , L K n (0 , χ n , b n ) can be written as112 q ( A χ ( r ) + kB χ ( r )) , where A χ ( r ) , B χ ( r ) ∈ Z [ χ (1) , χ (2) , · · · , χ ( q )] if a certain conditionon b n in terms of its continued fraction is satisfied. Furthermore,we write precisely A χ ( r ) and B χ ( r ) using values of the Bernoullipolynomials. We describe how the linearity is used in solving classnumber one problem for some families and recover the proofs insome cases. Finally, we list some families of real quadratic fieldswith the linearity. Contents
1. Introduction 2Acknowledgment 4Notations and conventions 42. Partial Hecke L -function 62.1. Shintani-Zagier cone decomposition 83. Proof of the main theorem 133.1. Plan of the proof 143.2. Periodicity and invariance 183.3. Summations 203.4. End of the proof 234. Bir´o’s method 245. A generalization 26References 26 The work of the first named author was supported by KRF-2007-341-C00006. Introduction
In this paper, we are mainly concerned with linear behaviour of thespecial values of Hecke’s L -function at s = 0 for families of real qua-dratic fields.Let { K n = Q ( p f ( n )) } n ∈ N be a family of real quadratic fields where f ( n ) is a positive square free integer for each n . For example f ( x ) canbe a polynomial with integer coefficients.For a Dirichlet character χ modulo q , we have a ray class character χ n := χ ◦ N K n for each n . Fixing an ideal b n in K n for each n , oneobtains an indexed family of partial Hecke L-functions { L K n ( s, χ n , b n ) } ,where the partial Hecke’s L-function for ( K, χ, b ) is defined as L K ( s, χ, b ) := X a ∼ b integral( q, a )=1 χ ( a ) N ( a ) − s . and a ∼ b means that a = α b for totally positive α ∈ K .Roughly speaking, if L K n (0 , χ n , b n ) can be written as linear polyno-mial in k with coefficients depending only on r for n = qk + r , we saythat L K n (0 , χ n , b n ) is linear. Definition 1.1 (Linearity) . When the special values of L K n ( s, χ n , b n ) at s = 0 is expressed as L K n (0 , χ n , b n ) = 112 q ( A χ ( r ) + kB χ ( r )) for n = qk + r , A χ ( r ) , B χ ( r ) ∈ Z [ χ (1) , χ (2) , · · · χ ( q )] , we say that L K n (0 , χ n , b n ) is linear . The “linearity” is originally observed by Bir´o in his proof of Yokoi’sconjecture([2]).
Theorem 1.2 (Yokoi’s conjecture solved by Bir´o) . If the class numberof Q ( √ n + 4) is then n ≤ . In Yokoi’s conjecture, we take K n = Q ( √ n + 4) and b n = O K n . Inpage 88, 89 of [2], Bir´o expressed the special value of Hecke’s L -functionfor ( K n , χ n , O K n ) at s = 0 for n = qk + r (1) L K n (0 , χ n , b n ) = 1 q ( A χ ( r ) + kB χ ( r )) , PECIAL VALUES OF HECKE’S L-FUNTION 3 where A χ ( r ) = X ≤ C,D ≤ q − χ ( D − C − rCD ) l rC − Dq m ( C − q ) ,B χ ( r ) = X ≤ C,D ≤ q − χ ( D − C − rCD ) C ( C − q ) . When K n is of class number 1, the unique ideal class can be repre-sented by any ideal b n . A priori the partial Hecke L -function equalsthe total Hecke L -function up to multiplication by 2(ie. L K n (0 , χ n ) = cL K n (0 , χ n , O K n )for c the number of narrow ideal classes).From this identification, one can find the residue of n by sufficientlymany primes p for which the class number of Q ( √ n + 4) is one. More-over, from the linearity, this residue depends only on r . Consequently,one can tell whether p inerts or not in Q ( √ n + 4). As we have abound for a smaller prime to inert depending on n , finally we haveenough conditions to list all K n of class number 1.Later in diverse works of Bir´o, Byeon, Kim and the second namedauthor ([3],[7],[8],[12],[11]), other families ( K n , χ n , b n ) that has linearityhave been discovered. Similarly, developing Biro’s method, one cansolve the associated class number one problems.In this paper, we give a criterion for ( K n , χ n , b n ) to have the linearityof the values L K n (0 , χ n , b n ) in terms of the continued fraction expres-sion of δ ( n ) where b − n = [1 , δ ( n )] := Z + δ ( n ) Z . Let [[ a , a , . . . , a n ]]be the purely periodic minus continued fraction[ a , a , a , . . . , a n , a , a , . . . ] , where [ a , a , a , . . . ] := a + 1 a + 1 a + · · · . Our main theorem is as follows:
Theorem 1.3 (Linearity Criterion) . Let { K n = Q ( p f ( n )) } n ∈ N be afamily of real quadratic fields where f ( n ) is a positive square free integerfor each n . Let χ be a Dirichlet character modulo q for a positive integer q and χ n be a ray class character modulo q defined by χ ◦ N K n . Suppose b n is an integral ideal relatively prime to q such that b − n = [1 , δ ( n )] .Assume the continued fraction expansion of δ ( n ) − δ ( n ) − a ( n ) , a ( n ) , · · · , a s − ( n )]] BYUNGHEUP JUN AND JUNGYUN LEE is purely periodic and of a fixed length s independent of n and a i ( n ) = α i n + β i for some fixed α i , β i ∈ Z .If N K n ( b n ( C + Dδ ( n ))) modulo q is a function only depending on C , D and r for n = qk + r , then L K n (0 , χ n , b n ) is linear. Furthermore, we give a precise description of A χ ( r ) and B χ ( r ) usingvalues of the Bernoulli polynomials (Proposition 3.8). From this de-scription, for n with h ( K n ) = 1, as in Bir´o’s case, one can compute theresidue of n modulo p depending on the mod- q residue r of n . Thereare possibly many ( q, p ) pairs. The more pairs of ( q, p ) we have, themore we can restrict possible n . There are known many families ofwhich class number one problem can be solved in this way. Many ofknown results can be recovered by ensuring the linearity from continuedfraction expansion and finding enough ( q, p ).There are still other families of real quadratic fields with linearitywhose class number one problems are not yet answered. Morally, oncewe obtain reasonable class number one criterion, finding sufficientlymany ( q, p )-pairs should solve it.This paper is composed as follows. In Section 2, we describe thespecial value at s = 0 of the partial Hecke L-function in terms of valuesof the Bernoulli polynomials. Ssection 3 is devoted to the proof of ourmain theorem. In Section 4, Bir´o’s method is sketched as a prototypeto apply the linearity. Finally in Section 5, we finish this paper with apossible generalization of the linearity criterion to polynomial of higherorder. Acknowledgment.
We would like to thank Prof. Dongho Byeon forhelpful comments and discussions. We also thank the anonymous ref-eree for careful reading and many invaluable suggestions. The firstnamed author wishes to thank Prof. Bumsig Kim and Prof. Soon-YiKang for warm supports and encouragements.
Notations and conventions
Throughout this article, we keep the following general notations andconventions. If we find it necessary, we rewrite the notations in concreteterms at the place where it is used.(1) K is a real quadratic field.(2) For a real quadratic field K , we fix an embedding ι : K → R . Ifthere is no danger of confusion, we denote ι ( α ) by an element α ∈ K . α ′ denotes the conjugate of α as well as ι ( α ′ ).(3) For α ∈ K , N K ( α ) denotes the norm of α over Q . If there is nodanger of confusion, we simply write N ( α ) to denote N K ( α ). PECIAL VALUES OF HECKE’S L-FUNTION 5
For an integral ideal a of K , N ( a ) denotes the norm of a definedto be [ o K , a ].(4) For two linearly independent elements α, β ∈ K as a vectorspace over Q , [ α, β ] denotes the lattice (ie. free abelian group)generated by α and β . A fractional ideal a of K seen as a latticeis denoted by [ α, β ] if { α, β } is a free basis of a .(5) For a subset A of K , we denote A + the set of totally positiveelements in A .(6) χ is a fixed Dirichlet character of modulus q .(7) For a real number x , h x i := ( x − [ x ] , for x Z , for x ∈ Z Equivalently, h−i is the unique composition R mod Z −−−→ R / Z → R that is identity on (0 , x , [ x ] := x − h x i .(9) For an integer m , h m i q denotes the residue of m in [1 , q ] by q (ie. m = qk + h m i q for k ∈ Z , h m i q ∈ [1 , q ] ∩ Z .).(10) [ a , a , a , .... ] for positive integers a i denotes the usual contin-ued fraction:[ a , a , a , . . . ] := a + 1 a + 1 a + · · · [ a , a , . . . , a i − , a i , a i +1 , . . . , a i + j ] denotes the continued fractionwith periodic part ( a i , a i +1 , . . . , a i + j ).[[ a , a , . . . , a n ]] is the purely periodic continued fraction[ a , a , . . . , a n , a , a , . . . ] . (11) ( a , a , a , . . . ) denotes the minus continued fraction:( a , a , a , . . . ) := a − a − a − · · · (( a , a , . . . , a n )) is the purely periodic minus continued frac-tion: ( a , a , a , . . . , a n , a , a , . . . )(12) For an integer s , µ ( s ) = 1(resp. ) if s is odd(resp. even). BYUNGHEUP JUN AND JUNGYUN LEE Partial Hecke L -function Throughout this section, K denotes a real quadratic field and b is afixed integral ideal of K relatively prime to q .A ray class character modulo q is a homomorphism χ : I K ( q ) /P K ( q ) → C ∗ , where I K ( q ) is a group of fractional ideals of K which is relatively primeto q and P K ( q ) is a subgroup of principal ideals ( α ) for totally positive α ≡ q ) . Throughout this section, b is an integral ideal such that b − = [1 , δ ]for δ ∈ K satisfying 0 < δ ′ < δ > F := { ( C, D ) ∈ Z | ≤ C, D ≤ q − , (( C + Dδ ) b , q ) = 1 } . Let E + (resp. E + q ) be the set of totally positive units (resp. the setof totally positive units congruent to 1 mod q ) in K . Then E + acts onthe set F by the rule ǫ ∗ ( C + Dδ ) = C ′ + D ′ δ where ǫ · ( C + Dδ ) + q b − = C ′ + D ′ δ + q b − for ǫ ∈ E + . Lemma 2.1. ( C, D ) in F is fixed by the action of ǫ if and only if ǫ isin E + q .Proof. ( C, D ) is fixed by ǫ ∈ E + if and only if ( C + Dδ )( ǫ − ∈ q b − . Since ( b ( C + Dδ ) , q ) = 1, the condition ( C + Dδ )( ǫ − ∈ q b − isequivalent to ǫ ≡ q ) . (cid:3) Lemma 2.2.
Suppose ≤ C, D ≤ q − . Then the following areequivalent: (1) ( C, D ) is in F . (2) For every α ∈ C + Dδq + b − , the ideal qα b is relatively prime to q . (3) For a α ∈ C + Dδq + b − , the ideal qα b is relatively prime to q .Proof. Suppose that ( q, ( C + Dδ ) b ) = 1.We have qαC + Dδ ∈ qC + Dδ b − for α ∈ C + Dδq + b − . Thus ( q, b ( C + Dδ )) = 1 implies that qαC + Dδ ≡ q ) . Since q b α = b ( C + Dδ ) qαC + Dδ ,
PECIAL VALUES OF HECKE’S L-FUNTION 7 we have ( q b α, q ) = 1 . If ( q, ( C + Dδ ) b ) = 1, then ( q, q b α ) = 1 for α ∈ C + Dδq + b − , sincefor α ∈ C + Dδq + b − , we have q b α ⊂ ( C + Dδ ) b + qO K . (cid:3) Let F ′ = F/E + be the orbit space by the action of E + on F . Let˜ F ′ a fundamental set of F ′ . Let ǫ be the totally positive fundamentalunit. The order of the action of ǫ is λ := [ E + : E + q ] by Lemma 2.1.Then we can decompose F as follows:(2) F = λ − G i =0 ǫ i ˜ F ′ . According to this decomposition of F , we can decompose further thepartial Hecke’s L -function: Proposition 2.3.
Let q be a positive integer. For an ideal b ⊂ K relatively prime to q and a ray class character χ modulo q , we have L K ( s, χ, b ) = X a ∼ b integral ( q, a )=1 χ ( a ) N ( a ) − s = X ( C,D ) ∈ ˜ F ′ χ (( C + Dδ ) b ) X α ∈ ( C + Dδq + b − ) + /E + q N ( q b α ) − s . Proof.
For α , α ∈ ( q − b − ) + , qα b = qα b if and only if α /α ∈ E + .So we have X a ∼ b integral ( q, a )=1 χ ( a ) N ( a ) s = X a ∼ q b integral ( q, a )=1 χ ( a ) N ( a ) s = X α ∈ ( q − b − ) + /E + ( q,qα b )=1 χ ( qα b ) N ( qα b ) s We also have for a totally positive fundamental unit ǫ > X α ∈ ( q − b − ) + /E + q ( q,q b α )=1 χ ( q b α ) N ( q b α ) s = X α ∈ ( q − b − ) + /E + ( q,q b α )=1 λ − X i =0 χ ( q b αǫ i ) N ( q b αǫ i ) s = λ · X α ∈ ( q − b − ) + /E + ( q,q b α )=1 χ ( q b α ) N ( q b α ) s . BYUNGHEUP JUN AND JUNGYUN LEE
And from Lemma 2.2, we have X α ∈ ( q − b − ) + /E + q ( q,q b α )=1 χ ( q b α ) N ( q b α ) s = X ( C,D ) ∈ F X α ∈ ( C + Dδq + b − ) + /E + q ( q,q b α )=1 χ ( q b α ) N ( q b α ) s = X ( C,D ) ∈ F X α ∈ ( C + Dδq + b − ) + /E + q χ ( q b α ) N ( q b α ) s . By equation (2), the above is equal to X ( C,D ) ∈ ˜ F ′ λ − X i =0 X α ∈ ( ( C + Dδ ) ǫiq + b − ) + /E + q χ ( q b α ) N ( q b α ) s . Since X α ∈ ( ( C + Dδ ) ǫiq + b − ) + /E + q χ ( q b α ) N ( q b α ) s = X α ∈ ( ( C + Dδ ) q + b − ) + /E + q χ ( q b αǫ i ) N ( q b αǫ i ) s , the above also equal to λ · X ( C,D ) ∈ ˜ F ′ X α ∈ ( C + Dδq + b − ) + /E + q χ ( q b α ) N ( q b α ) s . Note that for α ∈ ( C + Dδq + b − ) + , q b α and ( C + Dδ ) b are in the sameray class modulo q . Thus χ ( q b α ) = χ (( C + Dδ ) b ). This completes theproof. (cid:3) Shintani-Zagier cone decomposition.
We review briefly thedecomposition of ( R ) + into cones due to Shintani and Zagier in [18],[19], [20]. This depends on a real quadratic field K and a fixed ideal a inside. Here for the sake of computation, we fix a = b − where b is setas in the beginning of this section. K is embedded into R by ι = ( τ , τ ), where τ , τ are two realembeddings of K . Especially the totally positive elements of K landson ( R ) + . We are going to describe the fundamental domain of ( C + Dδq + b − ) + /E + q embedded into ( R ) + .The multiplicative action of E q + on K + induces an action on ( R ) + by coordinate-wise multiplication: ǫ ◦ ( x, y ) = ( τ ( ǫ ) x, τ ( ǫ ) y ) . A fundamental domain D R of ( R ) + /E + q is given by(3) D R := { xι (1) + yι ( ǫ − λ ) | x > , y ≥ } ⊂ ( R ) + PECIAL VALUES OF HECKE’S L-FUNTION 9 where E + q = (cid:10) ǫ λ (cid:11) for an integer λ and ǫ > ι ( b − ) ∩ ( R ) + in ( R ) + , the verticeson the boundary are { P i } i ∈ Z for P i ∈ ι ( b − ) and determined by theinequalities that P = ι (1) , P − = ι ( δ ) and x ( P i ) < x ( P i − ) where x ( P k ) denotes the first coordinate of P k for k ∈ Z . Since any twoconsecutive boundary points make a basis of ι ( b − ), we find that (cid:18) − b i (cid:19) (cid:18) P i − P i (cid:19) = (cid:18) P i P i +1 (cid:19) , for an integer b i . It is easy to see that b i ≥ x ( P i − ) + x ( P i +1 ) = b i x ( P i ) . Put δ i := x ( P i − ) x ( P i ) >
1. Note that δ = δ . δ i satisfies a recursiverelation: δ i = b i − δ i +1 , for i ∈ Z . Therefore δ i = b i − b i +1 − b i +2 − · · · = ( b i , b i +1 , b i +2 · · · ) . Let ǫ > ǫ moves a bound-ary point to another boundaty point preserving the order. Thus wehave(5) ǫ ◦ P i = P i − m , for a positive integer m . Therefore we obtain the following proposition. Proposition 2.4. (1) δ i + m = δ i for all i ∈ Z . (2) δ i = (( b i , b i +1 , · · · , b i + m − )) = b i − b i +1 − · · · b i + m − − b i − · · · . (3) ι ( ǫ − ) = P m (4) ǫ − ◦ P i = P i + m (5) ι ( ǫ − γ ) = P γm Proof. (1) δ i + m = x ( P i + m − ) x ( P i + m ) = ǫx ( P i − ) ǫx ( P i ) = δ i .(2) This is an immediate consequence of 1.(3) From Eq. (5), P m = ǫ − ◦ P . Since P = ι (1) and ǫ − ◦ ι (1) = ι ( ǫ − ).(4) This is immediate from (5).(5) It is trivial from (3) and (4). (cid:3) From (3) and (4) of Proposition2.4, D R the fundamental domain( R ) + /E + q is further decomposed into ( λ · m )-disjoint union of smallercones: D R = λm G i =1 { xP i − + yP i | x > , y ≥ } . Obviously the fundamental set of the quotient ( ι ( C + Dδq + b − ) T ( R ) + ) /E + q inside D R , which we denote by D is given by a disjoint union: D := λm G i =1 (cid:16) ι ( C + Dδq + b − ) \ { xP i − + yP i | x > , y ≥ } (cid:17) . Since { P i − , P i } is a Z -basis of ι ( b − ), there is a unique ( x iC + Dδ , y iC + Dδ ) ∈ (0 , × [0 ,
1) such that x iC + Dδ P i − + y iC + Dδ P i ∈ ι ( C + Dδq + b − ) , for each i, C, D ∈ Z . Thus ι (cid:0) C + Dδq + b − (cid:1) \ { xP i − + yP i | x > , y ≥ } = { ( x iC + Dδ + n ) P i − + ( y iC + Dδ + n ) P i | n , n ∈ Z ≥ } . (6)In [16], Yamamoto found a recursive relation satisfied by ( x iC + Dδ , y iC + Dδ ): x i +1 C + Dδ = h b i x iC + Dδ + y iC + Dδ i ,y i +1 C + Dδ = 1 − x iC + Dδ , (7)where h·i is as defined at the end of the introduction. (ie. h x i = x − [ x ](resp. 1) for x Z (resp. for x ∈ Z )).((2.1.3) of loc. sit. ).Let A i := x ( P i ) for all i ∈ Z . Then from Eq.(6), we obtain thefollowing: X α ∈ ( C + Dδq + b − ) + /E + q N ( α ) s = λm X i =1 X n ,n ≥ N (( x iC + Dδ + n ) A i − + ( y iC + Dδ + n ) A i ) − s = λm X i =1 X n ,n ≥ N (( x iC + Dδ + n ) δ i + ( y iC + Dδ + n )) − s A − si . (8) PECIAL VALUES OF HECKE’S L-FUNTION 11
In [19], Shintani evaluated P n ,n ≥ N (( x + n ) δ + ( y + n )) − s atnonpositive integers. In particular, the value at s = 0 is expressed byfirst and second Bernoulli polynomials as follows: Lemma 2.5 (Shintani) . X n ,n ≥ N (( x + n ) δ + ( y + n )) − s (cid:12)(cid:12)(cid:12) s =0 = δ + δ ′ B ( x ) + B ( x ) B ( y ) + 14 ( 1 δ + 1 δ ′ ) B ( y ) . Using this, we have X α ∈ ( C + Dδq + b − ) + /E + q N ( α ) s (cid:12)(cid:12)(cid:12) s =0 = λm X i =1 δ i + δ ′ i B ( x iC + Dδ ) + B ( x iC + Dδ ) B ( y iC + Dδ ) + 14 ( 1 δ i + 1 δ ′ i ) B ( y iC + Dδ )(9)Moreover, Yamamoto in the proof of Theorem 4.1.1 of [16] simplifiedthe above: Lemma 2.6 (Yamamoto) . λm X i =1 δ i + δ ′ i B ( x iC + Dδ ) + 14 ( 1 δ i + 1 δ ′ i ) B ( y iC + Dδ )= λm X i =1 b i B ( x iC + Dδ )Finally, we have X α ∈ ( C + Dδq + b − ) + /E + q N ( α ) s (cid:12)(cid:12)(cid:12) s =0 = λm X i =1 B ( x iC + Dδ ) B ( y iC + Dδ ) + b i B ( x iC + Dδ )(10) Lemma 2.7.
Let ǫ be the totally positive fundamental unit of K and λ := [ E + : E + q ] . Then we have x mi + jC + Dδ = x jǫ i ∗ ( C + Dδ ) and y mi + jC + Dδ = y jǫ i ∗ ( C + Dδ ) , for j = 0 , , , · · · , m − . Proof.
From (4) of Proposition 2.4, we have A mi + j = ǫ − i A j , for any integer i .Thus x mi + jC + Dδ A mi + j − + y mi + jC + Dδ A mi + j = x mi + jC + Dδ ǫ − i A j − + y mi + jC + Dδ ǫ − i A j ∈ C + Dδq + b − . Therefore, x mi + jC + Dδ A j − + y mi + jC + Dδ A j ∈ ǫ i · ( C + Dδ ) q + b − . (cid:3) From Lemma 2.7 and the periodicity of b i , we have Lemma 2.8. X α ∈ ( C + Dδq + b − ) + /E + q N ( α ) s (cid:12)(cid:12)(cid:12) s =0 = m X i =1 λ − X j =0 B ( x iǫ j ∗ ( C + Dδ ) ) B ( y iǫ j ∗ ( C + Dδ ) ) + b i B ( x iǫ j ∗ ( C + Dδ ) ) . Finally, we have
Proposition 2.9.
For a ray class character χ modulo q and an ideal b of K such that b − = [1 , δ ] for δ ∈ K with δ > and < δ ′ < , we have L K (0 , χ, b )= X ≤ C,D ≤ q χ (( C + Dδ ) b ) m X i =1 B ( x i ( C + Dδ ) ) B ( y iC + Dδ ) + b i B ( x iC + Dδ ) Proof.
From Proposition 2.3, we obtain L K (0 , χ, b )= X ( C,D ) ∈ ˜ F ′ χ (( C + Dδ ) b ) X α ∈ ( C + Dδq + b − ) + /E + q N ( q b α ) − s | s =0 . PECIAL VALUES OF HECKE’S L-FUNTION 13
Lemma 2.8 implies that the above is equal to X ( C,D ) ∈ ˜ F ′ χ (( C + Dδ ) b ) λ − X j =0 m X i =1 B ( x iǫ j ∗ ( C + Dδ ) ) B ( y iǫ j ∗ ( C + Dδ ) )+ b i B ( x iǫ j ∗ ( C + Dδ ) ) . Since ( C + Dδ ) ǫ b = ( C + Dδ ) b , the above is expressed as follows X ( C,D ) ∈ ˜ F ′ λ − X j =0 χ (( C + Dδ ) ǫ j b ) m X i =1 B ( x iǫ j ∗ ( C + Dδ ) ) B ( y iǫ j ∗ ( C + Dδ ) )+ b i B ( x iǫ j ∗ ( C + Dδ ) ) . From(11) F = λ − G i =0 ǫ i ˜ F ′ , we find that the above equals to X ( C,D ) ∈ F χ (( C + Dδ ) b ) m X i =1 B ( x i ( C + Dδ ) ) B ( y i ( C + Dδ ) ) + b i B ( x i ( C + Dδ ) ) . If (( C + Dδ ) b , q ) = 1 then χ (( C + Dδ ) b ) = 0 . Thus we complete theproof. (cid:3)
Remark . It is important to note that the summation running over
C, D ∈ [1 , q ] is actually supported on F . This is justified by the twistof the mod q Dirichlet character. Obviously, F depends on δ in K , butthe twisted sum has invariant form of δ and K . This is a subtle pointin the proof of the main theorem as we deal with family of the Hecke’s L -values with respect to a family ( K n , χ n , b ).3. Proof of the main theorem
In this section, we compute the special values of Hecke’s L-functionfor a family of real quadratic fields. The computation is made usingthe expression of the L-value in the previous section. After the compu-tation, it will be apparent that the linearity property comes sufficientlyfrom the shape of the continued fractions in the family. This will com-plete the proof of Theorem 1.3.This gives a criterion that will recover several approaches of classnumber problems for some families of real quadratic fields.Consider a family of real quadratic fields K n = Q ( √ d n ), where d n is a positive square free integer. For a fixed Dirichlet character χ ofmodulus q , we associate a ray class character χ n := χ ◦ N K n / Q for each n . Let us fix an ideal b n of K n for each n . Then we have a family ofthe Hecke’s L-functions associated to ( K n , χ n , b n ): L K n ( s, χ n , b n ) = X a χ n ( a ) N ( a ) s where a runs over integral ideals a in the ray class represented by b n .3.1. Plan of the proof.
Assume that b − n = [1 , δ ( n )]with δ ( n ) > , < δ ( n ) ′ <
1. As discussed in Prop.2.4, δ ( n ) has apurely periodic minus continued fraction expansion: δ ( n ) =(( b ( n ) , b ( n ) , · · · , b m ( n ) − ( n )))= b ( n ) − b ( n ) − · · · b m ( n ) − ( n ) − b ( n ) − · · · (12)with b k ( n ) ≥ b i ( n ) for all i ∈ Z by requiring that b i + m ( n ) ( n ) = b i ( n ) for i ∈ Z . Let δ k ( n ) = (( b k ( n ) , b k +1 ( n ) , · · · , b k + m ( n ) − ( n )))and we define { A k ( n ) } k ∈ Z by A − ( n ) = δ ( n ) , A ( n ) = 1 , . . . , A k +1 ( n ) = A k ( n ) /δ k +1 ( n ) . Then for fixed
C, D and n , there is a unique ( x iC + Dδ ( n ) , y iC + Dδ ( n ) ) suchthat(13) 0 < x iC + Dδ ( n ) ≤ , ≤ y iC + Dδ ( n ) < , (14) x iC + Dδ ( n ) A i − ( n ) + y iC + Dδ ( n ) A i ( n ) ∈ C + Dδ ( n ) q + b − n , for each i ∈ Z , as described in the previous section. This ( x iC + Dδ ( n ) , y iC + Dδ ( n ) )satisfies Yamamoto’s recursive relation (7) as follows:(15) x i +1 C + Dδ ( n ) = h b i ( n ) x iC + Dδ ( n ) + y iC + Dδ ( n ) i , y i +1 C + Dδ ( n ) = 1 − x iC + Dδ ( n ) . Now we recall a standard conversion formula of a plus continuedfraction expansion to minus continued fraction expansion:
Lemma 3.1.
Let δ − be a purely periodic continued fraction: [[ a , a , · · · , a s − ]] . Then the minus continued fraction expansion of δ is (( b , b , · · · , b m − )) , PECIAL VALUES OF HECKE’S L-FUNTION 15 where b i := ( a j + 2 , for i = S j , otherwisefor S j = ( , for j = 0 S j − + a j − , for j ≥ and the period m = ( a + a + a · · · + a s − = S s , for even sa + a + a · · · + a s − = S s , for odd s Proof. (See page 177, 178 of [18]). Actually if s is an odd integer, theperiod m is s X i =1 a i − = a + a + · · · + a s − = S s . Since a i has period s , we find that a + 1 + · · · + a s − = a + a + a · · · + a s − = s − X i =0 a i . (cid:3) For the family of δ ( n ) ∈ K , we assumed that δ ( n ) − a ( n ) , a ( n ) , a ( n ) , . . . , a s − ( n )]] , has the same period for every n .Then δ ( n ) has purely periodic minus continued fraction expansion δ ( n ) = (( b ( n ) , b ( n ) , · · · , b m ( n ) − ( n )))with b i ( n ), S j ( n ) and m ( n ) defined by the same manner as in theprevious lemma.One should be aware that m ( n ) vary with n , while the period ofpositive continued fraction s is fixed.From Proposition 2.9 and recursive relation (15) of ( x iC + Dδ ( n ) , y iC + Dδ ( n ) ),we have L K n (0 , χ n , b n ) = X ≤ C,D ≤ q χ n (( C + Dδ ( n )) b n ) m ( n ) X i =1 (cid:0) B ( x iC + Dδ ( n ) ) B ( y iC + Dδ ( n ) ) + b i ( n )2 B ( x iC + Dδ ( n ) ) (cid:1) . (16) To check the linear behavior, it suffices to show that(17) m ( n ) X i =1 (cid:0) B ( x iC + Dδ ( n ) ) B ( y iC + Dδ ( n ) ) + b i ( n )2 B ( x iC + Dδ ( n ) ) (cid:1) is linear in k with the coefficients determined only by r .Because b i ( n ) = 2 if i = S j ( n ) for some j , we can divide the aboveinto two parts: sµ ( s ) X l =1 (cid:0) − B ( x S l ( n ) C + Dδ ( n ) ) B ( x S l ( n ) − C + Dδ ( n ) ) + a l ( n ) + 22 B ( x S l ( n ) C + Dδ ( n ) ) (cid:1) + sµ ( s ) − X l =0 S l +1 ( n ) − X i = S l ( n )+1 F ( x iC + Dδ ( n ) , x i − C + Dδ ( n ) )(18)where µ ( s ) = or 1 for s even or odd, respectively, and F ( x, y ) := − B ( x ) B ( y ) + B ( x ).If C, D are fixed and there is no danger of misunderstand, x i ( n ) willsimply mean x iC + Dδ ( n ) .Below is the behavior of x i ( n ), when n varies. The proof will begiven later.1. { x i ( n ) } S j ( n ) ≤ i ≤ S j +1 ( n ) is an arithmetic progression mod Z with com-mon difference (cid:10) x S j ( n )+1 ( n ) − x S j ( n ) ( n ) (cid:11) .2. { x i ( n ) } S j ( n ) ≤ i ≤ S j +1 ( n ) has period q .3. x S j ( n ) ( n ), x S j ( n ) − ( n ) and x S j ( n )+1 ( n ) are invariant as k varies for n = qk + r .In short, { x i ( n ) } is a ‘piecewise arithmetic progression’.As we have constrained that a i ( n ) = α i n + β i , h a i ( n ) i q is independentof k for n = qk + r but depends only on i and r .Define γ i ( r ) as follows:(19) γ i ( r ) := h a i ( n ) i q PECIAL VALUES OF HECKE’S L-FUNTION 17
Then actually γ i ( r ) is h a i ( r ) i q . Since { F ( x i ( n ) , x i − ( n )) } S j ( n )+1 ≤ i ≤ S j +1 ( n ) − has period q from 2 above, we obtain S l +1 ( n ) − X i = S l ( n )+1 F ( x i ( n ) , x i − ( n ))= S l ( n )+ γ l +1 ( r ) − X i = S l ( n )+1 F ( x i ( n ) , x i − ( n )) + κ l +1 ( n ) S l ( n )+ q X i = S l ( n )+1 F ( x i ( n ) , x i − ( n ))(20)where a i ( n ) = κ i ( n ) q + γ i ( r ) for an integer κ i ( n ). Written precisely,(21) κ i ( n ) = a i ( n ) − γ i ( r ) q . Since α i r + β i = qτ i ( r ) + γ i ( r )for some integer τ i ( r ), we can write for n = qk + r (22) κ i ( n ) = kα i + τ i ( r )Using 3, x S l ( n ) ( n ) and x S l ( n )+1 ( n ) are sufficiently determined by theresidue r of n by q . A priori the summations P S l ( n )+ γ l +1 ( r ) − i = S l ( n )+1 F ( x i ( n ) , x i − ( n ))and P S l ( n )+ qi = S l ( n )+1 F ( x i ( n ) , x i − ( n )) are completely determined by x S l ( n ) ( n )and x S l ( n )+1 ( n ) and remain unchanged while k varies.Thus we conclude first thatI. For n = qk + r , P S l +1 ( n ) − i = S l ( n )+1 F ( x i ( n ) , x i − ( n )) is linear function of k .Using (21) and (22), we have − B ( x S l ( n ) ( n )) B ( x S l ( n ) − ( n )) + a l ( n ) + 22 B ( x S l ( n ) ( n ))= − B ( x S l ( n ) ( n )) B ( x S l ( n ) − ( n )) + α l qk + τ l ( r ) q + γ l ( r ) + 22 B ( x S l ( n ) ( n ))(23)Again after 3 we conclude thatII. For n = qk + r , − B ( x S l ( n ) ( n )) B ( x S l ( n ) − ( n )) + a l ( n )+22 B ( x S l ( n ) ( n ))is linear function on k .Additionally, we haveIII. s and µ ( s ) is independent of n . Altogether I,II and III imply the linearity of P m ( n ) i =1 − B ( x i ( n )) B ( x i − ( n ))+ b i ( n )2 B ( x i ( n )) in k and the coefficients are function in r for fixed C, D .In sequal, we will clarify the properties 1, 2, 3 of { x i ( n ) } . Also wegive precise description P m ( n ) i =1 − B ( x i ( n )) B ( x i − ( n )) + b i ( n )2 B ( x i ( n ))that will finish the proof of Theorem 1.3.3.2. Periodicity and invariance.
In this section, we will prove theproperties 1, 2, 3 of { x i ( n ) } in the previous section. Proposition 3.2.
For j ≥ , { x i ( n ) } S j ( n ) ≤ i ≤ S j +1 ( n ) is an arithmeticprogression mod Z with common difference h x S j ( n )+1 ( n ) − x S j ( n ) ( n ) i .Proof. Since b i ( n ) = 2 for S j ( n ) + 1 ≤ i ≤ S j +1 ( n ) −
1, we have that x i +1 ( n ) = h x i ( n ) − x i − ( n ) i . It implies that for S j ( n ) + 1 ≤ i ≤ S j +1 ( n ) − h x i +1 ( n ) − x i ( n ) i = hh x i ( n ) − x i − ( n ) i − x i ( n ) i = h x i ( n ) − x i − ( n ) i . (cid:3) Lemma 3.3.
For i ≥ − , we have (1) qx i ( n ) ∈ Z . (2) 0 < x i ( n ) ≤ . Proof.
Since A ( n ) = 1, A − ( n ) = δ ( n ) , from (13),(14) and (15) wefind that x ( n ) = (cid:28) Dq (cid:29) , x − ( n ) = 1 − Cq .
We also note that b i ( n ) ∈ Z for any i ≥
0. Thus (15) implies abovelemma. (cid:3)
Proposition 3.4.
For j ≥ and a j +1 ( n ) ≥ q , { x i ( n ) } S j ( n ) ≤ i ≤ S j +1 ( n ) has period q . Explicitly we have x S j ( n )+ q + i ( n ) = x S j ( n )+ i ( n ) for ≤ i ≤ a j +1 ( n ) − q. Proof.
Note that { x i ( n ) mod 1 } S j ( n ) ≤ i ≤ S j +1 ( n ) is an arithemetic progres-sion. Thus we have x S j ( n )+ q + i ( n ) = h x S j ( n )+ i ( n ) + q h x S j ( n )+ i ( n ) − x S j ( n )+ i − ( n ) ii , for 0 ≤ i ≤ a j +1 ( n ) − q. From Lemma 3.3, we find that q h x S j ( n )+ i ( n ) − x S j ( n )+ i − ( n ) i ∈ Z . Thus h x S j ( n )+ i ( n ) + q h x S j ( n )+ i ( n ) − x S j ( n )+ i − ( n ) ii = h x S j ( n )+ i ( n ) i . PECIAL VALUES OF HECKE’S L-FUNTION 19
Since 0 < x S j ( n )+ i ( n ) ≤ , we finally have that h x S j ( n )+ i ( n ) i = x S j ( n )+ i ( n ) . (cid:3) For 0 ≤ r ≤ q − , we defineΓ j ( r ) := ( , for j = 0Γ j ( r ) + γ j − ( r ) , for j ≥ , where γ i ( r ) is defined as in (19). For i ≥
0, we put c i ( r ) = ( γ j ( r ) + 2 , for i = Γ j ( r )2 , otherwiseConsider a sequence { ν iCD ( r ) } i ≥− with the initial value and the re-cursive relation as follows: ν − CD ( r ) = q − Cq , ν CD ( r ) = h Dq i and ν i +1 CD ( r ) = h c i ( r ) ν iCD ( r ) − ν i − CD ( r ) i . If C, D are fixed and clear from the context, we omit the subscriptand abbreviate ν iCD ( r ) to ν i ( r ) . Proposition 3.5.
With the notations above, for j ≥ and n = qk + r ,we have x S j ( n )+ i ( n ) = ν Γ j ( r )+ i ( r ) for ≤ i ≤ γ j +1 ( r ) Proof.
We use induction on j .When j = 0 . S ( n ) = Γ ( r ) = 0. We need to show x i ( n ) = ν i ( r ) for i ∈ [0 , γ ( r )]. As we have seen in the proof of lemma 3.3, x ( n ) = h Dq i = ν ( r ) , x − ( n ) = 1 − Cq = ν − ( r ) . Since a ( n ) − γ ( r ) ∈ q Z , using (15) and the recursive relation of ν i ( r ),one can easily check that x ( n ) = h ( a ( n ) + 2) h Dq i + Cq i = h ( γ ( r ) + 2) ν ( r ) − ν − ( r ) i = ν ( r )For 1 ≤ i ≤ γ ( r ) − x i ( n ) and ν i ( r ) satisfy the same recursiverelation x i +1 ( n ) = h x i ( n ) − x i − ( n ) i , ν i +1 ( r ) = h ν i ( r ) − ν i − ( r ) i . Thus we have x i ( n ) = ν i ( r ) for 0 ≤ i ≤ γ ( r ) . Now assume that the proposition holds true for j < j . From Propo-sition 3.4, we find that if a j − ( n ) ≥ q then(24) x S j − ( n )+ q + i ( n ) = x S j − ( n )+ i ( n ) for 0 ≤ i ≤ a j − ( n ) − q. Since a j − ( n ) − γ j − ( r ) ∈ q Z , we obtain x S j ( n ) − ( n ) = x S j − ( n )+ a j − ( n ) − ( n ) = x S j − ( n )+ γ j − ( r ) − ( n )= ν Γ j − r )+ γ j − ( r ) − ( r ) = ν Γ j ( r ) − ( r ) . and x S j ( n ) ( n ) = x S j − ( n )+ a j − ( n ) ( n )= x S j − ( n )+ γ j − ( r ) ( n ) = ν Γ j − r )+ γ j − ( r ) ( r ) = ν Γ j ( r ) ( r ) . Moreover from (15), we find that x S j ( n )+1 ( n ) = h ( a j ( n ) + 2) x S j ( n ) ( n ) − x S j ( n ) − ( n ) i = h ( γ j ( r ) + 2) ν Γ j ( r ) ( r ) − ν Γ j ( r ) − ( r ) i = ν Γ j ( r )+1 ( r ) . (25)Since for S j ( n ) + 1 ≤ i ≤ S j +1 ( n ) − x i +1 ( n ) = h x i ( n ) − x i − ( n ) i and for Γ j ( r ) + 1 ≤ i ≤ Γ j ( r ) + γ j +1 ( r ) − j +1 ( r ) − ,ν i +1 ( r ) = h ν i ( r ) − ν i − ( r ) i , we have x S j ( n )+ i ( n ) = ν Γ j ( r )+ i ( r ) for 0 ≤ i ≤ γ j +1 ( r ) . (cid:3) Summations.
In this section we express P m ( n ) i =1 − B ( x iC + Dδ ( n ) ) B ( x i − C + Dδ ( n ) )+ b i ( n )2 B ( x iC + Dδ ( n ) ) using { ν iCD ( r ) } . Lemma 3.6.
Let d l ( r ) := h ν Γ l ( r )+1 ( r ) − ν Γ l ( r ) ( r ) i and [ x ] := x − h x i .Then for ≤ γ ≤ q and n such that γ ≤ a l +1 ( n ) and n = qk + r , wehave S l ( n )+ γ X i = S l ( n )+1 ( x i ( n ) − x i − ( n )) = γd l ( r ) + (1 − d l ( r ))[ ν Γ l ( r ) ( r ) + d l ( r ) γ ] Proof.
Since 0 < x i ( n ) ≤
1, we have − < x i ( n ) − x i − ( n ) < . Thus x i ( n ) − x i − ( n ) = h x i ( n ) − x i − ( n ) i + ψ i ( n ) , PECIAL VALUES OF HECKE’S L-FUNTION 21 where ψ i ( n ) = ( − , x i ( n ) ≤ x i − ( n )0 , x i ( n ) > x i − ( n ) . As h x i +1 ( n ) − x i ( n ) i = hh x i ( n ) − x i − ( n ) i − x i ( n ) i = h x i ( n ) − x i − ( n ) i for S l ( n ) + 1 ≤ i ≤ S l +1 ( n ) −
1, we have h x i ( n ) − x i − ( n ) i = h x S l ( n )+1 ( n ) − x S l ( n ) ( n ) i = h ν Γ l ( r )+1 ( r ) − ν Γ l ( r ) ( r ) i = d l ( r ) . Hence we have x i ( n ) − x i − ( n ) = d l ( r ) + ψ i ( n ) . Thus we obtain S l ( n )+ γ X i = S l ( n )+1 ( x i ( n ) − x i − ( n )) = γd l ( r ) + (1 − d l ( r )) S l ( n )+ γ X i = S l ( n )+1 ψ i ( n ) . Note that P S l ( n )+ γi = S l ( n )+1 ψ i ( n ) equals the number of i ’s satisfying x i ( n ) ≤ x i − ( n ) for S l ( n ) + 1 ≤ i ≤ S l ( n ) + γ. Therefore S l ( n )+ γ X i = S l ( n )+1 ψ i ( n ) = [ x S l ( n ) ( n ) + d l ( r ) γ ] = [ ν Γ l ( r ) ( r ) + d l ( r ) γ ] . (cid:3) For simplicity, we let F ( x, y ) := − B ( x ) B ( y ) + B ( x ) = ( x −
12 )( 12 − y ) + x − x + 16 . Lemma 3.7. If l ≥ and a l +1 ( n ) ≥ q, S l ( n )+ q X i = S l ( n )+1 F ( x i ( n ) , x i − ( n )) = 112 (cid:20) (cid:18) qd l ( r ) +(1 − d l ( r ))[ ν Γ l ( r ) ( r )+ d l ( r ) q ] (cid:19) − q (cid:21) And if ≤ γ ≤ q − and a l +1 ( n ) ≥ γ , S l ( n )+ γ X i = S l ( n )+1 F ( x i ( n ) , x i − ( n ))= 112 (cid:20) (cid:18) γd l ( r ) + (1 − d l ( r ))[ ν Γ l ( r ) ( r ) + d l ( r ) γ ] + B ( x S l ( n )+ γ ( n )) − B ( x S l ( n ) ( n )) (cid:19) − γ (cid:21) where B ( x ) is the second Bernoulli polynomial. Proof.
We note that F ( x, y ) = 12 ( x − y ) −
112 + 12 ( B ( x ) − B ( y )) . Thus we have S l ( n )+ γ X i = S l ( n )+1 F ( x i ( n ) , x i − ( n ))= S l ( n )+ γ X i = S l ( n )+1 h
12 ( x i ( n ) − x i − ( n )) −
112 + 12 ( B ( x i ( n )) − B ( x i − ( n ))) i . We note that for 1 ≤ γ ≤ q − , S l ( n )+ γ X i = S l ( n )+1 B ( x i ( n )) − B ( x i − ( n )) = B ( x S l ( n )+ γ ( n )) − B ( x S l ( n ) ( n )) . and for γ = q from the periodicity of x i ( n ) we have S l ( n )+ q X i = S l ( n )+1 B ( x i ( n )) − B ( x i − ( n )) = 0 . (cid:3) Proposition 3.8.
Suppose δ ( n ) − a ( n ) , a ( n ) , · · · a s − ( n )]] , a i ( n ) = α i n + β i for α i , β i ∈ Z and a i ( r ) = qτ i ( r ) + γ i ( r ) for γ i ( r ) = h a i ( r ) i q .Let d lCD ( r ) := h ν Γ l ( r )+1 CD ( r ) − ν Γ l ( r ) CD ( r ) i . Then, for n = qk + r , we have m ( n ) X i =1 − B ( x iC + Dδ ( n ) ) B ( y iC + Dδ ( n ) )+ b i ( n )2 B ( x iC + Dδ ( n ) ) = 112 ( A CD ( r )+ kB CD ( r )) where A CD ( r ) := sµ ( s ) X l =1 − B ( ν Γ l ( r ) CD ( r )) B ( ν Γ l ( r ) − CD ( r )) + 6( a l ( r ) + 2) B ( ν Γ l ( r ) CD ( r ))+ sµ ( s ) − X l =0 h (cid:16) ( γ l +1 ( r ) − d lCD ( r ) + (1 − d lCD ( r ))[ ν Γ l ( r ) CD ( r ) + d lCD ( r )( γ l +1 ( r ) − + B ( ν Γ l +1 ( r ) − CD ( r )) − B ( ν Γ l ( r ) CD ( r )) (cid:17) − γ l +1 ( r ) + 1+ τ l +1 ( r ) (cid:16) qd lCD ( r ) + (1 − d lCD ( r ))[ ν Γ l ( r ) CD ( r ) + d lCD ( r ) q ] ) − q (cid:17)i PECIAL VALUES OF HECKE’S L-FUNTION 23 and B CD ( r ) := sµ ( s ) X l =1 qα l B ( ν Γ l ( r ) CD ( r ))+ sµ ( s ) − X l =0 α l +1 (cid:16) qd lCD ( r ) + (1 − d lCD ( r ))[ ν Γ l ( r ) CD + d lCD ( r ) q ] ) − q (cid:17) (26) Proof.
From equation (18), we have m ( n ) X i =1 B ( x iC + Dδ ( n ) ) B ( y iC + Dδ ( n ) ) + b i ( n )2 B ( x iC + Dδ ( n ) )= sµ ( s ) X l =1 [ − B ( x S l ( n ) C + Dδ ( n ) ) B ( x S l ( n ) − C + Dδ ( n ) ) + α l qk + τ l ( r ) q + γ l ( r ) + 22 B ( x S l ( n ) C + Dδ ( n ) )]+ sµ ( s ) − X l =0 S l ( n )+ qα l +1 k + qτ l +1 ( r )+ γ l +1 ( r ) − X i = S l ( n )+1 F ( x iC + Dδ ( n ) , x i − C + Dδ ( n ) )From lemma 3.7, we have12 S l ( n )+ qα l +1 k + qτ l +1 ( r ) γ l +1 ( r ) − X i = S l ( n )+1 F ( x iC + Dδ ( n ) , x i − C + Dδ ( n ) ) = 12 S l ( n )+ γ l +1 ( r ) − X i = S l ( n )+1 F ( x iC + Dδ ( n ) , x i − C + Dδ ( n ) )+ 12( α l +1 k + τ l +1 ( r )) S l ( n )+ q X i = S l ( n )+1 F ( x iC + Dδ ( n ) , x i − C + Dδ ( n ) ) =6 (cid:16) ( γ l +1 ( r ) − d lCD ( r ) + (1 − d lCD ( r ))[ ν Γ l ( r ) CD ( r ) + d lCD ( r )( γ l +1 ( r ) − + B ( x S l ( n )+ γ l +1 − C + Dδ ( n ) ) − B ( x S l ( n ) C + Dδ ( n ) ) (cid:17) − ( γ l +1 ( r ) − α l +1 k + τ l +1 ( r )) (cid:16) qd lCD ( r ) + (1 − d lCD ( r ))[ ν Γ l ( r ) CD ( r ) + d lCD ( r ) q ] ) − q (cid:17) Since x S l ( n ) C + Dδ ( n ) = ν Γ l ( r ) CD ( r ), x S l ( n ) − C + Dδ ( n ) = ν Γ l ( r ) − CD ( r ) and x S l ( n )+ γ l +1 ( r ) − C + Dδ ( n ) = ν Γ l +1 ( r ) − CD , we complete the proof. (cid:3) End of the proof.
Proof.
Since ν Γ l ( r ) CD ( r ) , ν Γ l ( r ) − CD ( r ) and d lCD ( r ) are in q Z , we find that q A CD ( r ) , q B CD ( r ) ∈ Z . Moreover, we have L K n (0 , χ n , b n ) = 112 q X C,D χ n ( C + Dδ ( n ))( q A CD ( r ) + kq B CD ( r )) . Since χ is a Dirichlet character of modulus q , if n = qk + r , we canwrite χ n ( b n ( C + Dδ ( n ))) = F CD ( r ) , for a function F CD . Note if K r is defined, χ n ( b n ( C + Dδ ( n ))) = χ r ( b r ( C + Dδ ( r ))) = F CD ( r ). We warn the reader that the aboveexpression does not make sense if K r and δ ( r ) are undefined.If we set A χ ( r ) := X C,D F CD ( r ) q A CD ( r )and B χ ( r ) := X C,D F CD ( r ) q B CD ( r ) , we obtain the proof. (cid:3) Bir´o’s method
Let K n be a family of real quadratic fields such that special valueat s = 0 of the Hecke L -function has linearity. In [2] and [3], Bir´odeveloped a way to find the residue of n with h ( K n ) = 1 by certainprimes using the linearity. In this section, we sketch Bir´o’s method.Let K n = Q ( √ d ) for a square free integer d = f ( n ) and D n bethe discriminant K n . For an odd Dirichlet character χ : Z /q Z → C ∗ , χ n denotes the ray class character defined as χ n = χ ◦ N K n : I n ( q ) /P n ( q ) + → C ∗ . χ D = ( D · ) denotes the Kronecker character. Thenthe special value of the Hecke L -function at s = 0 has a factorization L K n (0 , χ n ) = L (0 , χ ) L (0 , χχ D n )= ( 1 q q X a =1 aχ ( a ))( 1 qD n qD n X b =1 bχ ( b ) χ D n ( b )) , (27)Let b n = O K n . Suppose that L K n (0 , χ n , b n ) is linear in the form: L K n (0 , χ n , b n ) = 112 q ( A χ ( r ) + kB χ ( r )) . for A χ ( r ) , B χ ( r ) ∈ Z [ χ (1) , χ (2) · · · χ ( q )]. Let ǫ n be the fundamentalunit of K n . Form Proposition 2.2 in [9], we find that L K n (0 , χ n , b n ) = PECIAL VALUES OF HECKE’S L-FUNTION 25 L K n (0 , χ n , ( ǫ n ) b n ) . Thus if the class number of K n is one then we havefor n = qk + r L K n (0 , χ n ) = c q ( A χ ( r ) + kB χ ( r ))where c is the number of narrow ideal classes.Then we have B χ ( r ) k + A χ ( r ) = 12 qc · ( q X a =1 aχ ( a )) · (cid:16) qD n qD n X b =1 bχ ( b ) χ D n ( b ) (cid:17) . Let L χ be the cyclotomic field generated by the values of χ . Since qD n P qD n b =1 bχ ( b ) χ D n ( b ) is integral in L χ , for a prime ideal I of L χ divid-ing ( P qa =1 aχ ( a )), we have B χ ( r ) k + A χ ( r ) ≡ I ) . And if I ∤ B χ ( r ) then k ≡ − A χ ( r ) B χ ( r ) (mod I ) . Since n = qk + r , we have n ≡ − q A χ ( r ) B χ ( r ) + r (mod I ) . Moerover if O L χ /I = Z /p Z , the residue of n modulo p is expressed onlyin A χ ( r ) , B χ ( r ) and r as above.Below we arrange all the necessary conditions of q and p .Condition(*)1. q : odd integer2. p : odd prime3. χ : character with conductor q I : prime ideal in L χ lying over pI | ( P qa =1 aχ ( a )) and O L χ /I = Z /p Z Note that the condition is independent of the family { K n } , once thelinearity holds.Let S be the set of ( q, p ) satisfying Condition(*). S = [ q :odd integer S q where S q := { ( q, p ) ∈ S } .Finally we remark that for ( q, p ) ∈ S we obtain the residue of n = qk + r modulo p for which the class number of K n is one. The above method has been applied to find an upper bound of thediscriminant of real quadratic fields with class number one in some fam-ilies of Richaud-degert type where the linearity criterion is satisfied(cf.[3], [2], [7], [12]). Together with properly developed class number onecriteria for each cases, the class number problems could be solved.It is easily checked that in fact the criterion is fulfilled by generalfamilies of Richaud-Degert type. Furthermore, there are still abundantexamples such families of real quadratic fields satisfying the linearitycriterion(cf. [14]). For these, we have controlled behavior of the specialvalues of Hecke’s L-function at s = 0 and Biro’s method is directlyapplied for each cases. We expect many other meaningful problems forfamily of real quadratic fields than class number problem in arithmeticcan be studied in this line.5. A generalization
We conclude this section with a possible generalization of the linear-ity of the special value of the Hecke L -function. This generalizationwill be dealt in a separate paper [10].As in the criterion for linearity, we set K n = Q ( p f ( n )) and b n is anintegral ideal of K n . We assume b − n = [1 , δ ( n )] , for δ ( n ) − a ( n ) , a ( n ) , . . . , a s ( n )]with a i ( x ) ∈ Z [ x ].For a given conductor q , write n = qk + r for r = 0 , , , . . . , q − N = max i { deg( a i ( x )) } , then we obtain that the special valueof the partial ζ -function of the ray class of b n mod q at s = 0 is writtenas ζ K n ,q (0 , ( C + Dδ ( n )) b n ) = 112 q (cid:0) A ( r ) + A ( r ) k + · · · + A N ( r ) k N (cid:1) for some rational integers A i depending only on r .We have no application of this property in arithmetic. It will bevery interesting if one applies this in similar fashion as Bir´o’s methodas presented here. References [1] Barnes, E.W.,
On the theory of the multiple gamma function , Tran. CambridgePhil. Soc., (1904), 374-425.[2] Bir´o, A., Yokoi’s conjecture , Acta Arith. (2003), 85-104.[3] Bir´o, A.,
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Class number 2 citeria for real quadratic fields ofRichaud-Degert type , J. Number Theory , 257-272 (1997).[5] Byeon, D. and Kim, H., Class number 1 criteria for real quadratic fields ofRichaud-Degert type , J. Number Theory , 328-339 (1996).[6] Chowla, S. and Friedlandler, J., Class numbers and quadratic residues,
GlasgowMath. J. (1976), 47–52.[7] Byeon, D., Kim, M. and Lee, J., Mollin’s conjecture , Acta Arithmetica (2007)[8] Byeon, D. and Lee, J.,
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A complete determination of Rabinowitch polynomials ,preprint 20 pages[10] Jun, B. and Lee, J.,
Special values of partial zeta functions of real quadraticfields at s = 0, in preparation.[11] Lee, J., The complete determination of narrow Richaud-Degert type which isnot 5 modulo 8 with class number two , J. Number Theory
The complete determination of wide Richaud-Degert type which is not5 modulo 8 with class number one , to appear in Acta Arithmetica.[13] Mollin, R.A.,
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Proc. JapanAcad , 1987, 121-125.[16] Yamamoto, S., On Kronecker limit formulas for real quadratic fields , J. Num-ber Theory. , 2008, no.2, 426–450.[17] Yokoi, H.,
Class number one problem for certain kind of real quadratic fields, in:Proc. Internat. Conf. (Katata, 1986), Nagoya Univ., Nagoya, 1986, 125–137.[18] Zagier, D.,
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Math.Ann.213,153-184 (1975).[19] Shintani, T.,
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J. Fac.Sci.Univ. Tokyo. 63 (1976), 393-417.[20] Van der Geer, G.,
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Ergebnisse der Mathematik undihrer Grenzgebiete (3), Springer-Verlag (1980).
E-mail address : [email protected] E-mail address : [email protected]@kias.re.kr