Higher Kronecker limit formulas for real quadratic fields

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1 J. reine angew. Math. 679 (203), DOI 0.55/crelle Journal für die reine und angewandte Mathematik ( Walter de Gruyter Berlin Boston 203 Higher Kronecker limit formulas for real quadratic fields By Maria Vlasenko and Don Zagier at Bonn Abstract. For every integer k f 2 we introduce an analytic function of a ositive real variable and give a universal formula eressing the values zðb; kþ of the zeta functions of narrow ideal classes in real quadratic fields in terms of this function and its derivatives u to order k evaluated at reduced real quadratic irrationalities associated to B. We show that our functions satisfy functional equations and use these to deduce elicit formulas for the rational numbers zðb; kþ. We also give an interretation of our formula for zðb; kþ in terms of cohomology grous of SLð2; ZÞ with analytic coe cients and describe a twisted etension of the main formula that allows one to treat zeta values of zeta functions of ray classes rather than just ideal classes. Finally, we use our formulas to comute some zeta-values numerically and test that they are eressible as combinations of higher olylogarithm functions evaluated at algebraic arguments. Introduction ffiffiffiffi Let K ¼ Qð D Þ be a real quadratic field of discriminant D and B A Cl þ ðkþ be a narrow ideal class. The z-function of B is defined for ReðsÞ > as zðb; sþ ¼ NðaÞ s ; a A B where the summation is over all integral ideals in the class B. This function can be continued to a meromorhic function on the whole of C with its only singularity at s ¼, where it has a simle ole with residue D 2 logðeþ. Here e is the smallest totally ositive unit of K with the roerty e >. The Kronecker limit formula (hereafter abbreviated as KLF) for real quadratic fields is an eression for the 0th Laurent coe cient of zðb; sþ at s ¼. (The original KLF, of course, was for imaginary quadratic fields.) Such a formula was given in [3]: there is an analytic function ð; yþ of > y > 0 such that for all narrow ideal classes in all real quadratic fields ðþ lim D s=2 zðb; sþ logðeþ ¼ s! s w A RedðBÞ ðw; w 0 Þ: Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

2 24 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields Here RedðBÞ is the set of larger roots w ¼ B þ ffiffiffiffi D of all reduced quadratic forms 2A QðX; YÞ ¼AX 2 þ BXY þ CY 2 (A; C > 0, A þ B þ C < 0) of discriminant D which belong to the class B. Recall that narrow ideal classes corresond to SLð2; ZÞ-orbits on the set of integer quadratic forms: if b ¼ Zw þ Zw 2 A B with w w2 0 w 2w 0 ffiffiffiffi > 0, then the D quadratic form QðX; YÞ ¼ NðXw þ Yw 2 Þ is in the corresonding orbit. The set RedðBÞ is NðbÞ obviously finite and every w A RedðBÞ satisfies w >, > w 0 > 0. Sometimes we identify RedðBÞ with the set of reduced forms themselves and write Q A RedðBÞ for such a form. We denote lðbþ¼kredðbþ in the sequel. The function ð; yþ mentioned above is defined as y ð; yþ ¼FðÞ FðyÞþLi þ log g y 2 logð yþþ 4 log ; y where g is Euler s constant, is the dilogarithm function, and Li 2 ðzþ ¼ y n¼ z n n 2 ; jzj < ð2þ FðÞ ¼ y ¼ cðþ logðþ ; with cðþ ¼G 0 ðþ=gðþ (digamma function). The sum defining F converges since cðþ ¼log þ Oð=Þ as! y, and defines an analytic function on R þ that satisfies the functional equations ð3þ FðÞþF ¼ 2 þ þ 6 2 log2 þ C; FðÞ Fð ÞþF ¼ Li þ C; > ; 2 with the real constant C A: given elicitly in terms of the derivative of zðsþ ðs Þ at s ¼. In [3] these functional equations were used to deduce Meyer s theorem from the Kronecker limit formula (), namely ð4þ lim zðb; sþ zðb ; sþ ¼ zð2þ ffiffiffiffi lðbþ lðb Þ ; s! D where B is the class of any ideal ab with any b A B and a A K, aa 0 < 0. In the resent aer we generalize formulas () and (4) to the higher zeta values zðb; 2Þ; zðb; 3Þ;... : We begin by generalizing the function (2). Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

3 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 25 Definition. ð5þ For every integer k > 2 and real number > 0 let F k ðþ ¼ y ¼ cðþ k : Here we do not need to subtract logðþ from cðþ since the estimate cðþ ¼Oðlog Þ su ces for (absolute) convergence, and the functions F k are analytic on R þ. They already occurred incidentally in [7]. We also introduce a collection of di erential oerators D n (n ¼ 0; ; 2;...) that turn di erentiable functions of one variable into di erentiable functions of two variables: ðd 0 FÞð; yþ ¼FðÞ FðyÞ; ðd FÞð; yþ ¼F 0 ðþþf 0 ðyþ 2 FðÞ FðyÞ ; y ðd 2 FÞð; yþ ¼ F ð2þ ðþ F ð2þ ðyþ 2. 3 F 0 ðþþf 0 ðyþ þ 6 FðÞ FðyÞ y ð yþ 2 ; where the coe cients, given elicitly in Definition 6 below, are chosen so that D n kills olynomials of degree e 2n and sends FðÞ ¼ 2nþ to ð yþ nþ. Then we have Theorem 2. For any narrow ideal class B A Cl þ ðkþ and integer k f 2, ð6þ D k=2 zðb; kþ ¼ k ðw; w 0 Þ; w A RedðBÞ where k ð; yþ is the function of two variables ; y > 0 defined by ð7þ k ð; yþ ¼ðD k F 2k Þð; yþ: We call this formula the higher Kronecker limit formula because it has a form similar to (). The reason for the quotes is that for k f 2 no limit is involved, since the series defining zðb; kþ converges. For any di erentiable function F on R one can also consider the two-variable function D n F as a homogeneous function of degree n on the sace Q þ of real quadratic forms with ositive discriminant by setting ð8þ ffiffiffiffi ffiffiffiffi B þ D B D ðd n FÞðQÞ ¼D nþ 2 ðdn FÞ ; 2A 2A for QðX; YÞ ¼AX 2 þ BXY þ CY 2 with ffiffiffiffi B 2 4AC ¼ D > 0. (This is defined whenever F is smooth near ffiffiffiffithe roots of Q.) Here D denotes the ositive square root and D ðnþþ=2 is defined as ð D Þ n, but because D n F is ð Þ nþ -symmetric the right-hand side of (8) is Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

4 26 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields actually indeendent of the choice of the square root. By roosition 7 below we have the equivariance roerty D n ðfj 2n gþðqþ ¼ðdet gþ 2nþ ðd n FÞðQ gþ Eg A GLð2; RÞ;! a b where j w for w A Z is defined as usual by Fj w ðþ ¼ðc þ dþ w F a þ b. c d c þ d With this notation, Theorem 2 can be rewritten in the form zðb; kþ ¼ Q A RedðBÞ ðd k F 2k ÞðQÞ: In the first art of the aer we rove Theorem 2 and study the functions F k.in articular we show that they (and their derivatives) can be comuted easily to high accuracy and satisfy functional equations analogous to (3). These functional equations turn out to be su cient to deduce from our higher KLF elicit formulas, analogous to (4), for the numbers zðb; kþþð Þ k zðb ; kþ A 2k D =2 Q, or equivalently, if one uses the functional equations for zðb; sþ G zðb ; sþ, for the numbers zðb; kþ A Q, where the rationality statement is the well-known theorem of Klingen and Siegel. One of the formulas we obtain was already roved in [4]. The second art of the aer is devoted to the interretation of the higher KLF in terms of (co)homology of SLð2; ZÞ. We construct a cocycle class ½f k Š A H SLð2; ZÞ; V 2k with coe cients in the sace V 2k of continuous functions of weight 2k on the rojective real line and use the functional equations satisfied by F 2k to show (Theorem 3) that ½f k Š has the rational image Ið½f k ŠÞ A H SLð2; ZÞ; 2k V 2k 2 ðqþ under the integral ma I : f 7! Ð f ðþðx Þ 2k 2 d from V 2k to the sace V 2k 2 of real olynomials of degree e 2k 2. These ½f k Š and Ið½f k ŠÞ turn out to be the two eriods of the non-holomorhic Eisenstein series according to the definition given in [6], []. The class Ið½f k ŠÞ itself is the well-known Eisenstein cohomology class studied e.g. in [9] and for the general linear grou of degree n in [8], [0]. That is why we call our ½f k Š the generalized Eisenstein cocycle class. Further, for each narrow ideal class B we define a cycle class h B A H SLð2; ZÞ; V2 2k and show that evaluating ½fk Š on it gives the value zðb; kþ (Theorem 6). Then the rationality of Ið½f k ŠÞ imlies the Siegel Klingen theorem again. In the third art of our aer we consider the twisted version of our functions F w k for a Dirichlet character w and k f 2. Such twisted functions can be used to comute zeta values for ray classes. We should mention that a KLF was given for ray classes in real quadratic fields in full generality by Yamamoto [2], but even for this case (k ¼ ) our formulas are di erent. We do not resent everything in comlete detail but consider a Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

5 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 27 articular eamle, where we comute Stark s unit and its higher version (for k ¼ 2), which involves dilogarithms instead of logarithms. We mention that there are similar higher Kronecker limit formulas also for imaginary quadratic fields. Let K be an imaginary quadratic field of discriminant D < 4 and B an ideal class of K. Then zðb; sþ ¼jD=4j s=2 Eðw; sþ, where w is the root in H (uer halflane) of any quadratic form of discriminant D in the SLð2; ZÞ-orbit corresonding to B and ð9þ Eðz; sþ ¼ 2 m; n A Z 0 y s ; z ¼ þ iy A H; 2s jmz þ nj is the non-holomorhic Eisenstein series. For k A Z f2 we have Eðz; kþ ¼ð2iÞ k ðd k C 2k Þðz; zþ; where D k is the same di erential oerator as in Theorem 2 and C h ðzþ ¼zðhÞz h y ¼ cotðzþ h ; z A CnR: Therefore jd=4j k=2 zðb; kþ ¼D k C 2k ðw; wþ. The relation between C h and F h (which is defined in all of Cnð y; 0Š by the sum (5)) is simly F h ðzþ F h ð zþ ¼zðhÞðz h þ =zþ C h ðzþ: (This is an immediate consequence of cðþ cð Þ ¼ cotðþþ.) The reason that the real quadratic case is more comlicated than the imaginary quadratic case is that there one has to work with half-eisenstein series, i.e. sums over lattice oints in a quadrant rather than a whole lattice. This is discussed in Section 2.. The functions F k and zeta values.. roerties of the functions F k. We begin by giving the eansions of the functions F k ðþ defined in (5) near ¼ y, 0 and. The result will involve the double zeta values zðm; nþ ¼ >q>0 They satisfy the well-known shu e relations m ; m f 2; n f : qn ð0þ k n¼ zðm; k mþþzðk m; mþ ¼zðmÞzðk mþ zðkþ;! n n þ zðn; k nþ ¼zðmÞzðk mþ m k m Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

6 28 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields for each m ¼ ;...; k. Here and everywhere below the divergent values zðþ and zð; k Þ are to be interreted as g (Euler s constant) and zðk ; ÞþzðkÞ gzðk Þ, resectively. We denote by B n the nth Bernoulli number, defined by the eansion e ¼ y n¼0 B n n n! ; or alternatively by B 0 ¼ and B n ¼ð Þ n nzð nþ for n f. roosition 3. The function F k ðþ has the asymtotic eansions F k zðk Þ log z 0 ðk Þþ y r¼ zð rþzðk þ r Þ r as! y, F k zðkþ þ y r¼; r3k ð Þ r zðrþzðk rþ r þð Þ k ½zðk Þðg log Þ z 0 ðk ÞŠ k 2 as! 0, and as!. F k k ½zðr; k rþþzðkþšð Þ r r¼ y r¼k r k þ r k l¼0 ð Þ l r k þ B l zðk þ l Þ ð Þ r l (By an asymtotic eansion f y a n ð aþ n as! a we mean that n¼n 0 f ðþ N a n ð aþ n ¼ O ð aþ N n¼n 0 when! a for every N. We do not require that the series be convergent in any neighborhood of a. For a ¼ y one relaces a by =.) roof. The standard asymtotic eansion of the digamma function at infinity, log þ y r¼ zð rþ r ;! y (the derivative of Stirling s formula), immediately gives the first asymtotic eansion for F k. For the second, we use the eansion cðþ ¼ y g þ ð Þ r zðrþ r ; 0 < < ; r¼2 Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

7 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 29 and aly the Euler Maclaurin summation formula following the method elained in detail in [8] (see [8], roosition 6.5, and the second remark after it). When!, we use the recursion cð þ Þ ¼cðÞþ and the eansion of c near 0 again. r roosition 4. ðþ The function F k ðþ satisfies the two functional equations F k ðþþð Þ k 2 F k ¼ A k ðþþzðkþ ð Þ k and ð2þ F k ðþ F k ð þ Þþð Þ k 2 þ F k ¼ B k ðþþzðkþ ð Þ k þ! ; where A k ðþ and B k ðþ are the olynomials of degree k 2 given by A k ðþ ¼ k zðrþzðk rþð Þ r ; r¼ B k ðþ ¼ k zðk r; rþð Þ r r¼ and related by ð3þ B k ðþþð Þ k 2 B k ¼ A k ðþþzðkþ ð Þ k ; B k ðþþb k ð þ Þ ¼ð Þ k 2 þ A k : we get roof. and hence From cðþ ¼ y q¼0 þ q þ q ð Þ k ðk Þ! cðk Þ ðþ ¼ y q¼0 ð þ qþ k ð Þ k ðk Þ! Fðk Þ k ðþ ¼ f; qf0 ð þ qþ k : From this we find that ð Þ k ðk Þ! ¼ F ðk Þ k ðþ! k Fðk Þ k >0; qf0 f0;q>0 ð þ qþ k ¼ zðkþ k Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

8 30 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields and ð Þ k ðk Þ! ¼ F ðk Þ k ðþ F ðk Þ k ð þ Þ! þ k Fðk Þ k >0; qf0 qf >0 fq>0! ð þ qþ k ¼ zðkþ k ð þ Þ k : Integrating these equations k times) we obtain formulas () and (2) for some olynomials A k and B k of degree k 2, whose coe cients are then determined by the asymtotic eansions given in roosition 3. The relations (3) between B k and A k are equivalent to the shu e relations (0), and also follow easily from equations (), (2) and () with relaced by =. r Finally, we would like to mention that, although the series in (5) converges only olynomially quickly, we can comute values of the function F k ðþ (or its derivatives) to high accuracy with little e ort. To this end, for integers M; N f we define F k; N; M ðþ ¼ N ¼ þ M m¼ cðþ logðþ k z 0 ðk 2Þþzðk Þ log zð mþ m zðk þ m Þ N ¼ kþm ; which can be comuted in time OðMNÞ, the calculation of the digamma and zeta functions being standard. Then jf k ðþ F k; N; M ðþj ¼ e y ¼Nþ y ¼Nþ k cðþ logðþ M zð mþ m¼ ðþ m k C M ðþ Mþ < ðk þ M ÞC M Mþ N kþm! M M with C M ¼jzð MÞj ¼ O M 2 by Stirling s formula. From this we find that 2e for a given time T ¼ MN the best accuracy is OðT 5 4 k 2e ffiffiffiffiffiffiffiffiffiffiffi 8T=eÞ, achieved by choosing em A2N..2. The functions F k and olylogarithms. For m; n f consider the function ð4þ F m; n ðþ ¼ð Þ n Ðy 0 Li m ðe t Þ Li n ðe t Þ dt; > 0; ) Here we use Bol s identity to write k F ðk Þ k ðc þ =Þ as ð d=dþ k k 2 F k ðc þ =Þ. Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

9 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 3 where Li m ðþ ¼ y n =n m is the mth olylogarithm function. Integrating by arts and using n¼ Li 0 m ðþ ¼Li m ðþ, we obtain F mþ; n ðþ ¼ ð Þ n Ðy 0 Li mþ ðe t Þd Li nþ ðe t Þ ¼ð Þ n zðm þ Þzðn þ ÞþF m; nþ ðþ: Hence if one fies the sum m þ n, then there is essentially one such function u to a olynomial. Let us set for k f 3 Then F k ðþ ¼F k 2; ðþ: F m; n ðþ ¼F k ðþ n r¼2 zðrþzðk rþð Þ r with k ¼ m þ n þ. Substituting t! t in the integral (4) we get the equality F m; n ðþþð Þ nþm F n; m ¼ 0: Eressing F m; n and F n; m in terms of the function F k yields ð5þ F k ðþþð Þ k 2 F k ¼ k 2 r¼2 zðrþzðk rþð Þ r : This functional equation is very similar to (). And indeed, F k is the same as our old F k u to sign and the addition of simle functions: roosition 5. For k > 2 one has F k ðþ ¼ F k ðþ gzðk Þ zðkþ : roof. From cðþ ¼ Ðy 0 e t t e t e t dt and g ¼ Ðy 0 e t e t dt t ([]) we get cðþþg þ ¼ cð þ Þþg ¼ Ðy 0 e t e t dt ¼ Ðy ð e t Þd Li ðe t Þ: 0 Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

10 32 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields Therefore Ðy 0 e t Li ðe t Þ dt ¼ cðþþg þ ; so F k ðþ ¼ Ðy 0 y ¼ e t k 2 Li ðe t Þ dt ¼ y ¼ cðþ k gzðk Þ zðkþ : r This roosition together with (5) gives an alternative roof of (). Let us also sketch a similar roof of the second functional equation (2). The double olylogarithm functions ([4], [3]) are defined for jz i j < by the series Li m; n ðz ; z 2 Þ¼ Let us consider again for m; n f and > 0< <q z m z q 2 q n : G m; n ðþ ¼ð Þ n Ðy 0 Li m; n ðe t ; e t Þ dt: This integral is also convergent for m ¼ 0 and n f 2. Integrating for m f 0, n f the equality we get the relation d dt Li mþ; nþðe t ; e t Þ¼ Li m; nþ ðe t ; e t Þ Li mþ; n ðe t ; e t Þ; G mþ; n ðþ ¼ð Þ n zðn þ ; m þ ÞþG m; nþ ðþ: So, there is one such function G m; n u to a olynomial for a fied sum m þ n. We claim this function is again F mþnþ. Let k ¼ m þ n þ. Careful integration (aying attention to the singularity at t ¼ 0) of the equality Li k ; 0 ðe t ; e t Þ¼Li k ðe ðþþt Þ Li 0 ðe t Þ yields G k 2; ðþ ¼F k ð þ Þþzðk ; ÞþzðkÞ. Hence G m; n ðþ ¼F k ð þ Þþgzðk Þ n Now eressing everything in the equality r¼ zðr; k rþð Þ r : F m; n ðþ ¼ ; qfð Þ n Ðy 0 tð þqþ e m q n dt ¼ <q þ q< þ ¼q ¼ G m; n ðþ ð Þ mþn G n; m þ zðm þ n þ Þ ð Þn þ Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

11 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 33 in terms of F k gives the functional equation F k ðþ F k ð þ Þþð Þ k 2 þ F k ¼ k r¼ which, together with roosition 5, yields (2)..3. The di erential oerator D n. zðk r; rþð Þ r þ zðkþ þ gzðk Þð Þk 2 ; Definition 6. Let D n for every integer n f 0 be the di erential oerator from functions of one variable to functions of two variables defined by ðd n FÞð; yþ ¼ n i¼0 2n i F ðiþ ðþ ð Þ i F ðiþ ðyþ n i!ðy Þ n i : This can be written as ðd n FÞð; yþ ¼ðDn þ FÞð; yþþð Þnþ ðdn þ FÞðy; Þ, where ð6þ ðd þ n 2n i F ðiþ ðþ FÞð; yþ ¼n i¼0 n i!ðy Þ n i : (The oerator Dn þ will be also used later in this article.) However, note that, unlike D nf, the function Dn þ F is not di erentiable, because it has an nth order ole along the diagonal ¼ y, whereas, as we shall see, ðd n FÞð; yþ and even ðd n FÞð; yþ=ð yþ nþ remain finite near ¼ y. roosition 7. (i) For any matri g ¼ a b A GLð2; RÞ and any smooth function c d F we have Dn þ ðc þ dþ 2n F a þ b c þ d and similarly with D þ n relaced by D n. (ii) For any T A C we have! ð; yþ ¼ðdet gþ n ðd þ n ð Þ n 2n ðt ÞðT yþ n ¼ 2n n y (iii) For all integers m f 0 we have ð7þ D n ð m Þ¼ð yþ nþ rþs¼m ð Þ m m¼0 2n m r s r n y s n : n n a þ b FÞ c þ d ; ay þ b cy þ d D þ n ðm ÞT 2n m : Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M ;

12 34 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields In articular, D n ðþ ¼0 for all olynomials of degree e 2n and ð8þ D n ð 2nþ Þ¼ð yþ nþ : (iv) For any T we have y nþ D n ¼ : T ðt ÞðT yþ (v) If F is continuously di erentiable 2n þ times between and y then ð9þ ðd n FÞð; yþ ¼ Ð ð tþðt yþ n n! 2 F ð2nþþ ðtþ dt: y y roof. All of the arts of the roosition follow from the machinery of di erential oerators given in the aendi (see the remark after roosition 22). Here we give more direct roofs. (i) The formula Dn þ ¼ q n! q þ 2 q y q þ 4 ð20þ y q q þ 2n y can be verified by induction. The intertwining roerty (i) for Dn þ, and hence also for D n, follows immediately since the oerator q q þ k intertwines between weights k and y weight 2 k in the variable. (ii) From (6) one has obviously Dn þ ðþ ¼ 2n ðy Þ n. Alying (i) with n FðÞ ¼ and g ¼ 0 we get T which is equivalent to (ii). Dn þ ðt Þ2n ¼ð Þ n 2n ðt ÞðT yþ n ; n y find (iii), (iv) We first rove the secial case (8). Directly from the definition of D n we n! 2 ð2n þ Þ! ð yþ n D n ð 2nþ Þ¼g n y þ g n y y with g n ðuþ ¼ n ð Þ i n u 2nþ i i 2n þ i : i¼0 Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

13 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 35 Since g 0 n ðuþ ¼un ð uþ n is symmetric under u $ u, the eression g n ðuþþg n ð uþ is constant. Its value is given by g n ð0þþg n ðþ ¼0 þ Ð u n ð uþ n n! 2 du ¼ 0 ð2n þ Þ! (beta integral). This gives (8). Now (iv) follows using (i) with FðÞ ¼ 2nþ and g ¼ 0. To get (iii) one lets T! y and eands both sides of (iv) as series in =T. T Finally, for (v) we note that equation (9) can be rewritten as ðd n FÞð; yþ ¼ ð yþnþ Ð n! 2 t n ð tþ n F t þð tþy dt 0 (roving the divisibility by ð yþ nþ mentioned above). This formula is true if F is a olynomial by (7) and the beta integral, and then in the general case by olynomial aroimation. r.4. The higher Kronecker limit formula. roof of Theorem 2. The roof is almost immediate from roosition 7 and the results of [3], where it was shown that the artial zeta function zðb; sþ has the decomosition where ð2þ zðb; sþ ¼D s=2 Z s ð; yþ ¼ >0; qf0 w A RedðBÞ Z s ðw; w 0 Þ; y s : ð þ qþðy þ qþ In [3], the function ðw; w 0 Þ from the Kronecker limit formula () was obtained as the limiting value of Z s ðw; w 0 Þ after its ole has been removed. Here there is no ole and we can simly set s ¼ k. By art (iv) of roosition 7, y k ¼ D k ð þ qþðy þ qþ 2k ; ð þ qþ and since F 2k ðþ ¼ >0; qf0 2k þ q ; þ q we find that Z k ð; yþ ¼ðD k F 2k Þð; yþ. r Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

14 36 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields ffiffiffi We illustrate the theorem with a numerical eamle. The field K ¼ Qð 3 Þ has two narrow classes, B 0 and B, with the corresonding sets of reduced quadratic irrationalities being RedðB 0 Þ¼f2þ ffiffiffi ( 3 g; RedðB Þ¼ þ ffiffiffi ; 3 þ ffiffiffi ) 3 : 3 2 Recall that we can comute the values of F k and its derivatives with high accuracy. In Table we give the values involved in the higher KLF for zðb i ; 2Þ. F 4 ðþ F4 0 ðþ 2 þ ffiffiffi ffiffiffi þ = ffiffiffi = ffiffiffi ð3 þ ffiffi 3 Þ= ð3 ffiffi 3 Þ= Table F 6 ðþ F6 0 ðþ F00 6 ðþ 2 þ ffiffiffi ffiffiffi þ = ffiffi = ffiffi ð3 þ ffiffi 3 Þ= ð3 ffiffi 3 Þ= Table 2 With Table we comute the zeta values at k ¼ 2 by Theorem 2: zðb 0 ; 2Þ ¼: ; zðb ; 2Þ ¼0: : Similarly, to comute zðb i ; 3Þ we need the values in Table 2. Therefore zðb 0 ; 3Þ ¼: ; zðb ; 3Þ ¼0: : With the zeta values comuted above one can check numerically that ð22þ zðb 0 ; 2ÞþzðB ; 2Þ ¼ 4 8 ffiffiffiffiffi 2 Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

15 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 37 and ð23þ zðb 0 ; 3Þ zðb ; 3Þ ¼ ffiffiffiffiffi : 2 One sees that the combinations zðb; kþþð Þ k ffiffiffiffi zðb ; kþ in both cases belong to 2k D Q. This statement is true in general and is equivalent to the Siegel Klingen theorem by the functional equations for L-functions. Now, we show how to deduce elicit formulas for such rational combinations from our higher KLF..5. Rational zeta values. Let us briefly recall the relation between ideal classes and continued fractions given in [3]. Any real quadratic irrationality w has a continued fraction eansion w ¼ b b 2... with b i A Z (all i), b i f 2(i 3 ) and fb i g eventually eriodic. The condition of being a root of the quadratic form in a given class B fies the tail of fb i g. The condition of being a larger root of a reduced quadratic form, i.e. ð24þ w > ; > w 0 > 0; is equivalent to fb i g being urely eriodic. Let us denote by ðb ;...; b l Þ the eriod of the continued fraction eansion. It is actually defined u to a cyclic shift, so we will say it is a cycle of integers. Then the numbers in RedðBÞ are eactly w i ¼ b i b iþ... ; i A Z=lZ; where b i with i A Z stands for b i ðmod lþ. Wide ideal classes A A ClðKÞ corresond to GLð2; ZÞ orbits on the sace of integer a b quadratic forms of discriminant D, where A GLð2; ZÞ acts on quadratic forms by c d QðX; YÞ 7! GQðaX þ by; cx þ dyþ if ad bc ¼ G: The quadratic form Q ¼ AX 2 þ BXY þ CY 2 is said to be reduced in the wide sense if A > 0, C < 0 and ja þ Cj < B or, equivalently, when its root ¼ B þ ffiffiffiffi D satisfies 2A ð25þ > ; 0 > 0 > : Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

16 38 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields As in the case of narrow classes, the condition of being a root of the quadratic form in the class A fies the tail of fa i g in the ordinary continued fraction eansion ¼ a þ a 2 þ... ; a i A Z; a i f ði 3 Þ; while condition (25) is equivalent to fa i g being urely eriodic. Thus to A there corresonds a cycle of integers ða ;...; a m Þ with a i f and a cycle of numbers i ¼ a i þ a iþ þ... ; i A Z=mZ; just in the same way as ðb ;...; b l Þ with b i f 2 and w ;...; w l were assigned to a narrow ideal class. Further, since satisfies (25), w ¼ þ satisfies (24) and w > 2, so w has a urely eriodic minus continued fraction w ¼ b b 2 with b ¼ a þ 2 f 3. The narrow ideal class B corresonding to w then lies in the wide class A corresonding to (since both contain the ideal Z þ Z ¼ Zw þ Z) and we can reroduce all the numbers ðb ;...; b l Þ as follows. We consider the minimal even eriod ða ;...; a 2r Þ, where r ¼ m if m is odd and r ¼ m=2 ifm is even. The b s and a s are related by b ¼ a þ 2; b 2 ¼¼b a2 ¼ 2; ð26þ b a2 þ ¼ a 3 þ 2; b a2 þ2 ¼¼b a2 þa 4 ¼ 2;. b l a2r ¼ a 2r þ 2; b l a2r þ ¼¼b l ¼ 2; i.e. ðb ;...; b l Þ¼ða þ 2; 2;...; 2; a 3 þ 2; 2;...; 2;...; a 2r þ 2; 2;...; 2Þ: fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} a 2 fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} a 4 fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} a 2r In articular, lðbþ¼a 2 þ a 4 þþa 2r. If we started with the cyclic shift ða 2 ; a 3 ;...; a 2r ; a Þ, we would get instead the cycle ða 2 þ 2; 2;...; 2;...; a 2r þ 2; 2;...; 2Þ: fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} a 3 fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} a This cycle corresonds to the conjugate narrow class B and lðb Þ¼a þ a 3 þþa 2r. In articular, we have lðbþþlðb Þ¼ 2r a i. Notice that if the minimal eriod length m is i¼0 odd, then B ¼ B ; this haens if and only if the fundamental unit of K has negative norm. Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

17 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 39 Let us denote by Red w ðbþ for a narrow ideal class B the set of those larger roots of the quadratic forms in B reduced in the wide sense. (Equivalently, Red w ðbþ consists of the numbers w where w A RedðBÞ, w > 2.) Then in the notations above we have Red w ðbþ ¼f ; 3 ;...g and Red w ðb Þ¼f 2 ; 4 ;...g. We will also write Q A Red w ðbþ for such a form. Lemma 8. ð27þ Let B 7! IðBÞ be an invariant of narrow ideal classes defined by IðBÞ ¼ Fðw; w 0 Þ w A RedðBÞ for some function F : R R! C. Suose F has the form ð28þ Fð; yþ ¼Gð ; y Þ G ; y y for some function G. Then (i) For any narrow ideal class B we have IðBÞ ¼ A Red w ðbþ Gð; 0 Þ A Red w ðb Þ G ; 0 : (ii) If Gð; yþ has the form H ð yþþh 2 ð= =yþ for some functions H ; H 2 : R! C, then IðBÞ ¼0 for all classes B. roof. From (26) we have and therefore ð29þ a 2 þ j¼2 w ¼ þ ¼ a þ 2 w 2 ; w 2 ¼ 2 ;...; w a2 ¼ 2 ; w 3 w a2 þ w a2 þ ¼ 3 þ ¼ a 3 þ 2 w a2 þ2 Fðw j ; wj 0 Þ¼ a "!# 2þ Gðw j ; wj 0 Þ G ; j¼2 w j wj 0 ¼ Gðw a2 þ ; wa 0 2 þ Þ G ; w 2 w2 0 ¼ Gð 3 ; 3 0 Þ G ; : Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

18 40 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields Summing this cyclically gives statement (i). For statement (ii) we observe that if G has the given form, then Fð; yþ ¼H ð yþ H ð = þ =yþ. Now from it follows that i mod l Fðw i ; w 0 i Þ¼0. w i w 0 i ¼ w iþ þ w 0 iþ r Corollary. Let the notations be as above. Then IðBÞ G IðB Þ¼ A Red w ðbþ G A Red w ðb Þ G H ð; 0 Þ with G H ð; yþ ¼Gð; yþ H Gð=; =yþ. This corollary is useful because in the alication to the following theorem the relevant function G will be transcendental but either G þ or G (deending on the arity of k) will be a rational function. Theorem 9. For every narrow ideal class B and integer k f 2, D 2 k zðb; kþþð Þ k zðb ; kþ ¼ where W k ð; yþ ¼D k k r¼0 A Red w ðbþ zð2rþzð2k 2rÞjj 2r. þð Þ k A Red w ðb Þ W k ð; 0 Þ; Remark. In the definition of W k, the absolute value signs are necessary, since D k kills olynomials of degree e 2k 2, but the restriction of W k to f > 0 > yg, which is all that is used in the theorem, is a rational function (in fact, a olynomial in G, y G and ð yþ ). roof. We define two functions of one variable by V 0 ðþ ¼ 2k 2 F 2k zð2kþ 4 þ 2k ; > 0; and VðÞ ¼V 0 ðjjþ zð2kþ 2 þ 2k ; A Rnf0g: From roosition 4 we get, after a straightforward but lengthy comutation (one has to distinguish the three cases >, 0 < < and < 0 and to use the functional equations () (3) reeatedly), F 2k ðþ ¼Vð Þ V þðolynomial of degree e 2k 2Þ Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

19 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 4 and hence, since D k is equivariant and kills olynomials of low degree (arts (i) and (iii) of roosition 7), ðd k F 2k Þð; yþ ¼ðD k VÞð ; y Þ ðd k VÞ ; y y for > > y > 0. Therefore, by Theorem, the conditions of the lemma above are satisfied for IðBÞ ¼D k=2 zðk; BÞ and G ¼ D k V. Moreover, since D k ð 2k Þ¼ð yþ k and D k ð=þ ¼ ð = þ =yþ k by roosition 7, art (ii) of Lemma 8 tells us that we can relace G by G 0 ð; yþ ¼D k V0 ðjjþ. Finally, from the functional equation () we get V 0 ðjjþ þ 2k 2 V 0 ð=jjþ ¼ A 2k ðjjþ þ zð2kþ ð=jjþjj 2k Þ 2 ¼ k r¼0 zð2rþzð2k 2rÞjj 2r þðolynomial of degree e 2k 2Þ; so G 0 ð; yþþð Þ k G 0 ð=; =yþ ¼W k ð; yþ. The theorem now follows immediately from the corollary to Lemma 8. r We illustrate the theorem (for k ¼ 2 and k ¼ 3) using the eamle from the revious section. We have Red w ðb 0 Þ¼fþ ffiffiffi 3 g, B 0 ¼ B, Red w ðb Þ¼fðþ ffiffiffi 3 Þ=2g. One can easily comute W 2 ð; yþ ¼ 4 þ y ð y 2 y 2 þ Þð 2 4y þ y 2 Þþ0 2 y 2 ; ð yþ for > y > 0, so zðb 0 ; 2ÞþzðB ; 2Þ equals 0 W 2 ð þ ffiffiffi ffiffi þ ffiffi ; 3 ÞþW2 ; A ¼ ffiffiffiffiffi ; 2 in accordance with our numerical comutation (22). Similarly, using the rather long formula W 3 ð; yþ ¼ 6 þ y ð y 3 ð yþ 2 y 3 þ Þð y þ 6 2 y 2 6y 3 þ y 4 Þ 2ðy þ Þ 3 y 3 ; > y > 0; we find that zðb 0 ; 3Þ zðb ; 3Þ equals 0 2 ffiffiffiffiffi W 3 ð þ ffiffiffi ffiffi þ ffiffi ; 3 Þ W3 ; A ¼ ffiffiffiffiffi ; 2 in agreement with (23). Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

20 42 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields We end by discussing the relation of Theorem 2 with the Siegel Klingen theorem and with the functional equations of zðb; sþ G zðb ; sþ. Consider, for k f 2, the rational function k ðþ ¼ 2k B r B 2k r r¼0 r!ð2k rþ! r ¼ 4 k ð2iþ 2k zð2rþzð2k 2rÞ 2r ; r¼0 where B r are the Bernoulli numbers. This function aeared in the literature (e.g. in [5]) as a generalized eriod olynomial associated to the Eisenstein series of weight 2k. It satisfies the relations k ðþ ¼ 2k 2 k ; k ðþ k ð Þþ 2k 2 ð30þ k ¼ 0: For k f 2 define ~W k ð; yþ A Q½ G ; y G ; ð yþ Š by ~W k ð; yþ ¼ð Þ k ðk Þ! 2 2 ðd þ k kþð; yþþð Þ k ðd þ k kþðy; Þ : The function ~W k is related to the function W k of Theorem 2 by ð3þ W k ð; yþ ¼ ð2þ2k 2ðk Þ! 2 ~ W k ð; yþ if > 0 > y: Moreover, since ~W k is ð Þ k -symmetric in and y, the number defined by ~W k ðqþ :¼ D k B þ ffiffiffiffi ffiffiffiffi D B D 2 ~W k ; 2A 2A for an integer quadratic form Q ¼ AX 2 þ BXY þ CY 2 is rational. Note that the normalization here is di erent from the one in (8), reflecting the di erent symmetry. Now we can formulate the following Corollary. For any narrow ideal class B and integer k f 2, zðb; kþ ¼ þð Þ k ~W k ðqþ ¼ Q A Red w ðbþ Q A Red w ðb Þ Q A RedðBÞ ~W k ðqþ: roof. ð32þ From the functional equations s G s þ e 2 D s=2 zðb; sþþð Þ e zðb ; sþ ¼ same with s 7! s ðe ¼ 0; Þ 2 we have zðb; kþ ¼ð Þ k zðb ; kþ and D k=2 zðb; kþþð Þ k zðb ; kþ ¼ ¼ ð2þ2k 4ðk Þ! 2 Dð kþ=2 zðb; kþþð Þ k zðb ; kþ ð2þ2k 2ðk Þ! 2 Dð kþ=2 zðb; kþ: Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

21 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 43 Combining the latter with Theorem 9 and using (3) gives zðb; kþ ¼D ðk Þ=2 þð Þ k A Red w ðbþ A Red w ðb Þ ~W k ð; 0 Þ; what is eactly the first statement of the corollary. Further, using the equivariance roerty of D k (roosition 7 (i)) we get from (30) ~W k ð; yþ ¼ð Þ k ~W k ; ; y ~W k ð; yþ ~W k ð ; y Þþ ~W k ; y ¼ 0: y From Lemma and its corollary with Gð; yþ ¼ ~W k ð; yþ it follows that IðBÞ :¼ ¼ w A RedðBÞ A Red w ðbþ ~W k ðw; w 0 Þ¼ð Þ k IðB Þ þð Þ k since ~W G k ¼ 2 ~W k when G ¼ð Þ k. Therefore zðb; kþ ¼D ðk Þ=2 IðBÞ ¼ A Red w ðb Þ Q A RedðBÞ ~W k ð; 0 Þ; ~W k ðqþ: r Let us comute the rational values zðb; kþ elicitly. If we define homogeneous olynomials d r; n ða; B; CÞ and f n ða; B; CÞ by ðax 2 þ BXY þ CY 2 Þ n ¼ 2n d r; n ða; B; CÞX 2n r ð YÞ r ; r¼0 f n ða; B; CÞ ¼ n ð Þ r ðn rþ! r!ð2n þ 2rÞ! Ar B 2nþ 2r C r ; r¼0 then for Q ¼ AX 2 þ BXY þ CY 2 we have and hence D n=2 Dn þ ðr Þ B þ ffiffiffiffi ffiffiffiffi D B D ; 2A 2A ffiffiffiffi ffiffiffiffi D n=2 B þ D B D D n ; jj 2A 2A ¼ð Þ n r!ð2n rþ! n! 2 d r; n ða; B; CÞ; ð2n þ Þ! ¼ n! f n ða; B; CÞ C nþ ð33þ ~W k ðqþ ¼ 2k B r B 2k r rð2k rþ d r ; k ða; B; CÞ r¼ þ ð Þk ðk Þ!B 2k 4k f k ða; B; CÞ C k þ f k ðc; B; AÞ A k : Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

22 44 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields Eamle. Let us once more go back to the eamle from the revious section and comute zðb 0 ; kþ and zðb ; kþ for k ¼ 2; 3. From (33) ~W 2 ðqþ ¼ B 44 þ B 2 0 C 2 þ B2 A A C þ C! : A In B 0 there is only one reduced quadratic form Q 0 ¼ X 2 4XY þ Y 2 and in B there are two reduced forms Q ¼ 3X 2 6XY þ 2Y 2 and Q 2 ¼ 2X 2 6XY þ 3Y 2. Therefore zðb 0 ; Þ ¼ ~W 2 ðq 0 Þ¼ 2 ; zðb ; Þ ¼ ~W 2 ðq Þþ ~W 2 ðq 2 Þ¼ 24 þ 24 ¼ 2 : For the Dedekind zeta function we then have z ffiffi Qð 3Þ ð Þ ¼=6. When k ¼ 3, BðA þ CÞ ~W 3 ðqþ ¼ B AB3 C þ A2 BC A 3 þ C 3 and zðb 0 ; 2Þ ¼ ~W 3 ðq 0 Þ¼ 8 ; zðb ; 2Þ ¼ ~W 3 ðq Þþ ~W 3 ðq 2 Þ¼ ¼ 8 ; in accordance with the fact that the Dedekind zeta function z ffiffi Qð 3Þ ðsþ has a zero (of the second order) at s ¼ 2. Remark. The right-hand side of (33) already aeared in [4]. To formulate the statement given there, let us first rewrite the decomosition (2) as ð34þ zðb; sþ ¼ Q A RedðBÞ Z Q ðsþ for ReðsÞ > ; where Z Q ðsþ ¼D s=2 B þ ffiffiffiffi ffiffiffiffi D B D Z s ; : 2A 2A Theorem 2 from [4] states that Z Q ðsþ can be continued to a meromorhic function on the whole of C with its only ole at s ¼ and, in our notations2), one has Z Q ð kþ ¼ ~W k ðqþ: 2) Our notations slightly di er from [4], namely our olynomial d r; n is d 2n r; n there and our Z Q is Z ~ Q where ~ Q ¼ CX 2 BXY þ AY 2. Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

23 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 45 One of the statements of our corollary follows from this and we now see that the individual terms ~W k ðqþ come from analytic continuation of zeta functions of reduced quadratic forms Z Q ðsþ in the decomosition (34). Another way to say this is that the results of this aer and the results of [4] together give a roof of the functional equations (32) as a consequence of functional equations for the cone zeta functions Z Q ðsþ at integer arguments. It would be interesting to see whether one can give a roof of the functional equations for arbitrary s in the same way (i.e., by writing the di erence of the right and left sides of (32) as an invariant of the form IðBÞ ¼ Fðw; w 0 Þ and showing that F has the form required in Lemma to force I 0), but we did not succeed in doing this. 2. Homological asects One may notice that the values D k ðf 2k Þðw; w 0 Þ in the higher KLF deend only on the ð2k Þst derivative of F 2k (see (9)). In this section we construct a cocycle class using this derivative and reresent our formula as a homological airing. 2.. Functional equations and cohomology of the modular grou. Recall that the modular grou G ¼ SLð2; ZÞ is freely generated by the two elements S ¼ G 0 ; U ¼ G 0 0 of orders 2 and 3, resectively. A -cocycle on G with coe cients in a (left) G-module M is a function f : G! M satisfying the relation ð35þ fðghþ ¼gfðhÞþfðgÞ for any g; h A G: Hence, if m ¼ fðsþ and m 2 ¼ fðuþ then ð36þ ð þ SÞm ¼ 0; ð þ U þ U 2 Þm 2 ¼ 0: And conversely, as soon as the two elements m ; m 2 A M are given such that the conditions (36) are satisfied, the ma fðsþ ¼m and fðuþ ¼m 2 can be uniquely etended to a -cocycle on G by (35). Let us now take for M the sace of functions on ðrþ with the action of G given by gf ¼ Fj 2k g. Then a -cocycle f is the same as a air of functions F ¼ fðsþ and G ¼ fðuþ satisfying ð37þ GðÞþ FðÞþ 2k F ¼ 0; ð Þ 2k G þ G ¼ 0: 2k Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

24 46 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields Consider T ¼ G. Then U ¼ TS and therefore GðÞ ¼Fð ÞþfðTÞðÞ. Using 0 this and (37), we rewrite the second of the above equations as ð38þ FðÞ FðþÞþ þ F ¼ BðÞ; 2k where B ¼ T ð þ U þ U 2 ÞfðTÞ. Equations (37) and (38) remind us corresondingly of equations () and (2) for the function F 2k. And indeed, one can observe that the roof of roosition 4 is based on the fact that similar equations are satisfied by the ðk Þst derivative of F k. Consider the function c 2k ðþ ¼signðÞ ; qf0 ðjjþqþ k ; where here and below on the summation sign means that we omit the term with ¼ q ¼ 0 and take the boundary terms (where either ¼ 0orq ¼ 0) with the coe cient =2. For ositive arguments c 2k ðþ equals d 2k F zð2kþ! 2kðÞþ ð2k Þ! d 2 þ 2k : This function satisfies (37) and (38) with B 0. Therefore we have the following roosition 0. For every k f 2 the ma f k ðsþ ¼c 2k ; f k ðtþ ¼0 can be uniquely etended to a -cocycle for G with coe cients in the sace of functions on ðrþ with the action in weight 2k. Let us give f k ðgþ for g A G elicitly. To this end let us consider for a 3 b A ðrþ the function on ðrþnfa; bg defined by ð39þ 8 >< A ½a; bš ðq Þ 2k ; q f a; b ðþ ¼ >: ðq Þ 2k ; q A ½b; aš A ðb; aþ; A ða; bþ; where ½a; bš, ½b; aš, ðb; aþ, ða; bþ are to be taken on ðrþ, e.g., ½a; bš has its usual meaning if y e a < b e y but means ½a; yš W ð y; bš if y < b < a e y, and where * now means that the terms with =q equal to a or b are to be counted with multilicity =2. Then with the convention that f a; a 0, we have the equality f a; b ðþþ f b; g ðþ ¼ f a; g ðþ; A ðrþnfa; b; gg: Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

25 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 47 Also there is an equivariance f ga; gb ðþ ¼ðgf a; b ÞðÞ for g A G. Therefore for any a A ðrþ the ma ð40þ g 7! f a; ga is a -cocycle. Since f k ðsþ ¼c 2k ðþ ¼ f y; Sy and f k ðtþ ¼0 ¼ f y; Ty, we have f k ðgþ ¼ f y; gy for any g A G. And for any a A ðrþ the cocycle (40) is homologous to f k. Finally, we remark that the roerties of f a; b given above mean eactly that the ma ða; bþ 7! f a; b is a modular seudo-measure on ðrþ in the sense of [7] taking values in the sace V2k 0 defined in the net section The generalized Eisenstein cocycle class. Definition. For k A Z, let V 2k be the sace of continuous functions f : R! R such that the limit lim 2k f ðþ eists and is finite.!y This is a reresentation of SLð2; RÞ with the action in weight 2k.3) Let V 2k 2 be the sace of real olynomials of degree e 2k 2. This is a finitedimensional reresentation of SLð2; RÞ (with the action in weight 2 2k, so V 2k 2 H V 2 2k ) and there is an equivariant ma I : V 2k! V 2k 2 given on f A V 2k by ð4þ ðif ÞðXÞ ¼ Ðþy f ðþðx Þ 2k 2 d: y Definition 2. Let V2k be the sace of functions f : R! R continuous at all but a finite number of oints with the action of SLð2; RÞ in weight 2k. The saces V 2k H V2k are reresentations of G, and the -cocycle f k from Lemma 0 takes values in V2k since c 2k A V2k. The net theorem shows that f k can be modified by a coboundary to a V 2k -valued cocycle with a rational image under the ma (4). Before formulating it, let us notice that in the long eact sequence!h 0 ðg; V 2k =V 2kÞ!H ðg; V 2k Þ!H ðg; V 2k Þ! the term H 0 ðg; V2k =V 2kÞ vanishes (since a G-invariant function can t have a finite nonzero number of oints of discontinuity), and therefore H ðg; V 2k Þ is a subsace of H ðg; V 2k Þ. f k Theorem 3. Let ½f k Š A H ðg; V2k Þ be the cohomology class of the -cocycle from Lemma 0. Then ½f k Š A H ðg; V 2k Þ, and I½f k Š A H G; 2k V 2k 2 ðqþ, where 3) There is a more symmetric definition of this reresentation as the sace of those continuous functions F : R 2 nf0g!r which are homogeneous of degree 2k, i.e. Fðl; lyþ ¼l 2k Fð; yþ. Then the right action of SLð2; RÞ is given by ðfgþð; yþ ¼Fða þ by; c þ dyþ, and f ðþ ¼Fð; Þ is an element of V 2k from Definition. One can find several other resentations for V 2k in []. Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

26 48 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields V 2k 2 ðqþ H V 2k 2 is the subsace of olynomials with rational coe cients. Namely, I½f k Š is the class of the -cocycle ð42þ roof. T 7! zð2kþ X 2k ðx Þ 2k ; 2k S 7! 2 2k k r¼ zð2rþzð2k 2rÞX 2r : From the asymtotic eansion of F 2k at infinity we conclude that c 2k ðþ ¼ zð2kþ zð2k Þ þ 2 2k 2k þ O 2kþ ;!þy: Let f : R! R be any continuous function such that the limit ð43þ lim!y 2k f ðþþ zð2kþ 2 zð2k Þ signðþþ 2k 2k eists and is finite. Then from the equivariance roerty of the functions f a; b in the revious section it follows that f k ðgþþð gþ f A V 2k for any g A G. Hence, f k þ qf is a V 2k -valued cocycle, so ½f k Š A H ðg; V 2k Þ. Let F be any smooth function on R with FðÞ ¼ zð2k Þlogjj zð2kþ jj 2k þ 2 2 C as! y with C smooth near 0. Then the limit (43) for f ðþ ¼ d 2k FðÞ ð2k Þ! d equals 0, and using the asymtotic eansion of F 2k near 0 we check that the function F 2k ðjjþ þ zð2kþ 2 jj þjj2k A 2k ðjjþ þ ð SÞFðÞ is continuously di erentiable 2k times on R (including ¼ 0). Its ð2k Þst derivative is obviously c 2k ðþþð SÞ f times ð2k Þ!. With this, by formula (54) from the aendi we find that I c 2k ðþþð SÞf ¼ A2k ðxþþa 2k ð XÞ 2k ¼ 2 2k k r¼ zð2rþzð2k 2rÞX 2r Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

27 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 49 and I ð TÞ f ¼ zð2kþ X 2k ðx Þ 2k ; 2k verifying that the cocycle Iðf k þ qf Þ A Z ðg; V 2k Þ is given by equation (42). r 2.3. Two eriods of the non-holomorhic Eisenstein series. The cocycle (42) is (a rational multile of) the reresentative of the cohomological class which corresonds to the holomorhic Eisenstein series E 2k ðzþ ¼ m; n 0 ðmz þ nþ 2k under the Eichler Shimura isomorhism between M 2k l S 2k and H G; V 2k 2 ðcþ (see [5],. 453). Let us briefly elain the relation of our theory to the Eisenstein series. The non-holomorhic Eisenstein series (9) is a Maass form in the sense that it is a G-invariant function in the uer half-lane satisfying the Lalace equation D z Eðz; sþ ¼sð sþeðz; sþ where D z ¼ðz zþ 2 q 2. qzqz Following [6], [], one can define the two eriods c A H ðg; V 2k Þ and c 2 A H ðg; V 2k 2 Þ of Eðz; kþ. Reresentatives of c and c 2 are defined by ðgz 0 2! 3 jx zj 2 k ðgz 0 2! 3 g 7! 4 jx zj Eðz; kþ; 5; k g 7! 4Eðz; kþ; 5; y y z 0 resectively, where z 0 is an arbitrary oint in the uer half-lane, and the bracket ½ ; Š is Green s form, ½uðzÞ; vðzþš ¼ qu qv vdzþ u dz. The Green form has the following roerties: qz qz (i) ½u; vš is closed if both u and v are eigenfunctions of D with the same eigenvalue. (ii) If z 7! gðzþ is any holomorhic change of variables, then ½u g; v gš ¼½u; všg. For every fied X both ðjx zj 2 =yþ k and ðjx zj 2 =yþ k are eigenfunctions of D with the eigenvalue kð kþ. Therefore both Green forms above are closed due to (i). Notice that! k! k jx zj2 jx zj2 z! and z! y y are SLð2; RÞ-equivariant mas from the uer halflane to V 2k and V 2k 2, corresondingly. Thus it follows from (ii) that both forms are G-equivariant (with values in the corresonding saces), so the above integrals indeed define -cocycles. Furthermore, one can check that for every fied z z 0 I! k jx zj2 ð2k 3Þ!! ¼ y ð2k 4Þ!!! k jx zj2 ; y Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

28 50 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields with n!! ¼ ð44þ Q 0ei<n=2 We claim that ðn 2iÞ as usual, so the eriods are related by Ic ¼ c ¼ i k 2 ð2k 3Þ!! ð2k 4Þ!! c 2: ð2k Þ!! ½fŠ; ð2kþ!! where ½fŠ is the cocycle class from Theorem 3. Indeed, let us take the reresentative of c given above and let z 0! y. Then the limiting -cocycle sends T 7! 0 and S 7! ð 0 y 2 4Eðz; kþ; jx zj 2 y! k 3 5 ¼ ikx ¼ i k 2 ðy 0 t k Eðit; kþ dt ðx 2 þ t 2 kþ Þ ð2k Þ!! ð2kþ!! signðxþ ; qf0 ðjxjþqþ 2k ; and we see it is a multile of the cocycle f from Lemma 0. Here the first equality is roven in [6] and the second calculation is due to Chang and Mayer [2], [6]. ið2k Þ c 4 2. We now use this to show the fol- So far, from (44) we have that I½fŠ ¼ lowing roosition. roosition 4. For k A Z f2 we have 2! 3 jx zj 2 k ð45þ 4Eðz; kþ; 5 a k ie 2k ðzþðx zþ 2k 2 dz; y with a k ¼ 2 2k ð 2kÞ A Q. Here o o 2, for G-equivariant V 2k 2 -valued -forms o and o 2, means that they di er by the full di erential of a G-equivariant V 2k 2 -valued function. This roosition imlies that I½fŠ is a rational multile of the class of E 2k ðzþðx zþ 2k 2 dz, so we get an alternative roof of Theorem 3. roof. The oerators d w ¼ q qz þ w z z ; d ¼ðz zþ2 q qz intertwine the action of SLð2; RÞ on the functions in the uer halflane in weights w, w þ 2 and w, w 2, corresondingly. Then dd w d w 2 d ¼ w. Let D w ¼ dd w, and denote by H w; l ¼fu j D w u ¼ lug the sace of eigenfunctions in weight w. We have d w : H w; l! H wþ2; lþwþ2 ; d : H w; l! H w 2; l w : Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

29 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields 5 For u A H w; l and v A H w; l w we introduce the generalized Green form as dz ½u; vš w ¼ d w ðuþvdzþ udðvþ ðz zþ 2 ¼ d wðuþvdzþ u q ðvþ dz: qz It is a closed form, and for g A SLð2; RÞ one has ½gu; gvš ¼g½u; vš. Also and ½u; vš w þ½v; uš w ¼ d w ðuþv þ d w ðvþu dz þ q ðuvþ dz qz Therefore we have roven that ¼ q qz ðuvþþ w z z uv w z z uv dz ½dv; d w uš w 2 ¼ d w 2 dvd w udzþ dvdd w u ðz zþ 2 dz þ q ðuvþ dz ¼ dðuvþ qz dz ¼ lvd w udzþ dvlu ðz zþ 2 ¼ l½u; vš w : ½u; vš w l ½d wu; dvš wþ2 ; u A H w; l ; v A H w; l w : If we aly this transformation k times to the functions u ¼ Eðz; kþ; v ¼! k jx zj2 A H 0; kð kþ; y we get (45) since d k 0 Eðz; kþ ¼ð 2iÞk ð2k Þ! ðk Þ! E 2kðzÞ and d k! k jx zj2 ¼ð2iÞ k ðk Þ!ðX zþ 2k 2 : r y 2.4. Homological formulation of the higher Kronecker limit formula. Notice that there is an SLð2; RÞ-equivariant airing V 2k n V 2 2k! R given by h f ; gi ¼ Ð R f ðtþgðtþ dt: Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M

30 52 Vlasenko and Zagier, Higher Kronecker limit formulas for real quadratic fields If g A V 2k 2 H V 2 2k is a olynomial, then we have h f ; gi ¼ðIf ; gþ where I is the ma (4) and the SLð2; RÞ-equivariant airing on olynomials is given by the formula 2k 2 a i X i ; 2k 2 b i X i ¼ 2k 2 ð Þ i a i b 2k 2 i : i¼0 i¼0 i¼0 2k 2 i We denote by the same brackets the airings on homology and cohomology h ; i : H ðg; V 2k Þ n H ðg; V 2 2k Þ!R; ð ; Þ : H ðg; V 2k 2 Þ n H ðg; V 2k 2 Þ!R: Let Q ¼ AX 2 þ BXY þ CY 2 be an integer quadratic form of discriminant D. We also denote by Q the function Qð; Þ of A R. Then Q k A V 2k 2. The stabilizer of Q in G is isomorhic to Z, and we ick that generator g Q A Stab G Q for which g Q ðþ < whenever QðÞ < 0. With this condition we then have g gq ¼ gg Q g, and therefore the class of the -cycle g Q n Q k in H ðg; V 2k 2 Þ deends only on the G-orbit of Q. Since G-orbits corresond to narrow ideal classes, we can give the following definition. Definition 5. Let for a narrow class of ideals B the elements B A H ðg; V 2k 2 Þ and h B A H ðg; V 2 2k Þ be the classes of -cycles g Q n Q k and g Q n Q k fq<0g, resectively, where Q is any integer quadratic form in B. Notice that Q corresonds to B y ¼ B 0 if Q corresonds to B, and g Q ¼ g Q. Therefore B y ¼ð Þ k B and ð46þ Theorem 6. ideal class B h B þð Þ k h B y ¼ B : Let ½f k Š be the cocycle class from Theorem 3. Then for every narrow h½f k Š; h B i ¼ð Þ k ðk Þ! 2 ð2k 2Þ! Dk 2 zðb; kþ: roof. Let us ick a form Q in the class B and a number a such that QðaÞ > 0. Due to Theorem 3 there eists an f A V2k such that the -cocycle ð47þ g 7! f a; ga þð gþ f (with f a; b defined by (39)) is V 2k -valued. Notice that we can choose f so that it vanishes outside of any small neighborhood of a. The -cocycle (47) is a reresentative of ½f k Š, and airing it with g Q n Q k fq<0g gives h½f k Š; h B i ¼ Notice that (9) can be rewritten as Ð Q<0 QðÞ k a< q <g QðaÞ d: 2k ðq Þ D n FðQÞ ¼ ð Þn signðaþ Ð QðÞ n F ð2nþþ ðþ d n! Q A <0 Brought to you by Ma-lanck-Gesellschaft - WIB647 Authenticated Download Date 6/4/3 2:57 M