Hammond, W.

Hirzebruch, F.

English

Hammond, W. F.

Hirzebruch, Friedrich

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L-Series, modular imbeddings, and signatures

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Math. Ann. 204, 263--270 (1973) @ by Springer-Verlag 1973 L-Series, Modular Imbeddings, and Signatures* W. F. Hammond and F. Hirzebruch Introduction For each cusp singularity of the Hilbert modular variety of a totally real algebraic number field there is associated an L-series which deter- mines the corresponding "parabolic contribution" to the version of Selberg's trace formula worked out by Shimizu. For a real quadratic field Meyer [9, 10] has given an elementary algorithm by Dedekind sums for the computation of the parabolic contribution (see also Siegel [14]) from which a simple formula for the parabolic contribution in terms of the continued fraction associated with the cyclic resolution of the corresponding cusp singularity can be derived ([5, 7, 8]). In fact, the parabolic ontribution equals ~2(3r-bo -b l  . . . . .  br-1) where bo ..... br-1 are the characteristic numbers of the dual normal bundles of the rational curves which form the excep- tional fibre of the cyclic resolution. The first purpose of this note is to give a simple formula for the total parabolic contribution, i.e., the sum of the parabolic ontributions for the different cusps. This formula is deduced from a general identity satisfied by certain L-series in any real algebraic number field. In the quadratic ase one obtains an expression for the total parabolic ontribu- tion in terms of the class numbers of imaginary quadratic fields. Further, in the quadratic ase the total parabolic ontribution vanishes ifand only if the discriminant of the field is the sum of two squares and is negative otherwise. The total parabolic contribution is of additional interest in the quadratic case for two reasons: first, it is one-fourth the signature (adjusted for defects coming from quotient singularities in certain special cases) of the open rational homology 4-manifold obtained as the quotient of the product of two upper-half planes by the non-symmetric Hilbert modular group. This result, announced to some extent in [5], will be proved here. Thus, from [2] the vanishing of this (adjusted) signature is a necessary and sufficient condition for the existence of modular im- beddings. Second, it will be shown that this signature is twice the difference of the arithmetic genera for the ordinary and mixed Hilbert modular groups. * The first author was supported in part by U.S. NSF grant SD GU3171. 264 W. F, Hammond and F. Hirzebruch: 1. A Relation between/..-Series Let K be a real algebraic number field, i.e., an extension of finite degree of the field of rational numbers which is isomorphic to a subfield of the real field. Let G stand for the group of ideal classes of K, and let Gn denote the norm class group of K: the quotient of the group of frac- tional ideals by the subgroup of principal ideals generated by elements of positive norm. A norm class character (being any character of Gn) is said to be of norm-signature type if it induces a non-trivial character on the multiplicative group of K. Such characters of Gn exist if and only if K has no unit of norm -1.  If norm-signature characters exist, then the kernel Pn of the canonical homomorphism Gn ~ G is a group of order 2 since K contains elements of negative norm (which follows, as Igusa has pointed out to the authors, from the "approximation theorem" for a finite number of inequivalent valuations - in this case the archimedean valuations of the field K), and conversely. In this case let 0 denote the unique non-trivial element of the group Pn; then norm-signature characters are those norm class characters satisfying Z(0)= - 1. Follow- ing traditional notation, for any norm class character X the symbol L(s, X) denotes the L-series ~,x(a)(Nma)-" (*) in which the summation is taken over all non-zero integral ideals a of K. Further, for a class A in G the symbol L(s, X, A) denotes the series of type (.) in which the summation is taken over the integrat ideals in A. Note that the square of an element of G is a well-defined element of Gn; if A is in G, A (2) will denote its square in Gn. For any character X the complex-conjugate character will be denoted by ;(*. Lemnm 1.1. For any ideal class A the expression z*(A (2)) L(s, X, A2) does not depend on the choice of the norm-signature character ~. The proof is obvious; the statement is recorded only for ease of reference. Theorem 1.2. Let ~ be any character of norm-signature type, and let X1 ..... 7., denote the real characters of norm-signature type. Then Z L(s, z, A = LO, ZJ)- AeG ./=1 14'hen r = 0, the right-hand side is understood to be zero. Proof. The argument is given in two cases according to whether or not r = 0. First suppose r = 0. Then X(0)= 1 for every real character Xof Gn. For any finite abelian group the real characters are precisely the characters of order 2. Hence, by the duality theory of finite abelian L-Series, Modular Imbeddings, and Signatures 265 groups 0 must be an element of the group Sqn of squares of elements of Gn. Letting Sq denote the group of squares in G, the left-hand side of the formula to be proved may be written Z B) BESq AA=B The inner sum may be re-written as x*(A g ') E C where A0 is a fixed element of G whose square is B and where the summa- tion is taken over the group of elements C of order 2 in G. This latter group is mapped surjectively to Pn by the homomorphism C~C ez) since 0 is in Sqn. Hence, the sum is zero, as required. Next suppose that r is positive. Then one may take ~ real in the left- hand side by the lemma, and the left-hand side becomes r Y L(s, x, B) B~Sq since r is easily seen to be the order of the group of elements of order 2 in G. The right-hand side of the formula to be proved is AeG 1=i and this sum may be split according to whether or not A lies in the group Sq of squares in G. If A is a square, then the fact that Z/is real yields L(s, ZI, A)= L(s, X, A} by the lemma, so that the part of the latter sum involving A in Sq agrees with the left-hand side. It remains to be seen that ~ L(s, xj, A)=O )=!  when A is not a square. This follows from the assertion that xj(a) = 0 j= l  for every ideal a in A. To see this last fact let X,+ 1 .. . . .  X2, be the remaining real characters of Gn. Then these are the real characters of the group G, and, therefore, the sum of x~(a) asj varies from r + 1 to 2r is zero since A is not a square. The proof is completed by the similar observation that the sum of ~j(a) as j varies from 1 to 2r is also zero since the norm class of a is also not a square. Supplement 1.3. The number r of real norm-signature characters vanishes if and only if the non-trivial element 0 of the group Pn is the square of an element of Gn. 266 w.F. Hammond and F. Hirzebruch: 2. The Total Parabolic Contribution In the case where K is a totally real number field of degree n with discriminant d the ideal classes are in bijective correspondence withthe cusp singularities of the Hilbert modular variety, and the parabolic contribution, according to Shimizu [13], for a given cusp has the value w A = ( -  1)"/2n-"d½z*(A(2) ) L(1, Z, A2) whenever norm-signature characters )C xist (and the value zero other- wise). Consequently, in view of Theorem 1, the total parabolic contribu- tion is given by W=(--1)'~:27c-nd ½ ~ L(1, Xj), j= l  which generalizes the formula given by Shimizu for the case where the ideal class group G has odd order (hence r = 1). In the quadratic ase the real norm-signature characters are given in avery explicit form (see Siegel [14]). Precisely, the real norm-signature characters ~( correspond bijectively to factorizations d= d 1 d 2 of the discriminant d into the product of two negative discriminants dl, d 2. For such X L(1, X) = 4n2 d-½ h(dt) h(d2) u(dl)- 1 u(d2 )- 1, where h(m) denotes the ideal class number and u(m) the order of the unit group for the quadratic field of discriminant m. Thus, Theorem 2.1. The total parabolic contribution for the real quadratic field of discriminant d is given by the formula: w = - 4 Y~ h(dl) h(d2) u(dl)- 1 u(d2 )- 1 with the summation taken over all decompositions d = dl d2 in which dl, d2 are the discriminants of imaginary quadratic f elds. (The decomposition d= d2d 1 is identified with d= did2.) Coronary 2.2. The total parabolic contribution for the real quadratic field of discriminant d vanishes if and only if d is the sum of two squares, i.e., d contains no prime -3rood4.  Otherwise it is negative. Proof. In the first place, the norm class group Gn coincides with the narrow class group, which is defined for any totally real field as the quotient of the group of fractional ideals by the subgroup of principal ideals generated by totally positive lements. Secondly the different of K, which is the principal ideal generated by d ½, defines the non-trivial norm ideal class 0. Thus, from the supplement (1.3), the number of real norm- signature characters vanishes if and only if the narrow ideal class of the different is a square. By a result of Igusa (see [2]) this is equivalent to the condition that the discriminant d should be the sum of two squares. For L-Series, Modular Imbeddings, and Signatures 267 the convenience ofthe reader a special case available from Shimizu [13] and Siegel [14] is stated: Corollary 2.3. For the field Q(]/~), p prime p= 3 (mod 4) one has: {-_ 1/6, p= 3. w= ½h(-p), p>3.  3. Signatures In earlier articles [3, 6] the authors have obtained two expressions for the arithmetic genus X of the minimal desingularization f the normal projective surface which compactifies the quotient H x H/F of the product H × H of two copies of the upper-half plane by the Hilbert modular group F of a real quadratic field K. Combining these two expressions enables one to prove quickly the following theorem: Theorem 3.1. Let d be the square-free part of the discriminant of the real quadratic field K. Then the signature of the rational homology 4-mani- fold H x H/F is given by I ~, d<6,  sgn(II x H/F)= 4w-~h( -d /3 ) ,  d> 6, d=O(mod3), [4w, d > 6, d ~ 0(mod3), where w is the total parabolic contribution and h(-  d/3) denotes the ideal class number of the field Q(~/3) .  Proof. Denoting by e( ) the Euler number, one has from [6] sgn(H x H/F)= 4z -e (H  x H/F), where r -  1 e(H × H/F) = 2~r{- 1) + ~ a(r, q ) - - ,  (r,q) r with aft, q) denoting the number of quotient singularities of type (r, q), i.e., arising from the cyclic group action on the comptex plane with eigenvalues 2,2 ~, (2'= 1). From [3] (see also [1]) 4 Z -- 2(r( - 1) + 4 ~. a(r, q) c(r, q) + 4w, (r, q) where c(r, q) is the contribution to Shimizu's formula for a quotient singularity of type (r, q). From Shimizu [13] one has c(r, q) = r- ~ £gq(2), where the summation is extended over the non-trivial r th roots of unity 2: 2"= 1, 2 =~ 1, and g~ denotes the rationat function of one variable g~(z)=(1- z)-~ (1 -  #) -~ . 268 W.F.  Hammond and F. Hirzebruch: The c(r, q) are very close to Dedekind sums and to the signature defects def(r; q, 1) introduced in [4], formula (27). We have def(r; q, 1) = X(lq- 2) (1 + 2 q) gq(2) where the summation isextended as before. The following trivial formula holds: 4 Z g~(2) = r - 1 + def(r; q, 1). Therefore we get Theorem 3.2. sgn (H x H/F) = 4w + ~ a(r, q) r-  1 def(r; q, 1). (r, q) The calculation of signature defects can be done by the following classical formulas ([12], compare also [4], p. 16-17): For any real x let Ix] denote the greatest integer not larger than x, and let ((x)) be defined by f0 if x is an integer ( i x ) )= ' x-  [x]--½, otherwise. For r, q coprime integers with r positive let then ' s(q, r)= ~, k=l  def(r; q, 1) = - 4r s(q, r). (3.1.1) (3.1.2) To finish the proof of (3.1) it is sufficient to apply the formulae (3.1.t) and (3.1.2) together with the results of Prestel [11] on the computation of the a(r, q). When d < 6, one must also compute w using, for example, (2.1). (It vanishes for d = 2 and d = 5.) For d __> 6 the only values of r are 2 and 3. One may forget the case r= 2 since s(1, 2)= 0, while for r= 3 one has s(1, 3) = - s(2, 3) = 1/18. From Prestel ~3h(-  d/3), d -  0 (mod3). a(3, 1) - a(3, 2) = [0, d ~ 0 (mod 3). As a consequence of (3.1) and (2.3) one has immediately: Corollary 3.3. I f  d = p, a prime, with p - 3 (mod4) and p > 3, then sgn(H x H/F) = - 2h( -  p). 4. Application to the Mixed Hilbert Modular Group The Hilbert modular group F of a real quadratic field K acts also on the product H x H' of the upper-half plane and the lower-half plane. L-Series, Modular Imbeddings, and Signatures 269 The rational homology manifolds H × H'/F and H x H/F are homeo- morphic with opposite orientation, and, therefore, e(n x H'/F)= e(n × H/F) and sgn(H x H'/F) = - sgn(H × H/F). Denoting by Z' the arithmetic genus of a smooth model for the field of modular functions associated to H × H'/F, one has as in Section 3: 4x'=e(n x H'/F)  + sgn(n  x H'/F). If n and n' denote the dimensions of the spaces of cusp forms of lowest weight, then one knows from Freitag [1] that n = Z-  1 and n '= Z ' -  1. Thus, Theorem 4.1. The difference of the arithmetic #enera o , equivalently, the difference of the dimensions of the spaces of cusp forms is given by the formula: X' -- X = n' - n = - ½sgn (H x H/F) >= O. Moreover, this difference vanishes if and only if d = 3 or d is not divisible by a prime = 3 mod 4. Finally, one has the special case: Corollary 4.2. I f  d = p, a prime, with p = 3 (rood4) and p > 3, then X ' - ; t=n ' -n=h( -p ) .  References 1. Freitag, E.: Lokale und globale Invarianten der Hilbertschen Modulgruppe. Invent. Math. 17, 106---134 (1972). 2. Hammond, W.F.: The modular groups of Hilbert and Siegel. Amer. J. Math. 88, 497--516 (1966). 3. Hammond, W. F.: The HUbert modular surface of a real quadratic field. Math. Ann. 200, 25---45 (1973). 4. Hirzebruch, F.: The signature theorem: reminiscences and recreation. Prospects in Mathematics, Ann. Math. Stud., no. 70, Princeton, 1971. 5. Hirzebrnch, F.: The Hilbert modular group, resolution of he singularities at thecusps and related problems. S6minaire Bourbaki, exp. 396 (1971). 6. Hirzebruch, F.: The Hilbert modular group and some algebraic surfaces, International Symposium in Number Theory, Moscow, 1971 (to appear). 7. Hirzebruch, F.: Hilbert modular surfaces. IMU-leetures, Tokyo, 1972 (to appear in L'Enseignement Math6matique). 8. Hirzebrueh, F., Zagier, D.: Class numbers, continued fractions and the Hilbert modular group (in preparation). 9. Meyer, C. : Die Bereehnung derKlassenzahl abelseher K6rper fiber quadratisehen Zahl- k6rpern. Berlin 1957. 10. Meyer, C.: 1Dber die Bildung yon dementar-arithmetischen Klasseninvarianten in reell-quadratischen ZahlkSrpern. Aigebraiscbe Zahlentheorie, Oberwolfach, pp. 165--215. Mannheim: Bibliogr. Institut 1966. 270 W. F. Hammond and F. Hirzebruch: L-Series, Modular Imbeddings, and Signatures 11. Prestel, A.: Die elliptisehen Fixpunkte der Hilbertsehen Modulgruppen. Math. Ann. 177, 181--209 (1968). t2. Rademacher, H., Grosswald,E.: Dedekind Sums. Math. Assoc. of America, 1972. 13. Shimizu, H. : On discontinuous groups operating on the product of the upper half planes. Ann. of Math. 77, 33--71 (1963). 14. Siegel, C.L.: Lectures on advanced analytic number theory. Tata Inst., Bombay, 1961 (re-issued 1965). W. F. Hammond Dept. of Mathematics State Univ. of N.Y. at Albany 1400 Washington Ave. Albany, N.Y. 12222, USA F. Hirzebruch Mathematisches Institut Universit~it Bonn, WegelerstraBe 10 D-5300 Bonn Federal Reputic of Germany (Received January 9, 1973) 

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