1. A refinement of a classical class number relation We give a refinement, and a new proof, of the following classical result [1, 2, 3].

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1 A COMBINATORIAL REFINEMENT OF THE KRONECKER-HURWITZ CLASS NUMBER RELATION rxiv:604.08v [mth.nt] Apr 06 ALEXANDRU A. POPA AND DON ZAGIER Astrct. We give refinement of the Kronecker-Hurwitz clss numer reltion, sed on tesseltion of the Eucliden plne into semi-infinite tringles leled y PSL (Z) tht my e of independent interest.. A refinement of clssicl clss numer reltion We give refinement, nd new proof, of the following clssicl result [,, 3]. Theorem (Kronecker, Gierster, Hurwitz). For ny n we hve mx(,d). t 4nH(4n t ) = n=d,d>0 Here H(D) (D 0,D 0,3 mod 4) is the Kronecker-Hurwitz clss numer, which hs initil vlues D H(D) nd for D > 0 equls the numer of PSL (Z)-equivlence clsses of positive definite integrl inry qudrtic forms of discriminnt D, with those clsses tht contin multiple of x + y or of x xy + y counted with multiplicity / or /3, respectively. Let Γ = PSL (Z). By the Γ-equivrint ijection ( ) c d cx +(d )xy y etween integrl mtrices of determinnt n nd trce t nd qudrtic forms of discriminnt t 4n, the clss numer reltion cn e written s () M M n M elliptic χ(z M ) = n=d,d>0 mx(,d) + { /6 if n is squre, 0 otherwise, where M n is the set of integrl mtrices of determinnt n modulo ±, z M is the fixedpoint ofnelliptic M in the upperhlf-plne H, ndχ : H Qis the modified chrcteristic function of the stndrd fundmentl domin F = {z H : / Re(z) /, z } The first uthor ws prtly supported y CNCSIS grnt TE He would like to thnk the MPIM in Bonn nd the IHES in Bures-sur-Yvette for providing support nd stimulting reserch environment while working on this pper.

2 ALEXANDRU A. POPA AND DON ZAGIER of Γ cting on H such tht χ(z) is /π times the ngle sutended y F t z (so χ is in the interior of F, 0 outside of F, / on the oundry points different from the corners ρ = e πi/3 nd ρ, nd /6 t the corners). We will prove refinement of () sying tht the susum of the expression on the left over ll M in given orit of the right ction of Γ on M n lwys tkes on one of the vlues 0,, (or 7/6 for the orit nγ if n is squre). Specificlly, let us define for ny right coset K in M n /Γ (more precisely, K is right coset in PGL (Q)/Γ, since M n is not group) two positive integers δ K nd δ K y δ K = gcd(c,d), δ K = n/δ K, where ( c d) is ny representtive of K. Then we hve: Theorem. For ech right coset K M n /Γ we hve M K M elliptic χ(z M ) = +sgn(δ K δ K )+ cosetsin the disjoint decomposition M n = ( δ β 0 δ { /6 if K = n Γ, 0 otherwise. Eqution () follows immeditely y summing the) reltions in Theorem over ll Γ with n = δ δ nd 0 β < δ. Theorem provides correspondence etween right cosets nd Γ-conjugcy clsses in M n, which genericlly ssigns two conjugcy clsses to ech coset with δ > δ. We will deduce it from similr sttement, Theorem 3, which is shrper in two respects (it counts the numer of mtrices with fixed point in smller domin, nd it llows rel coefficients), nd which gives genericlly one-to-one correspondence etween cosets nd conjugcy clsses. To stte it, we consider hlf-fundmentl domin F = {z H : / Re(z) 0, z }, nd define function α : GL + (R) Q y χ (z M ) if M is elliptic with fixed point z M H, α(m) = if M is sclr, 0 if M is prolic or hyperolic, where χ is defined in the sme wy s χ (nd hence equls in the interior of F, 0 outside F, / on the internl oundry points of F, nd /4 nd /6 t the corners i nd ρ, respectively). Note tht α( M) = α(m), so α is well-defined on MΓ. Theorem 3. For M = ( y x 0 ) GL (R) with y > 0, we hve () α ( Mγ ) +sgn(y ) =. γ Γ Since ech coset K M n /Γ contins representtive M with M =, Theoremimmeditelyfollowsfrom(), ndthefctthtthemp± ( ) ( c d ± ) c d is ijection etween the sets of elements in M n hving fixed point in the left hlf nd in the right hlf of the stndrd fundmentl domin for Γ. Theorem 3 is proved in Section 3, s n esy consequence of tringultion of Eucliden hlf-plne y tringles ssocited to elements of Γ (Theorem 4). This tringultion my e of independent interest, nd we give self-contined tretment in the next section.

3 REFINEMENT OF THE KRONECKER-HURWITZ CLASS NUMBER RELATION 3. A tringultion of Eucliden hlf-plne Let Γ = {γ Γ g = }. We identify Γ Γ with suset of SL (Z) y choosing representtives γ = ( c d) with c > 0, nd for such γ we define semi-infinite tringle (3) (γ) = {(x,y) R 0 dx y c dx y}. (The motivtion for this definition is tht (x,y) (γ) if nd only if ( y x 0 )γ hs fixed point in F.) Note tht (γ) is contined in the hlf-plne H = {(x,y) R y }, since y = c( dx y)+d d(dx y) c +d d. Theorem 4. We hve tesseltion H = γ Γ Γ (γ) of the hlf-plne H into semi-infinite tringles with disjoint interiors. Remrk. We cn extend the tringultion of Theorem 4 to tringultion of ll of R y tringles leled y ll of Γ if we define (γ) lso for γ Γ y ( ( n 0 )) = [ n, n] (,], nd cn then interpret the extended tringultion s giving piecewise-liner ction of Γ on R, with ech tringle eing fundmentl domin. However we will not use this in the sequel. Proof. The groupγis free product ofits twosugroupsgenertedy the elements S = ( ) ( 0 0 nd U = 0 ) of orders nd 3, respectively, which fix the two corners of F. Therefore we cn view elements of Γ s words in S,U,U or s vertices of the tree shown in Figure. The proof of oth Theorems 3 nd 4 will SUS SU S USU USU U SU U SU SU SU US U S S U U = S = U = U Figure. A tree ssocited to Γ = PSL (Z): the vertices re leled y the elements of Γ nd the edges y the genertors S, U nd U s shown. follow from the following decomposition into tringles with disjoint interiors: (4) R := {(x,y) R 0 x y } = (γ), γ T

4 4 ALEXANDRU A. POPA AND DON ZAGIER where T Γ is the set of words strting in U. The regions H nd R nd few tringles corresponding to words of smll length re pictured in Figure. (SU ) y USU SU SU USU SU SU S S U US U SU USU U SUS SUSU SUS U SUSUS y = Figure. The region R (shded) nd few tringles (γ). The finite side of tringle (γ) hs een leled y the finl letter of γ s word in S,U,U, with the sme convention s in Figure. To prove (4), let T = T + T, where T + consists of the elements of T ending in U or U, while T := T + S consists of those elements ending in S. The set T + cn e enumerted recursively y strting t U nd replcing γ = ( c d) t ech step y γsu = ( ) + c c+d, γsu = ( + c+d d). From this description we esily otin the following equivlent chrcteriztions γ T + 0 c < d, γ T 0 d < c. Alterntively, T + consists of those γ T hving d > 0. For γ Γ Γ, the tringle (γ) hs two vertices given y P 3 ( d+c,c +d d), P ( d,c +d ), connected y line segment of slope d, nd it hs two infinite prllel sides of slope. Forγ T wedenoteyc(γ) Hthehlf-conecontining (γ), ounded y hlf-lines of slopes / nd /d, nd hving s vertex P 3 or P, depending on whether γ T + or γ T respectively (see Figure 3). Using this informtion, it is esy to check tht for γ T + nd γ = γs T we hve the following decompositions into sets with disjoint interiors (see the right picture in Figure 3): C(γ) = (γ) C(γ ), C(γ ) = (γ ) C(γ U) C(γ U ). By induction we otin tht R = C(U) is the union of the tringles indexed y T, proving (4). Recll our convention tht c > 0.

5 REFINEMENT OF THE KRONECKER-HURWITZ CLASS NUMBER RELATION 5 d d + d + d d γsu γsu γ γs P 3 P P P 3 P 3 Figure 3. Left: The cone C(γ) nd the tringle (γ) C(γ) in the cse γ T +. Right: The cone C(γ) decomposes into two tringles nd two smller, higher-up cones. On top of ech line we hve mrked its slope. Finlly we show tht the decomposition in (4) implies the decomposition of H given in Theorem 4. From the prentheticl remrk following (3) it is cler tht (Tγ) = T (γ), where T = SU = ( 0 ) nd Γ cts on H y T n (x,y) = (x ny,y). The region (5) R = R (U ) = {(x,y) H : 0 x < y} (see Figure ) is fundmentl domin for this ction of Γ on H, nd we otin the following decompositions into tringles with disjoint interiors {(x,y) H y x} = (γ), {(x,y) H x 0} = (γ), γ T γ T where T consists of words strting with U, ut different from (U S) n = T n with n > 0, while T consists of words strting with S, ut different from (SU) n = T n with n > 0. Theorem 4 follows since Γ Γ = T T T. 3. Proof of Theorem 3 Since () is invrint under multiplying M = ( y x 0 ) on the right y elements in Γ, we ssume without loss of generlity tht 0 x < y. If Mγ is sclr for γ Γ, the only possiility is esily seen to e M =. In this cse, α(γ) 0 for γ {,S,U,U }, nd () holds since =. Assuming tht M, it follows tht α(mγ) 0 if nd only if Mγ hs fixed point in F, tht is (x,y) (γ). We conclude from Section tht y, so the point (x,y) elongs to the region R in (5), nd γ = U or γ T y (4). Therefore the elements γ such tht α(mγ) 0 depend on the position of the point (x,y) with respect to the tringultion of R s follows (see Figure 3): y = nd 0 < x < : α(mu ) = / ; (x,y) is in the interior of tringle (γ): α(mγ) = ;

6 6 ALEXANDRU A. POPA AND DON ZAGIER (x,y) is on common side etween (γ) nd (γ ), ut it is not vertex: α(mγ)+α(mγ ) = + = ; (x,y) R is the P vertex of the tringle (γ) for γ T + : α(mγ)+α(mγs)+α(mγu) = = ; (x,y) R is the P 3 vertex of (γ ) with γ T : α(mγ )+α(mγ U)+α(Mγ U )+α(mγ S) = =. References. J. Gierster, Üer Reltionen zwischen Klssenzhlen inärer qudrtischer Formen von negtiver Determinnte. Mth. Ann. (880), 50. A. Hurwitz, Üer Reltionen zwischen Klssenzhlen inärer qudrtischer Formen von negtiver Determinnte. Mth. Ann. 5 (885), L. Kronecker, Üer die Anzhl der verschiedenen Klssen qudrtischer Formen von negtiver Determinnte. J. Reine Angew. Mth. 57 (860), Institute of Mthemtics Simion Stoilow of the Romnin Acdemy, P.O. Box - 764, RO Buchrest, Romni E-mil ddress: lexndru.pop@imr.ro Mx-Plnck-Institut für Mthemtik, Vivtsgsse 7, 53 Bonn, Germny E-mil ddress: don.zgier@mpim-onn.mpg.de