Geometriae Dedicata 88: 37^53, 2001. # 2001 Kluwer Academic Publishers. Printed in the Netherlands.
37
Lattice Points Inside Rational Simplices and the Casson Invariant of Brieskorn Spheres LIVIU I. NICOLAESCU Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A. e-mail:
[email protected] (Received: 21 January 2000; in ¢nal form: 20 November 2000) Abstract. We express the number of lattice points inside certain simplices with vertices in Q3 or Q4 in terms of Dedekind ^ Rademacher sums. This leads to an elementary proof of a formula relating the Euler characteristic of the Seiberg^ Witten-Floer homology of a Brieskorn Z-homology sphere to the Casson invariant. Mathematics Subject Classi¢cations (2000). 11F20, 11P21 57M27. Key words. Seiberg^ Witten^ Floer homology, Casson invariant, Brieskorn spheres, lattice points, Dedekind sums.
0. Introduction From the very beginning, it was apparent that the Seiberg^Witten analogue of the instanton Floer homology of a (Z-)homology 3-sphere is no longer a topological invariant, since it can vary with the metric. W. Chen [2], Y. Lim [7] and Marcolli^Wang [8] have explained the metric dependence of the Euler characteristic of the SWF (¼ Seiberg^Witten^Floer) homology. More precisely, if gi ði ¼ 0; 1Þ are two generic Riemann metrics on a homology 3-sphere N and lSW ðN; gi Þ is the Euler characteristic of the SWF homology of ðN; gi ), the results of [2, 7, 8] imply that wSW ðN; g1 Þ wSW ðN; g0 Þ ¼ 18 Fðg1 Þ 18 Fðg0 Þ; where FðgÞ ¼ 4Zdir ðgÞ þ Zsign ðgÞ Zdir ðgÞ denotes the eta invariant of the Dirac operator of ðN; gÞ while Zsign ðgÞ denotes the eta invariant of the odd signature operator on ðN; gÞ. In particular, the above equality shows that the quantity aðNÞ ¼ wSW ðN; gÞ þ 18 FðgÞ is independent of g and thus is a topological quantity.
38
LIVIU I. NICOLAESCU
In 1996, W. Chen, P. Kronheimer and T. Mrowka have conjectured that this quantity coincides (up to a sign) with the Casson invariant of N. This has been established recently by Y. Lim, [6]. In the present paper we consider in greater detail the special case of Brieskorn homology spheres Sða1 ; . . . ; an Þ with at most 4 singular ¢bers, n W 4 and explore the rich arithmetic hiding behind these objects. More precisely, the results [13] show that for a certain natural metric g0 on Sða1 ; a2 ; . . . ; an Þ (realizing the Thurston geometric of this Seifert manifold), and we have wSW ðSða1 ; . . . ; an Þ; g0 Þ ¼ 2Ca1 ;...;an ;
n ¼ 3; 4
where Ca1 ;...;an is the number of lattice points in the simplex ( !) n n X X x 1 1 i n2 Dða1 ; . . . ; an Þ :¼ ðx1 ; . . . ; xn Þ 2 Rn ; xi X 0; < : ai ai 2 i¼1 i¼1 The ¢rst goal of this paper is the direct, explicit and elementary determination of Ca1 ;...;an when n ¼ 3; 4. This arithmetic problem intrigued the author since the vertices of the simplex Dða1 ; . . . ; an Þ are not lattice points and some of the counting techniques using Riemann^Roch theorem on toric varieties seem to apply.* Still, this is not the most general rational simplex. It displays a miraculous symmetry (Lemma 2.2) which, when used in conjunction with a generalization of an idea of Mordell (see [10, 15]), leads to an explicit formula of this number in terms of Dedekind^Rademacher sums (see below). In fact, very little additional effort is needed to compute the entire Ehrhart polynomial of this polytope. On the topological side of the story, R. Fintushel and R. Stern have shown in [3] that the Casson invariant lðSða; b; c; ÞÞ of the Brieskorn sphere Sða; b; cÞ is 1 8 sða; b; cÞ where sða; b; cÞ denotes the signature of the Milnor ¢ber of Sða; b; cÞ. This result was extended to arbitrary Sða1 ; . . . ; an Þ by Neuwmann^Wahl in [11], lðSða1 ; a2 ; . . . ; an ÞÞ ¼ 18 sða1 ; . . . ; an Þ:
ð0:1Þ
The Chen^Kronheimer^Mrowka formula in this case is equivalent to 2Ca1 ;...;an 18 Fðg0 Þ ¼ 18 sða1 ; . . . ; an Þ:
ð0:2Þ
According to Zagier (see [4, 11]) the signature of sða1 ; . . . ; an Þ can be expressed in terms of Dedekind sums. In [13] we have expressed Fðg0 Þ in terms of Dedekind^Rademacher sums as well. Thus (0.2) becomes an identity between Dedekind sums. The second goal of this paper is to prove the identity between Dedekind^Rademacher sums by elementary means. *M. Vergne has kindly pointed out to the author that the paper [1] contains a description of the Ehrhart polynomial of a general rational polytope. However, we believe that the amount of work required to translate the general formulae of [1] into the very explicit language of Dedekind–Rademacher sums needed for the topological applications in this paper would involve about the same amount of work as our ad-hoc proof based on Mordell’s trick.
LATTICE POINTS AND THE CASSON INVARIANT
39
The Dedekind^Rademacher sums are de¢ned for every coprime positive integers h, k, and any real numbers x; y by sðh; k; x; yÞ ¼
k1 X m þ y hðm þ yÞ k
m¼0
k
þx
where for any r 2 R we set
ððrÞÞ ¼
0; frg 12 ;
r 2 Z; r 2 R n Z;
ðfrg :¼ r ½r Þ:
The above description is theoretically very convenient but computationally cumbersome. Due to the reciprocity law (see [14] or the Appendix) the numerical determination of these sums in concrete cases is as computationally complex as the Euclid’s algorithm for the pair ðh; kÞ. The sums sðh; k; 0; 0Þ are precisely the Dedekind sums sðh; kÞ discussed in great detail in [4,15]. The present paper consists of three sections and an appendix. In the ¢rst section we survey the results of [13] which express F in terms of Dedekind^Rademacher sums and reduce the computation of wSW to a lattice point count. In the next section, we describe a generalization of an idea of Mordell which reduces the lattice point count to a certain arithmetic expression. The third section describes this arithematic expression in terms of Dedekind^Rademacher sums and completes the proof of (0.2). For the reader’s convenience we have included a brief appendix containing the basic properties of Dedekind^Rademacher sums used in this paper.
1. Geometric Preliminaries For pairwise coprime integers a1 ; . . . ; an X 2, n X 3 we denote by Sð~aÞ, a~ ¼ ða1 ; . . . ; an Þ, the Brieskorn homology sphere Sða1 ; . . . ; an Þ with n singular ¢bers (see [5] for a precise de¢nition). We orient Sð~aÞ as the boundary of a complex manifold. Sð~aÞ is a Seifert manifold. With respect to the above orientation it is a singular S 1 ¢bration over the orbi-sphere S 2 ð~aÞ which has n cone points of isotropies Zai , 1 W i W n. This ¢bration has rational degree ‘¼
1 ; A
A :¼ a1 a2 ; . . . ; an :
~ ¼ ðb ; . . . ; b Þ (normalized as in [13]) are Set bi :¼ A=ai . The Seifert invariants b 1 n de¢ned by bi bi 1 ðmod aÞ;
0 W b i < ai :
2 aÞ has Set qi :¼ b1 i ¼ bi (mod ai ), i ¼ 1; . . . ; n. The canonical line bundle of S ð~
40
LIVIU I. NICOLAESCU
rational degree k ¼ kð~aÞ :¼ ðn 2Þ
n X 1 : ai i¼1
The universal covering space of Sð~aÞ is the Lie group G ¼ Gð~aÞ
SUð2Þ; kð~aÞ < 0; Gð~aÞ ¼ g PSL2 ðRÞ; kð~aÞ > 0; g 2 ðRÞ denotes the universal cover of PSL2 ðRÞ. Moreover, Sð~aÞ ffi G=G where PSL where G is a discrete subgroup of G. The natural left invariant metrics on G (see [17]) induce a natural metric g0 on Sð~aÞ. All the geometric quantities discussed in the sequel are computed with respect to this metric and for simplicity we will omit g0 from the various notations. Thus Fð~aÞ is Fðg0 Þ. Set nko 1 ; A even; r :¼ ¼ 2 0; A odd; 2‘ and de¢ne ~g ¼ ðg1 ; . . . ; gn Þ by the equalities gi ¼ mbi ðmod ai Þ;
1 W i W n;
where m is the integer m :¼
k u ðn 2ÞA 2r r¼ ; 2‘ 2
u :¼
X
bi :
i
In [13] we have proved the following:
If A is even then Fð~aÞ ¼ 1 4
X
sðbi ; ai Þ
i
4
Xqi g þ r g þ bi r i ; r : þ 2s bi ; ai ; i ai ai i
ð1:3Þ
The above expression can be further simpli¢ed using the identities sðbi ; ai Þ ¼ sðbi ; ai Þ; gi þ bi =2 1 s bi ; ai ; ; ai 2 qi gi þ 12 1 1 1 ¼ s bi ; ai ; ; : 2 2 2 ai
ð1:4Þ
ð1:5Þ
The identity (1.4) is elementary and can be safely left to the reader. The identity (1.5)
LATTICE POINTS AND THE CASSON INVARIANT
41
is proved in the Appendix. Putting the above together we deduce that when A is even we have X X 1 1 Fð~aÞ ¼ 1 þ 4 sðbi ; ai Þ þ 8 s bi ; ai ; ; : ð1:6Þ 2 2 i i
If A is odd then X 1 4 sðbi ; ai Þ A i n X g i þ bi r qi g i þ r 2s bi ; ai ; ; r þ 4 : ai ai i¼1
Fð~aÞ ¼ 1
Similarly, we have an identity g 1 qi g i 1 1 ¼ s bi ; ai ; ; ; s bi ; ai ; i ; 0 þ 2 2 2 ai ai
ð1:7Þ
ð1:8Þ
and we deduce Fð~aÞ ¼ 1
X X 1 1 1 þ4 sðbi ; ai Þ þ 8 s bi ; ai ; ; : A 2 2 i i
ð1:9Þ
The signature sð~aÞ of the Milnor ¢ber of Sð~aÞ can be expressed in terms of Dedekind sums as well (see [11, Section 1]) sð~aÞ ¼ 1
X ðn 2ÞA 1 1 X bi þ þ 4 sðbi ; ai Þ: 3 3A 3 i ai i
ð1:10Þ
We deduce that Fð~aÞ þ sð~aÞ ¼
X ðn 2ÞA e 1 X bi 1 1 þ þ þ8 s bi ; ai ; ; ; 3 3A 3 i ai 2 2 i
where
e¼
1 2
A even; A odd:
The description of wSW requires a bit more work. Introduce the simplex ( ) X xi k n < : Dð~aÞ ¼ x~ 2 Z ; xi X 0; ai 2 i For each x~ 2 Dð~aÞ set X xi dð~xÞ ¼ ; ai i
ð1:11Þ
42
LIVIU I. NICOLAESCU
and denote by S x~ the symmetric product of dð~xÞ copies of S 2 . Note that if n ¼ 3; 4 then dð~xÞ ¼ 0 for all x~ 2 Dð~aÞ so that S x~ consists of single point. The irreducible part of the adiabatic Seibert^Witten equations on Sð~aÞ was studied S in [12, 13] and can be described as Ma~ ¼ x~ Mx~ where Mx~ ¼ Mþ ~ [ Mx~ , Mx~ ffi S x~ . x Moreover, the virtual dimensions of the spaces of ¢nite energy gradient £ows originating at the unique reducible solution and ending at one of the Mx~ are all odd. Using the adiabatic argument in ‰3.3 of [13] we deduce that if all dð~xÞ are zero the Seiberg^Witten^Floer homology obtained using the usual Seiberg^Witten equations is isomorphic with the Seiberg^Witten^Floer homology obtained using the adiabatic equations. Moreover, all the even dimensional Betti numbers of the Seiberg^Witten^Floer homology are zero and we deduce wSW ð~aÞ ¼ 2Ca~ :¼ 2#Dð~aÞ:
ð1:12Þ
The main result of this paper is the following: THEOREM 1.1. If a~ ¼ ða1 ; . . . ; ; an Þ 2 Znþ , n ¼ 3; 4 has mutually coprime entries then 16Ca~ ¼ Fð~aÞ þ sð~aÞ ¼
X ðn 2ÞA e 1 X bi 1 1 þ þ þ8 s bi ; ai ; ; : 3 3A 3 i ai 2 2 i
ð1:13Þ
According to (0.1), (1.12) and (1.11) this is equivalent to 1 aÞ 8 Fð~
þ 2Ca~ ¼ aðSð~aÞÞ ¼ lðSð~aÞÞ;
where l denotes the Casson invariant. Remark 1.2. As indicated in [13], Rohlin’s theorem implies that the term Fð~aÞ is divisible by 8. The results of [11] show the signature sð~aÞ is also divisible by 8. Thus the right-hand side of (1.13) is an integer divisible by 8. The above theorem shows that Fð~aÞ þ sð~aÞ is in effect divisible by 16!
2. The Mordell Trick Let a~ 2 Zn be as in the previous section. Denote by P ¼ P a~ the parallelipiped P :¼ ð½0; a1 1 ½0; an 1 Þ \ Zn : When n ¼ 3 we will use the notation a~ ¼ ða; b; cÞ. De¢ne q: P ! R by qð~xÞ ¼
X xi þ 1 2
i
ai
¼
X xi u : þ 2A a i i
LATTICE POINTS AND THE CASSON INVARIANT
43
Remark 2.1. (a) Suppose n ¼ 3, a~ ¼ ða; b; cÞ. Note that qðpÞ 2 12 Z for some p 2 P if and only if abc is odd and a1 b1 c1 ; ; p ¼ p0 :¼ : 2 2 2 In this case qðp0 Þ ¼ 32. (b) Suppose n ¼ 4, a~ ¼ ða1 ; . . . ; a4 Þ. Then qðpÞ 2 Z for some p 2 P if and only if A is odd and p ¼ p0 ¼
a1 1 a4 1 ;...; 2 2
in which case qðp0 Þ ¼ 2. For every interval I R we put NI :¼ #q1 ðIÞ. Note that if n ¼ 3 Ca;b;c ¼ Nð0;12Þ :
ð2:1Þ
while if n ¼ 4 Ca1 ;...;a4 ¼ Nð0;1Þ :
ð2:2Þ
For every r 2 R de¢ne krk ¼ ½r þ 12 where ½ denotes the integer part function. Note that krk is the integer closest to r. We now discuss separately the two cases, n ¼ 3 and n ¼ 4
The case n ¼ 3, a~ ¼ ða; b; cÞ. Imitating Mordell (see [10, 15) we introduce the P quantity a :¼ P ðkqk 1Þðkqk 2Þ. Observe that a ¼ 2N½0;12Þ þ 2N½52;3Þ ¼ 2Nð0;12Þ þ 2Nð52;3Þ :
ð2:3Þ
The importance of the last equality follows from the following elementary result. LEMMA 2.2. Nð0;1Þ ¼ Nð52;3Þ . 2 Proof. Consider the involution o : P ! P; ðx; y; zÞ 7 ! ða 1 x; b 1 y; c 1 zÞ: It has the property qðoðpÞÞ ¼ 3 qðpÞ from which the lemma follows immediately. &
44
LIVIU I. NICOLAESCU
Using the lemma and the equalities (2.1), (2.3) we deduce X 4Ca;b;c ¼ ðkqk 1Þðkqk 2Þ:
ð2:4Þ
P
The case n ¼ 4, a~ ¼ ða1 ; . . . ; a4 Þ. Arguing exactly as above we deduce X 4Ca~ ¼ 4Nð0;1Þ ¼ ð½q 1Þð½q 2Þ:
ð2:5Þ
p
The proof of Theorem 1.1 will be completed by providing an expression for the above sums in terms of Dedekind^Rademacher sums. This will be achieved in the next section following the strategy of [10] (see also [15]).
3. The Proof of Theorem 1.1 We will consider separately the two cases n ¼ 3 and n ¼ 4.
3.1. THE CASE n ¼ 3 Set a~ ¼ ða; b; cÞ so that A ¼ abc, u ¼ ab þ bc þ ca. We will distinguish two cases: A is even and A is odd. A is even. In this case qðpÞ þ 12 62 Z so that kqk ¼ q þ 12 q þ 12 ¼ q q þ 12 : The sum can be rewritten as X q q þ 12 1 q q þ 12 2 P
¼
X
q2 3
P
X P
2
qþ2
X
1
P
X X 2 X q q þ 12 þ q þ 12 þ3 q þ 12 : P
P
P
We compute each of these 6 sums separately. X 1 ¼ #P ¼ abc; P
X
q¼
P
X xþ1 X yþ1 X zþ1 2 2 2 þ þ a b c P P P
¼
a1 b1 c1 bc X ca X ab X ð2x þ 1Þ þ ð2y þ 1Þ þ ð2z þ 1Þ 2a x¼0 2b y¼0 2c z¼0
¼
3abc ; 2
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LATTICE POINTS AND THE CASSON INVARIANT
X
q2 ¼
P
Xx þ 1 2 Xy þ 1 2 Xz þ 1 2 2 2 2 þ þ þ a b c P P P ! ! ! ! a1 b1 a1 c1 X X x þ 12 X y þ 12 x þ 12 X z þ 12 þ 2c þ 2b þ a b a c x¼0 y¼0 x¼0 z¼0 ! ! c1 b1 X z þ 12 X y þ 12 þ 2a : c b z¼0 y¼0
Using basic properties of Bernoulli polynomials (see [16]) we deduce 2 a1 X x þ 12 1 ¼ 2 B3 ða þ 12Þ B3 ð12Þ ; 3a a x¼0 where B3 ðtÞ ¼
tð2t 1Þðt 1Þ 2
is the third Bernoulli polynomial. Note that B3 ð12Þ ¼ 0 and B3 t þ 12 ¼ t t2 14 : Using the identity n1 X k þ 12 n ¼ ; 2 n k¼0
we conclude X
q2 ¼
P
5abc 1 bc ca ab þ þ : 2 12 a b c
Next X P
qþ
1 2
Xx
y z u þ abc þ þ þ ¼ a b c 2abc P abc1 X k u þ abc þ ¼ abc 2abc k¼0
According to the Kubert identity (A.4) in the Appendix the last sum is equal to ðððu þ abcÞ=2ÞÞ which is zero. Thus X q þ 12 ¼ 0: P
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LIVIU I. NICOLAESCU
The sum
P
qððq þ 12ÞÞ requires a bit more work. Note ¢rst that it decomposes as
a1 X x þ 12 X x y z u þ abc þ þ þ þ a b c 2abc a y;z x¼0 b1 X y þ 12 X x y z u þ abc þ þ þ þ þ a b c 2abc b z;x y¼0 c1 X z þ 12 X x y z u þ abc þ þ þ þ þ a b c 2abc c x;y x¼0 ¼ S1 þ S2 þ S3 : We describe in detail the computation of S1 . The other two sums are entirely similar. Note ¢rst that Xx
y z u þ abc þ þ þ a b c 2abc y;z X y z x u þ abc þ þ þ ¼ b c a 2abc y;z bX c1 k x u þ abc þ þ ¼ bc a 2abc k¼0
(use the Kubert identity (A.4)) ¼ ¼
bcx u þ abc þ a 2a
:
bcðx þ 12Þ bc þ b þ c þ a 2
¼
bcðx þ 12Þ 1 þ : a 2
We conclude a1 X x þ 12 bcðx þ 12Þ 1 þ a 2 a x¼0 a1 a1 X x þ 12 bcðx þ 12Þ 1 bcðx þ 12Þ 1 1 X þ þ ¼ þ a a 2 2 x¼0 2 a x¼0
S1 ¼
(use the Kubert identity) ¼
a1 X xþ1 bcðx þ 1Þ 2
a 1 1 ¼ s bc; a; 2 ; 2 : x¼0
2
a
þ
1 2
LATTICE POINTS AND THE CASSON INVARIANT
47
Hence X q q þ 12 ¼ s bc; a; 12 ; 12 þ s ca; b; 12 ; 12 þ s ab; c; 12 ; 12 : P
Finally X
q þ 12
2
P
y z u þ abc 2 þ þ þ a b c 2abc P abc1 X k þ ðu þ abc=2Þ 2 ¼ abc k¼0 ¼
Xx
(use the fact that u þ abc is odd in this case)
¼
abc1 X
2 k þ 12 ¼ sð1; abc; 0; 12Þ abc
k¼0
(use (A.1)
¼
abc 1 : 12 12abc
Putting together the above information we deduce that if abc is even then
4Ca;b;c ¼
abc 1 1 bc ca ab þ þ 12 12abc 12 a b c 2ðsðbc; a; 12 ; 12Þ þ sðca; b; 12 ; 12Þ þ sðab; c; 12 ; 12ÞÞ:
The identity (1.13) is now obvious.
A is odd. In this case, using Remark 2.1 we deduce
kqðpÞk ¼
q ððq þ 12ÞÞ; q ðq þ 12ÞÞ þ 12 ;
p 6¼ p0 ; p ¼ p0 :
Thus
ðkqk 1Þðkqk 2Þ ¼
ðq ððq þ 12ÞÞ 1Þðq ððq þ 12ÞÞ 2Þ; p 6¼ p0 ; ðq ððq þ 12ÞÞ 12Þðq ððq þ 12ÞÞ 32Þ; p ¼ p0 :
ð3:1Þ
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LIVIU I. NICOLAESCU
Hence 4Ca;b;c ¼
X ðq ððq þ 12ÞÞ 1Þðq ððq þ 12ÞÞ 2Þþ P
þ ðq ððq þ 12ÞÞ 12Þðq ððq þ 12ÞÞ 32Þ jp0 ðq ððq þ 12ÞÞ 1Þðq ððq þ 12ÞÞ 2Þ jp0 X ¼ ðq ððq þ 12ÞÞ 1Þðq ððq þ 12ÞÞ 2Þ þ 14 :
ð3:2Þ
P
The above sum can be computed exactly as in the even case with one notable difference, namely abc1 X X k þ ðu þ abc=2Þ 2 ððq þ 12ÞÞ2 ¼ abc k¼0
(u þ abc is even)
¼
abc1 X k¼0
k abc
2 ¼ sð1; abc; 0; 0Þ ¼
abc 1 1 þ : 12 6abc 4
Thus, when abc is odd we have 4Ca;b;c
abc 1 1 bc ca ab þ þ þ ¼ 12 6abc 12 a b c 2ðsðbc; a; 12 ; 12Þ þ sðca; b; 12 ; 12Þ þ sðab; c; 12 ; 12ÞÞ:
This completes the proof of Theorem 1.1 when n ¼ 3.
3.2. THE CASE n ¼ 4 We follow a similar strategy with some obvious modi¢cations. Set a~ ¼ ða1 ; . . . ; a4 Þ, A ¼ a1 a2 a3 a4 , u ¼ b1 þ . . . þ b4 and Sa~ ¼
X ð½q 1Þð½q 2Þ: P a~
As in the previous subsection will distinguish two situations.
A is even. Note that for every p 2 P we have qðpÞ 62 Z so that ½q ¼ q ððqÞÞ 12 :
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LATTICE POINTS AND THE CASSON INVARIANT
Thus Sa~ ¼
X ðq ððqÞÞ 3=2Þðq ðqÞÞ 5=2Þ P
X X X X ¼ ðq2 4q þ 15=4Þ 2 qððqÞÞ þ ððqÞÞ2 þ 4 ððqÞÞ: P
P
P
P
The computation of the above terms follows the same pattern as in the previous subsection. X ððqÞÞ ¼ 0; P
X
15=4 ¼ 15#P=4 ¼ 15A=4;
P
X
q¼
4 X
bi
aX i 1
4 xi þ 12 X bi ai ¼ 2A; ¼ ai 2 i¼1
P
i¼1
xi ¼0
X
4 X
aX i 1
q2 ¼
p
bi
xi ¼0
i¼1
¼
¼
xi þ 12 ai
4 X
bi ai ða2i 14Þ 3a2i i¼1
4 X i¼1
2
þ
1 !0a 1 j i 1 X X A aX xi þ 12 xj þ 12 @ A þ2 a a a a i j i j i y 2 Z; < þ ; 12 6k 4 sð1; k; 0; yÞ ¼ > > : k þ 1 B2 ðfygÞ; y 2 R n Z; 12 k
ðA:1Þ
where B2 ðtÞ ¼ t2 t þ 16 is the second Bernoulli polynomial. Perhaps the most important property of these Dedekind^Rademacher sums is their reciprocity law which makes them computationally very friendly: their computational complexity is comparable with the complexity of the classical Euclid’s algorithm. To formulate it we must distinguish two cases.
Both x and y are integers. Then sðb; a; x; yÞ þ sða; b; y; xÞ ¼ 14 þ
a 2 þ b2 þ 1 : 12ab
ðA:2Þ
x and/or y is not an integer. Then sðb; a; x; yÞ þ sða; b; y; xÞ ¼ ððxÞÞ ððyÞÞ þ
b2 c2 ðyÞ þ c2 ðby þ axÞ þ a2 c2 ðxÞ 2ab
ðA:3Þ
where c2 ðxÞ :¼ B2 ðfxgÞ. An important ingredient behind the reciprocity law is the following identity ([14, Lemma 1]) k1 X m þ w m¼0
k
¼ ððwÞÞ
8w 2 R:
ðA:4Þ
Following the terminology in [9], we will call the above equality the Kubert identity. We conclude with a proof of the identity (1.5). For simplicity we consider only the case n ¼ 3 and i ¼ 1. Set a~ ¼ ða; b; cÞ. Thus A ¼ abc is even, u ¼ bc þ ca þ ab and b1 ¼ bc. For arbitrary n the proof is only notationally more complicated. The proof of (1.5) goes as follows: g1 þ b1 =2 1 ; s b1 ; a; a 2 a 1 X x1 b1 x þ g1 2 ¼ a a x¼0
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LIVIU I. NICOLAESCU
ðg1 ¼ b1 ðu abc 1Þ=2 mod aÞ ¼
a1 X x 12 b1 ðx ðabc u þ 1=2Þ a a x¼0
ðy :¼ x ðabc u þ 1=2Þ mod aÞ a1 X y þ ðabc u þ 1=2Þ 12 b1 y ¼ a a y¼0 (use y ¼ bcz mod a and b1 bc 1 mod a1 ) ¼
a1 X bcz ðabc u=2Þ z
a a a1 X bcðz þ 12Þ b þ c bc z þ ¼ a 2 a z¼0 a1 X bcðz þ 12Þ 1 z ¼ þ : 2 a a z¼0 z¼0
At this point we use the elementary identity z z þ 1 1 1 2 ¼ þ dðzÞ; a 2a 2 a where
dðzÞ ¼
1 0
z 0 ðmodÞ a; otherwise:
We deduce g1 þ b1 =2 1 s b1 ; a1 ; ; a1 2 a1 1 X bcðz þ 2Þ 1 z þ 12 þ ¼ þ a 2 a z¼0 a1 bcðz þ 12Þ 1 1 X 1 bc 1 þ þ þ : a 2a z¼0 2 2 2a 2 The Kubert identity shows that the second sum above vanishes. Also q1 g1 þ 12 u abc=2 b þ c bc bc þ ¼ ¼ a 2 2a a bc 1 þ ¼ : 2a 2 The identity (1.5) is proved. The proof of (1.8) is similar and is left to the reader
LATTICE POINTS AND THE CASSON INVARIANT
53
Acknowledgement I want to thank the referee for carefully reading this paper and suggesting several corrections.
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