The Genus One Gromov-Witten Invariants of Calabi-Yau Complete Intersections

We obtain mirror formulas for the genus 1 Gromov-Witten invariants of projective Calabi-Yau complete intersections. We follow the approach previously used for projective hypersurfaces by extending the scope of its algebraic results; there is little change in the geometric aspects. As an application, we check the genus 1 BPS integrality predictions in low degrees for all projective complete intersections of dimensions 3, 4, and 5.

genus 0 GW-invariants of a quintic threefold has since been verified and shown to be a special case of mirror formulas satisfied by GW-invariants of complete intersections; see [Gi] and [LLY]. Mirror formulas for the genus 1 GW-invariants of projective Calabi-Yau hypersurfaces are obtained in [Z4] and [Z5], in particular confirming the prediction of [BCOV] for a quintic threefold. In this paper, we obtain mirror formulas for the genus 1 GW-invariants of all projective Calabi-Yau complete intersections following the approach in [Z5], extending [ZaZ], and using [PoZ] in place of [Z3].
Throughout this paper, n, a 1 , a 2 , . . . , a l ≥ 2 will be fixed integers. 1 Let a ≡ (a 1 , a 2 , . . . , a l ), a ≡ l k=1 a k , and a a ≡ l k=1 a a k k .
Let ε 0 (a) and ε 1 (a) be the coefficients of w n−1−l and w n−2−l , respectively, in the power series expansion of (1+w) n l r=1 (1+ar w) around w = 0. We denote by X a a smooth complete intersection in P n−1 of multi-degree a. This complete intersection is Calabi-Yau if and only if l r=1 a r = n; from now on it will be assumed that this condition holds. Let N d 1 (X a ) denote the degree d genus 1 GW-invariant of X a . Note that ε 0 (a) and ε 1 (a) describe the top two Chern classes of X a : c n−1−l (X a ) = ε 0 (a)H n−1−l | Xa , c n−2−l (X a ) = ε 1 (a)H n−2−l | Xa , (1.1) where H ∈ H 2 (P n−1 ) is the hyperplane class. As in [ZaZ], we denote by P ⊂ 1 + qQ(w) q the subgroup of power series in q with constant term 1 whose coefficients are rational functions in w which are holomorphic at w = 0. Thus, the evaluation map P → 1 + qQ q , F (w, q) → F (0, q) , is well-defined. We define a map M : P → P by MF (w, q) ≡ 1 + q w d dq F (w, q) F (0, q) .
(1.2) LetF ∈ P be the hypergeometric series (1.6) The map q −→ Q is a change of variables; it will be called the mirror map.
Theorem 1. The genus 1 GW-invariants of a multi-degree a CY CI X a in P n−1 are given by: where Q ≡ q e J(q) .
Since dropping a component of a equal to 1 has no effect on the power seriesF in (1.3), this also has no effect on the right-hand side of the formula in Theorem 1 as expected from the relation N d 1 X 1,a 1 ,a 2 ,...,a l = N d 1 X a 1 ,a 2 ,...,a l .
Thus, using the formula [Z5, (B.12)] for the number of degree r unramified covers of a torus, we obtain: This identity together with Theorem 1 implies that The same argument is applied to X 3 in [Z5, Section 0.3] to obtain The latter identity is verified directly in [Sc]; we expect that similar modular-forms techniques can be used to verify the former identity directly as well.
If l = n−3, X a is a K3 surface, either X 4 ⊂ P 3 , X 2,3 ⊂ P 4 , or X 2,2,2 ⊂ P 5 . Since a ε 0 (a) = χ(X a ) = 24 and ε 1 (a) = 0, by (1.1), the right hand-side of the formula in Theorem 1 is zero in all 3 cases, as expected (all GW-invariants of K3 surfaces vanish). If l = n − 4, X a ⊂ P n−1 is a CY threefold. Since CY 3-folds are of a particular interest in GW-theory, we restate the l = n−4 case of Theorem 1 as a corollary below. In this case, Corollary 2. The genus 1 GW-invariants of a CY CI threefold X a ⊂ P n−1 are given by: where S p (a) ≡ l r=1 a p r and Q = q e J(q) .   Tables 1-3 below show low-degree genus 1 BPS numbers for all CY CI 3, 4 and 5-folds obtained from Theorem 1 using [MirSym,(34.3)], [KlPa,(3)], and [PaZ,(0.5)], respectively. 2 Using computer programs 3 , we verified the predicted integrality of these numbers up to degree 100 for all CY CI 3, 4, and 5-folds. While the degree 1 and 2 genus 1 BPS numbers are 0 as expected, the degree 3 BPS numbers match the classical Schubert calculus on G(3, n). It should be possible to obtain the degree 4 numbers using the approach of [ESt], which provides such numbers for hypersurfaces.
I would like to express my deep gratitude to Aleksey Zinger for explaining [Z4] and [Z5] to me, for proposing the questions answered in this paper, and for his invaluable suggestions.
2 Genus 1 BPS counts in higher dimensions are yet to be defined. 3 based on Aleksey Zinger's programs for hypersurfaces 2 Outline of the proof We prove Theorem 1 following the approach used to prove [Z4,Theorem 2]. In particular, we compute the reduced genus 1 GW-invariants N d;0 1 (X a ) of X a defined in [Z1]; these are related to the standard genus 1 invariants by Lemma 2.1 below.
The genus 1 hyperplane theorem of [LiZ] and the desingularization construction of [VaZ] express the reduced genus 1 GW-invariants of X a in terms of integrals over smooth spaces of maps to P n−1 . We use this in Section 3.2 to package the numbers N d;0 1 (X a ) into a power series X(α, x, Q), in a formal variable Q and with coefficients in the equivariant cohomology of P n−1 . As X(α, x, Q) involves integrals on smooth moduli spaces, the Atiyah-Bott Localization Theorem [ABo] can be applied as in [Z5]. This leads to Proposition 3.1 of Section 3.3; the latter expresses X(α, x, Q) in terms of residues of some genus 0 generating functions.
We extract "the non-equivariant part" of X(α, x, Q) in Section 5, using [Z5,Lemma 3.3] and mirror formulas for genus 0 generating functions. This reduces the problem of computing the numbers N d;0 1 (X a ) to purely algebraic questions concerning the power series (1.3). These are addressed in Section 4, which significantly extends [ZaZ]; this section can be read independently of the rest of the paper.
All cohomology groups in this paper will be with rational coefficients. We will denote by [n], whenever n ∈ Z ≥0 , the set of positive integers not exceeding n: [n] ≡ 1, 2, . . . , n .
Whenever g, d, k and n are nonnegative integers and X is a smooth subvariety of P n−1 , M g,k (X, d) will denote the moduli space of stable degree d maps into X from genus g curves with k marked points and ev i : M g,k (P n−1 , d) [C, y 1 , . . . , y k , f ] −→ f (y i ), i = 1, 2, . . . , k, for the evaluation map at the i-th marked point; see [MirSym,Chapter 24]. For each m ∈ Z >0 , define where i is any element of [m]. For each i ∈ [m], let Thus, η p is the sum of all degree p monomials in π * i ψ 1 : i ∈ [m] . The symmetric group on m elements, S m , acts on M (m) (X, d) by permuting the elements of each m-tuple of stable maps. Let Since the map ev 1 and the cohomology class η p on M (m) (X, d) are S m -invariant, they descend to the quotient: Let U be the universal curve over M (m) (P n−1 , d), with structure map π and evaluation map ev: The orbi-sheaf is locally free; it is the sheaf of (holomorphic) sections of the vector orbi-bundle where L −→ P n−1 is the total space of the vector bundle corresponding to the sheaf l r=1 O P n−1 (a r ).
By the (genus-zero) hyperplane-section relation, where H ∈ H 2 (P n−1 ) is the hyperplane class. There is a natural surjective bundle homomorphism is a vector orbi-bundle. 4 It is straightforward to see that If f = f (w) admits a Laurent series expansion around w = 0, for any p ∈ Z we denote by Theorem 1 follows immediately from from (3.9), Theorem 3 stated at the beginning of Section 5, and Lemma 2.1 below, which extends [Z4, Lemma 2.2] to complete intersections.
Lemma 2.1. If X a ⊂ P n−1 is a complete intersection of multi-degree a, where H ∈ H 2 (P n−1 ) is the hyperplane class, Q and q are related by the mirror map (1.6), and F (w, q) and I 0 (q) are given by by (1.3) and (1.5), respectively.
Proof. The first identity above is a special case of [Z4, (2.15)]. The second identity is immediate from It remains to verify the third identity. For each r ∈ Z ≥0 , let By (2.1), (2.2), and the decomposition along the small diagonal in (P n−1 ) m , the left-hand side of the third identity in Lemma 2.1 above equals The third statement of Lemma 2.1 now follows from the last identity is obtained from [Gi,Theorem 11.8] using the string relation [MirSym,Section 26.3].

Equivariant cohomology
This section reviews the basics of equivariant cohomology following [Z5, Section 1.1] closely and setting up related notation. The classifying space for the n-torus T is BT ≡ (P ∞ ) n . Thus, the group cohomology of T is where α i ≡ π * i c 1 (γ * ), γ −→ P ∞ is the tautological line bundle, and π i : (P ∞ ) n −→ P ∞ is the projection to the i-th component. In the remainder of the paper, The field of fractions of H * T will be denoted by We which is characterized by the property that Throughout this paper, T will act on P n−1 in the standard way: e iθ 1 , . . . , e iθn · [z 1 , . . . , z n ] = e iθ 1 z 1 , . . . , e iθn z n .
This action has n fixed points: For each i = 1, 2, . . . , n, let By the Atiyah-Bott Localization Theorem [ABo], The standard action of T on P n−1 lifts to an action on the tautological bundle . The equivariant cohomology of P n−1 is given by The restriction map on the equivariant cohomology induced by the inclusion P i −→ P n−1 is given by and so (3.6)

Generating function for reduced genus 1 GW-invariants
As in [Z5], the reduced genus 1 GW-invariants N d;0 1 (X a ) of X a are packaged into a generating function X; this is a power series in the formal variable Q with coefficients in the equivariant cohomology of P n−1 . In this section, we define X and explain what its relationship with N d;0 1 (X a ) is; see (3.8) and (3.9).
Let π : U −→ M g,k (P n−1 , d) be the universal curve with evaluation map ev as before and the vector bundle corresponding to the locally free sheaf The Euler class e(V 0 ) relates genus 0 GW-invariants of X a ⊂ P n−1 to genus 0 GW-invariants of P n−1 ; it also appears in the genus 0 2 point generating functions (3.11)-(3.13) which are used in the proof in Theorem 3.
The genus 1 GW-invariants of X a are related to the GW-invariants of P n−1 in a more complicated way. This is partly because M 1,k (P n−1 , d) is not an orbifold and is not locally free. However, it is shown in [VaZ] that there exists a natural desingularization of the main component of M 1,k (P n−1 , d), whose generic element is a map from smooth domain. There is also a vector orbi-bundle V 1 over M 0 1,k (P n−1 , d) so that the diagram The standard T-action on P n−1 induces T-actions on the moduli spaces of M g,k (P n−1 , d) and lifts to an action on M 0 1,k (P n−1 , d). The evaluation maps, are T-equivariant. The natural T-action on O P n−1 (−1) −→ P n−1 induces T-actions on the sheafs π * ev * O P n−1 (a) and the vector bundle With ev 1,d the evaluation map on M 0 1,1 (P n−1 , d), let for some X 0 ∈ Q Q and power series X p ∈ Q[α 1 , . . . , α n ] Q , whose coefficients are symmetric degree p homogeneous polynomials in α 1 , . . . , α n . By (3.7) and (3.1), (3.9) By (3.5), (3.3), and (3.1), for each i = 1, 2, . . . , n. Since X is symmetric in α 1 , . . . , α n , X 0 ∈ Q Q is completely determined by either of the n power series in (3.10). We use this to obtain the explicit formula for X 0 given in Theorem 3.

A localization proposition
As in [Z5], we express X(α, α i , Q) in terms of residues of genus 0 generating functions. Proposition 3.1 below is the analogue of [Z5, Propositions 1.1, 1.2]; its proof is essentially identical to the proof of [Z5, Propositions 1.1, 1.2] in [Z5,Section 2]. In this section, we set up the notation needed to state Proposition 3.1, motivate it, and describe the few minor changes needed in [Z5, Section 2] for a complete proof of this proposition. In the remainder of this paper, we will use Proposition 3.1 to obtain an explicit formula for X 0 . If f is a rational function in and possibly other variables and 0 ∈ S 2 , let R = 0 f ( ) denote the residue of the one-form f ( )d at = 0 ; thus, If f involves variables other than , R = 0 f ( ) is a function of the other variables. If f is a power series in Q with coefficients that are rational functions in and possibly other variables, let R = 0 f ( ) denote the power series in Q obtained by replacing each of the coefficients by its residue at = 0 . If 1 , . . . , k is a collection of points in S 2 , not necessarily distinct, we define If 0 ∈ C or 0 is one of the "other" variables in f , let For instance, if a = (2, 2, 3, 3, 3, 3) and α i is one of the other variables, then Since the T-equivariant bundle homomorphism is surjective, its kernel is a T-equivariant vector bundle. Since the T-action on M g,k (P n−1 , d) lifts naturally to the tautological tangent line bundles L i , there are well-defined equivariant ψ-classes see [MirSym,Section 25.2]. For all i, j = 1, 2, . . . , n, let Explicit formulas for these generating functions are given explicitly in [Gi,Theorem 11.8] and [PoZ,Theorem 4]. These theorems show that in particular Thus, the -residues of these power series are well-defined. Since the Q degree 0 term of the power series Z * i ( , Q) is 0, the residue By [Z5,Lemma 2.3], the power series e −η i (Q)/ (1+Z * i ( , Q)) is holomorphic at = 0; thus Φ 0 (α i , Q) is its value at = 0. 5 Note that the degree-zero term of Φ 0 (α i , Q) is 1.
Proposition 3.1 below is obtained by applying the Atiyah-Bott Localization Theorem [ABo] to the last expression in (3.10). As described in detail in [Z5,Sections 1.3,1.4], the fixed loci of the T-action on M 0 1,1 (P n−1 , d) are indexed by decorated graphs with one marked point. The vertices are decorated by elements of [n], indicating the T-fixed point of P n−1 to which the node or component corresponding to the vertex is mapped to. These graphs have either zero loops and one distinguished vertex (as in Figure 2) or one loop (as in Figure 1), depending on whether the stable maps they describe are constant or not on the principal component of the domain. 6 The graphs with no loops are called B-graphs in [Z5], while the graphs with one loop are called A-graphs. In a B-graph, the distinguished vertex corresponds to the contracted principal component. As every graph has a marked point, even the A-graphs have a distinguished vertex: the vertex in the loop closest to the vertex to which the marked point is attached. The distinguished vertices are indicated by thick dots in the four graphs in Figures 1 and 2. Within each of the 2 types, there are 2 sub-types of graphs, depending on whether the marked point is attached to the distinguished vertex or some other vertex. In the former case, a graph has one special vertex label: the number decorating the vertex to which the marked point is attached. In the latter case, a graph has two special vertex labels: the number decorating the vertex to which the marked point is attached and the number decorating the distinguished vertex. Since φ i | P j = δ ij , only the graphs that describe stable maps taking the marked point to P i contribute to (3.10); in these graphs the first special vertex label is i. Thus, the types of graphs that contribute to (3.10) can be described as A i ,Ã ij , B i , andB ij , with the first subscript describing the label of the vertex to which the marked point is attached and the second describing the label of the distinguished vertex if this vertex is different from the first (the label may still be the same).
The approach of [Z5] to computing the total contribution to (3.10) of all graphs of a fixed type is to break every graph at the distinguished vertex, adding a marked point to each of the resulting "strands" so that each graph is completely encoded by its strands. In the case of B i -graphs, all strands are graphs with one marked point. In the case of A i and B ij -graphs, there is precisely one strand with two marked points; in the former case it contributes to Z * ii , while in the latter it contributes to Z * ji . In the case of A ij -graphs, there are two strands with two marked points, one of which contributes to Z * jj , while the other to Z * ji . Each of the one-pointed strands contributes to Z * j . While the number of one-pointed strands can be arbitrary large, it is possible to sum up over all arrangements of such strands because of a special property of the power series Z * i described in [Z5,Section 2.2]. This reduces the total contribution, A i ,Ã ij , B i , orB ij of all graphs of a fixed type, A i ,Ã ij , B i , orB ij , to a fairly simple expression involving Z * i , Z * ij , and/or Z * ij .
Proposition 3.1. For every i = 1, 2, . . . , n, This proposition is essentially proved in [Z5, Sections 1.3, 1.4, 2], which treats the l = 1 case. In the general case, the T-fixed loci and their normal bundles remain the same. The only required changes involve the Euler classes of the bundles V ′ 0 and V 1 , which are now products of the Euler classes of the bundles in [Z5]. These changes are: a r α µ(v 0 ) +ψ Γ +λ ; (ii) nα i + is replaced by (iii) nα i is replaced by a α l i in [Z5, (2.24)] and nα j is replaced by a α l j in the last equation in [Z5, Section 2.3], leading to the corresponding modification in the final expressions for B i andB ij above.

Some properties of hypergeometric seriesF
In this section we study properties of the hypergeometric seriesF of (1.3) which are used in Section 5.3 to deduce Theorem 3 from Proposition 3.1. The results in this section extend most of the statements and proofs in [ZaZ], which treats the l = 1 case.
Let M : P −→ P be as in (1.2) and define F ∈ P by . (4.1) Some advantages of the power series F overF are illustrated by Lemmas 4.1 and 4.2 below.
Lemma 4.1. The hypergeometric series F satisfies M n F = F.
Lemma 4.2 ([ZaZ, Lemma 1.3 and its proof]). If F ∈ P and M k F = F for some k > 0, then every coefficient of the power series log M p F (w, q) ∈ Q(w) q is O(w) as w → ∞ for all p ≥ 0. Moreover, R w=0 log M p F (w −1 , q) does not depend on p.
Applying this lemma to F = F, we find that F(w, q) has an asymptotic expansion Proposition 4.3. The power series µ, Φ 0 , and Φ 1 in (4.3) are given by The last proposition of this section concerns properties ofF and F around w = 0.

Proof of Lemma and Proposition 4.4
We will repeatedly use the following lemma.
Lemma 4.5 ([ZaZ, Corollary 2.2]). Suppose F (w, q) ∈ P satisfies m r=0 C r (q) D r w F (w, q) = A(w, q) (4.12) for some power series C 0 (q), . . . , C m (q) ∈ Q q and A(w, q) ∈ Q(w) q with A(0, q) ≡ 0. Then (4.14) Using (4.11), we find that It is also straightforward to check that F −l solves the differential equation This equation is of the form (4.12) with F = F −l , A = w n F −l , m = n, (4.17) Applying Lemma 4.5 repeatedly, we obtain (4.18) where by the first identity in (4.15) and by (4.2) Using (4.17) and induction on p, we find that the top two coefficients in (4.18) are given by (4.20) Setting p = n−l in (4.18) and (4.19) thus gives (4.21) Setting w = 0 in (4.21) and using F −l (0, q) = 1 gives (4.9). Substituting (4.9) back into (4.21) gives F n−l /I n−l = F −l and thus F n−l+1 = F −l+1 . Applying M to both sides of the last identity l−1 times and using (4.15), we obtain Lemma 4.1 and equation (4.7). Similarly, setting p = n−l−1 in (4.18)-(4.20) and then taking w = 0 gives Integrating this identity and then exponentiating, we obtain (4.10). We next prove the reflection symmetry (4.8). The function F ∈ P defined in (1.3) satisfies the differential equation This equation is of the form (4.12) with F =F, A = w n−l , m = n−l, and C n−l (q) = 1 − a a q.
Applying Lemma 4.5 repeatedly, we obtain n−l−p given by (4.19). Setting p = n−l in (4.22) and using (4.19) and (4.9), we find that M n−l F(w, q) = I n−l (q) is independent of w. Using (4.11) and downward induction on p, we then find that Comparing the coefficients of q d on the two sides of (4.23), we find that . Thus, the substitution w → −w−d acts by (−1) n−l on the left-hand side of (4.24), and so c p (d) = c n−l−p (d) for all 0 ≤ p ≤ n−l, as needed for the inductive step.

Proof of (4.5)
By Lemmas 4.1 and 4.2, the functions F p (w, q) ≡ M p F(w, q) admit asymptotic expansions with the same function µ(q) in the exponent for all p. Since F 0 = F and F p+1 = MF p , Taking s = 0 in (4.27), we find by induction that (4.28) Since F n = F 0 by Lemma 4.1 and Φ 0 (0) = 1, setting p = n in the above identity we obtain (1 + Dµ) n = I 0 . . . I n−1 .
Proposition 4.6. The power series Φ s ∈ Q q , s ≥ 0, defined by (4.3), are determined by the first-order ODEs together with the initial conditions Φ s (0) = δ 0,s and Φ s = 0 for s < 0.

Computation of reduced genus 1 GW-invariants
In this section, we deduce Theorem 3 below from Proposition 3.1, using Lemmas 5.1 and 5.3 and the properties of the hypergeometric series F(w, q) described by Proposition 4.3 and 4.4. Lemma 5.1 is used to drop purely equivariant terms from the power series X, while Lemma 5.3 provides the relevant information about the genus 0 generating functions Z * i , Z * ji , and Z * ii .
Theorem 3. The generating function X 0 (Q) defined by (3.8) is given by where Q and q are related by the mirror map (1.6) and We will show that the terms A i andÃ ij , with j ∈ [n], in (3.16) together contribute 1 2 Q d dQ A(q) to X 0 (Q), where log I r (q) = − 1 2 log (1−a a q) , if 2 |(n−l), by (4.8) and (4.9), the expression on the right-hand side of (5.1) equals twice the right-hand side in the first equation in Theorem 3.

Some algebraic notation and observations
This section recalls the statement of [Z5,Lemma 3.3], which shows that most terms appearing in the computation of X(α, x, Q) have no effect on X 0 (Q). We then set up additional related notation and make a few algebraic observations that help streamline computations in the remainder of the paper.
For each p ∈ [n], let σ p be the p-th elementary symmetric polynomial in α 1 , . . . , α n . Denote by the subspace of symmetric polynomials, by I ⊂ Q[α] Sn the ideal generated by σ 1 , . . . , σ n−1 , and bỹ the subalgebra of symmetric rational functions in α 1 , . . . , α n whose denominators are products of α j and (α j −α k ) with j = k. For each i = 1, . . . , n, let be the subalgebra consisting of rational functions symmetric in {α k : k = i} and with denominators that are products of α i and (α i −α k ) with k = i. Let Lemma 5.1 ([Z5, Lemma 3.3] 7 ). If n ≥ 2, the linear span of α n−2 i is disjoint from K i : For each i = 1, . . . , n, let be the subalgebra consisting of rational functions symmetric in {α k : k = i} and with denominators that are a product of a polynomial with rational coefficients in and α i and of linear factors of the form (α i −α k +r ), r ∈ Z. Denote by The definition of Ki in [Z5] is missing α n−2 i IQ[α] Sn , but the proof of [Z5,Lemma 3.3] still goes through. This change adds the term α n−1 i gn−1, with gn−1 ∈ I, to the second numerator in [Z5, (3.13)] and gn−1 to the right-hand side of [Z5, (3.15)]. As g ∈ I, this addition has no effect on the concluding sentence in the proof of Lemma 3.3 in [Z5]. the subalgebra consisting of rational functions of the form A+B l k=1 (a k α i + ) with A, B ∈Q i [α] S n−1 both regular at = −a k α i for every k ∈ [l] and the denominator of A an element of Q[α i , ]. We defineQ All statements in the next lemma follow immediately from the definitions.

The genus zero generating functions
We will now express the genus 0 generating functions Z * i , Z * ij , and Z * ii defined in Section 3.3 in terms of the hypergeometric series F of (4.1) and the operator M of (1.2).
Lemma 5.3. The genus 0 generating functions Z * i , Z * ij , and Z * ii satisfy and Q and q are related by the mirror map (1.6).
Proof. By [Gi,Theorem 11.8], for some C ∈ qQ q and .
There is no term n k=1 (x − α k ) in the generating function used in place of Y in [Gi], but putting it does not effect the validity of [Gi,Theorem 11.8] as it vanishes under all evaluations x −→ α i . On the other hand, with this extra term in place Y becomes a function of , x, σ 1 , . . . , σ n−1 , and not σ n . Since all denominators in Y( , α i , q) are products of α i −α k +r with r ∈ Z, Y( , α i , q) − F(α i / , q) ∈ I ·Q i [α] S n−1 q .
If a k = 1, 2 or n is odd, the denominators in the above expression do not vanish at = −a k α i , and so the difference lies in I · qS i, q . Otherwise, the denominators have a simple zero at = −a k α i (in q-degree at least 2 if a k = 1). If a k = 2 and n is even, the factor [(α i + ) n − α n i ] has a zero at = −a k α i as well, and so The case a k = 1 is excluded by the assumption on a in Section 1. Thus, (5.3) follows from (5.6). By (3.2), [PoZ,Theorem 4], and the same reasoning as in the previous paragraph, p+r+s=n−1 p,r,s≥0 8 Note that ( where Y r ∈ Q α (x) q is a power series such that Y r ( , α j , q) ∈ S j, q and (α j + ) n − α n j Y r ( , α j , q) − α r j M r F(α j / , q) I r (q) ∈ I · qS j, q . 9 (5.8) The claim (5.4) thus follows from (5.7). Finally, by [PoZ,Theorem 4], p+r+s=n−1 p,r,s≥0 (5.9) s 1 +s 2 =k 0≤s 1 ≤n By (5.10) and (5.11), Thus, (5.5) follows from (5.9), (5.11), and (5.8).

Proof of Theorem 3
We will use Lemmas 5.1-5.3 to extract the coefficients of α n−2 i from the expressions of Proposition 3.1 modulo K i q . In the notation of Theorem 3 and Proposition 3.1, Let D ≡ q d dq as in Section 4. We begin by computing residues of the transforms of F that appear in the description of the generating functions in Lemma 5.3.
Thus, by (5.17) and the second statement in (4.5), The second identity in the lemma follows from the last two equations along with (4.5), (4.6), DL = L(L n −1)/n, and (4.4).
Since 1 I 1 (q) D = Q d dQ , this proves the claim stated in the sentence after Theorem 3. We next compute (5.13). LetB By the last two equations, (5.22), and (5.23), This concludes the proof of Theorem 3.