Another Look at the Dwork Family

Nicholas M. Katz

Princeton University, Mathematics, Fine Hall, NJ 08544-1000, USA

nmk@math.princeton.edu

dedicated to Yuri Manin on his seventieth birthday

1 Introduction and a bit of history

After proving [Dw-Rat] the rationality of zeta functions of all algebraic varieties over finite fields nearly fifty years ago, Dwork studied in detail the zeta function of a nonsingular hypersurface in projective space, cf. [Dw-Hyp1] and [Dw-HypII]. He then developed his “deformation theory”, cf. [Dw-Def], [Dw-NPI] and [Dw-NPII], in which he analyzed the way in which his theory varied in a family. One of his favorite examples of such a family, now called the Dwork family, was the one parameter (λ) family, for each degree n ≥ 2, of degree n hypersurfaces in ℙ^{n−1} given by the equation

∑_{i=1}^{n} X_{i}^{n} − nλ ∏_{i=1}^{n} X_{i} = 0,

a family he wrote about explicitly in [Dw-Def, page 249, (i),(ii),(iv), the cases n = 2, 3, 4], [Dw-HypII, section 8, pp. 286-288, the case n = 3] and [Dw-PC, 6.25, the case n = 3, and 6.30, the case n = 4]. Dwork of course also considered the generalization of the above Dwork family consisting of single-monomial deformations of Fermat hypersurfaces of any degree and dimension. He mentioned one such example in [Dw-Def, page 249, (iii)]. In [Dw-PAA, pp. 153-154], he discussed the general single-monomial deformation of a Fermat hypersurface, and explained how such families led to generalized hypergeometric functions, cf. also [Dw-BP, 6.1].

My own involvement with the Dwork family started (in all senses!) at the Woods Hole conference in the summer of 1964 with the case n = 3, when I managed to show in that special case that the algebraic aspects of Dwork’s deformation theory amounted to what would later be called the Gauss-Manin connection on relative de Rham cohomology, but which at the time went by the more mundane name of “differentiating cohomology classes with respect to parameters”.

That this article is dedicated to Manin on his seventieth birthday is particularly appropriate, because in that summer of 1964 my reference for the notion of differentiating cohomology classes with respect to parameters was his 1958 paper [Ma-ACFD]. I would also like to take this opportunity to thank, albeit belatedly, Arthur Mattuck for many helpful conversations that summer.

I discussed the Dwork family in [Ka-ASDE, 2.3.7.17-23, 2.3.8] as a “particularly beautiful family”, and computed explicitly the differential equation satisfied by the cohomology class of the holomorphic n − 2 form. It later showed up in [Ka-SE, 5.5, esp. pp. 188-190], about which more below. Ogus [Ogus-GTCC, 3.5, 3.6] used the Dwork family to show the failure in general of “strong divisibility”. Stevenson, in her thesis [St-th],[St, end of Section 5, page 211], discussed single-monomial deformations of Fermat hypersurfaces of any degree and dimension. Koblitz [Kob] later wrote on these same families. With mirror symmetry and the stunning work of Candelas et al [C-dlO-G-P] on the case n = 5, the Dwork family became widely known, especially in the physics community, though its occurence in Dwork’s work was almost (not entirely, cf. [Ber], [Mus-CDPMQ]) forgotten. Recently the Dwork family turned out to play a key role in the proof of the Sato-Tate conjecture (for elliptic curves over ℚ with non-integral j-invariant), cf. [H-SB-T, Section 1, pp. 5-15].

The present paper gives a new approach to computing the local system given by the cohomology of the Dwork family, and more generally of families of single-monomial deformations of Fermat hypersurfaces. This approach is based upon the surprising connection, noted in [Ka-SE, 5.5, esp. pp. 188-190], between such families and Kloosterman sums. It uses also the theory, developed later, of Kloosterman sheaves and of hypergeometric sheaves, and of their behavior under Kummer pullback followed by Fourier Transform, cf. [Ka-GKM] and [Ka-ESDE, esp. 9.2 and 9.3]. In a recent preprint, Rojas-Leon and Wan [RL-Wan] have independently implemented the same approach.

2 The situation to be studied: generalities

We fix an integer n ≥ 2, a degree d ≥ n, and an n-tuple W = (w_{1}, ..., w_{n}) of strictly positive integers with ∑_{i} w_{i} = d, and with gcd(w_{1}, ..., w_{n}) = 1. This data (n, d, W) is now fixed. Let R be a ring in which d is invertible.

Over R we have the affine line 𝔸_{R}^{1} := Spec(R[λ]). Over 𝔸_{R}^{1}, we consider certain one parameter (namely λ) families of degree d hypersurfaces in ℙ^{n−1}. Given an n + 1-tuple (a, b) := (a_{1}, ..., a_{n}, b) of invertible elements in R, we consider the one parameter (namely λ) family of degree d hypersurfaces in ℙ^{n−1},

X_{λ}(a, b) : ∑_{i=1}^{n} a_{i}X_{i}^{d} − bλX^{W} = 0,

where we have written

X^{W} := ∏_{i=1}^{n} X_{i}^{w_{i}}.

More precisely, we consider the closed subscheme 𝕏(a, b)_{R} of ℙ_{R}^{n−1} ×_{R} 𝔸_{R}^{1} defined by the equation

∑_{i=1}^{n} a_{i}X_{i}^{d} − bλX^{W} = 0,

and denote by

π(a, b)_{R} : 𝕏(a, b)_{R} → 𝔸_{R}^{1}

the restriction to 𝕏(a, b)_{R} of the projection of ℙ_{R}^{n−1} ×_{R} 𝔸_{R}^{1} onto its second factor.

Lemma 2.1. The morphism

π(a, b)_{R} : 𝕏(a, b)_{R} → 𝔸_{R}^{1}

is lisse over the open set of 𝔸_{R}^{1} where the function

(bλ/d)^{d} ∏_{i}(w_{i}/a_{i})^{w_{i}} − 1

is invertible.

Proof. Because d and the a_{i} are invertible in R, a Fermat hypersurface of the form

∑_{i=1}^{n} a_{i}X_{i}^{d} = 0

is lisse over R. When we intersect our family with any coordinate hyperplane X_{i} = 0, we obtain a constant Fermat family in one lower dimension (because each w_{i} ≥ 1). Hence any geometric point (x, λ) ∈ 𝕏 at which π is not smooth has all coordinates X_{i} invertible. So the locus of nonsmoothness of π is defined by the simultaneous vanishing of all the X_{i}d/dX_{i}, i.e., by the simultaneous equations

da_{i}X_{i}^{d} = bλw_{i}X^{W} , for i = 1, ..., n.

Divide through by the invertible factor da_{i}. Then raise both sides of the i’th equation to the w_{i} power and multiply together right and left sides separately over i. We find that at a point of nonsmoothness we have

X^{dW} = (bλ/d)^{d} ∏_{i}(w_{i}/a_{i})^{w_{i}}X^{dW} .

As already noted, all the X_{i} are invertible at any such point, and hence

1 = (bλ/d)^{d} ∏_{i}(w_{i}/a_{i})^{w_{i}}

at any geometric point of nonsmoothness. ⊓ ⊔

In the Dwork family per se, all w_{i} = 1. But in a situation where there is a prime p not dividing dℓ but dividing one of the w_{i}, then taking for R an 𝔽_{p}-algebra (or more generally a ring in which p is nilpotent), we find a rather remarkable family.

Corollary 2.2. Let p be a prime which is prime to d but which divides one of the w_{i}, and R a ring in which p is nilpotent. Then the morphism

π(a, b)_{R} : 𝕏(a, b)_{R} → 𝔸_{R}^{1}

is lisse over all of 𝔸_{R}^{1}

Remark 2.3. Already the simplest possible example of the above situation, the family in ℙ^{1}/𝔽_{q} given by

X^{q+1} + Y^{q+1} = λXY^{q},

is quite interesting. In dehomogenized form, we are looking at

x^{q+1} − λx + 1

as polynomial over 𝔽_{q}(λ); its Galois group is known to be P SL(2, 𝔽_{q}), cf. [Abh-PP, bottom of p. 1643], [Car], and [Abh-GTL, Serre’s Appendix]. The general consideration of “p|w_{i} for some i” families in higher dimension would lead us too far afield, since our principal interest here is with families that “start life” over ℂ. We discuss briefly such “p|w_{i} for some i” families in Appendix II. We would like to call the attention of computational number theorists to these families, with no degeneration at finite distance, as a good test case for proposed methods of computing efficiently zeta functions in entire families.

3 The particular situation to be studied: details

Recall that the data (n, d, W) is fixed. Over any ring R in which d ∏_{i} w_{i} is invertible, we have the family π : 𝕏 → 𝔸_{R}^{1} given by

X_{λ} := X_{λ}(W, d) : ∑_{i=1}^{n} w_{i}X_{i}^{d} − dλX^{W} = 0;

it is proper and smooth over the open set U := 𝔸_{R}^{1}[1/(λ^{d} − 1)] ⊂ 𝔸_{R}^{1} where λ^{d} − 1 is invertible.

The most natural choice of R, then, is ℤ[1/(d ∏_{i} w_{i})]. However, it will be more convenient to work over a somewhat larger cyclotomic ring, which contains, for each i, all the roots of unity of order dw_{i}. Denote by lcm(W) the least common multiple of the w_{i}, and define d_{W} := lcm(W)d. In what follows, we will work over the ring

R_{0} := ℤ[1/d_{W}][ζ_{d_{W}}] := ℤ[1/d_{W}][T]/(Φ_{d_{W}} (T)),

where Φ_{d_{W}} (T) denotes the d_{W} ’th cyclotomic polynomial.

We now introduce the relevant automorphism group of our family. We denote by μ_{d}(R_{0}) the group of d’th roots of unity in R_{0}, by Γ = Γ_{d,n} the n-fold product group (μ_{d}(R_{0}))^{n}, by Γ_{W} ⊂ Γ the subgroup consisting of all elements (ζ_{1}, ..., ζ_{n}) with ∏_{i=1}^{n} ζ_{i}^{w_{i}} = 1, and by Δ ⊂ Γ_{W} the diagonal subgroup, consisting of all elements of the form (ζ, ..., ζ). The group Γ_{W} acts as automorphisms of 𝕏/𝔸_{R_{0}}^{1}, an element (ζ_{1}, ..., ζ_{n}) acting as

((X_{1}, ..., X_{n}), λ) ↦ ((ζ_{1}X_{1}, ..., ζ_{n}X_{n}), λ).

The diagonal subgroup Δ acts trivially.

The natural pairing

(ℤ/dℤ)^{n} × Γ → μ_{d}(R_{0}) ⊂ R_{0}^{×},

(v_{1}, ..., v_{n}) × (ζ_{1}, ..., ζ_{n}) → ∏_{i} ζ_{i}^{v_{i}},

identifies (ℤ/dℤ)^{n} as the R_{0}-valued character group DΓ := Hom_{group}(Γ, R_{0}^{×}). The subgroup

(ℤ/dℤ)_{0}^{n} ⊂ (ℤ/dℤ)^{n}

consisting of elements V = (v_{1}, ..., v_{n}) with ∑_{i} v_{i} = 0 in ℤ/dℤ is then the R_{0}- valued character group D(Γ/Δ) of Γ/Δ. The quotient group (ℤ/dℤ)_{0}^{n}/ < W > of (ℤ/dℤ)_{0}^{n} by the subgroup generated by (the image, by reduction mod d, of) W is then the R_{0}-valued character group D(Γ_{W} /Δ) of Γ_{W} /Δ.

For G either of the groups Γ/Δ or Γ_{W} /Δ, an R_{0}-linear action of G on a sheaf of R_{0}-modules M gives an eigendecomposition

M = ⊕_{ρ∈D(G)}M(ρ).

If the action is by the larger group G = Γ/Δ, then DG = (ℤ/dℤ)_{0}^{n}, and for V ∈ (ℤ/dℤ)_{0}^{n} we denote by M(V) the corresponding eigenspace. If the action is by the smaller group Γ_{W} /Δ, then DG is the quotient group (ℤ/dℤ)_{0}^{n}/<W >; given an element V ∈ (ℤ/dℤ)_{0}^{n}, we denote by V mod W its image in the quotient group, and we denote by M(V mod W) the corresponding eigenspace.

If M is given with an action of the larger group Γ/Δ, we can decompose it for that action:

M = ⊕_{V ∈(ℤ/dℤ)_{0}^{n}}M(V).

If we view this same M only as a representation of the sugroup Γ_{W} /Δ, we can decompose it for that action:

M = ⊕_{V ∈(ℤ/dℤ)_{0}^{n}/<W >}M(V mod W).

The relation between these decompositions is this: for any V ∈ (ℤ/dℤ)_{0}^{n},

M(V mod W) = ⊕_{r mod d}M(V + rW).

We return now to our family π : 𝕏 → 𝔸_{R_{0}}^{1}, which we have seen is (projective and) smooth over the open set

U = 𝔸_{R_{0}}^{1}[1/(λ^{d} − 1)].

We choose a prime number ℓ, and an embedding of R_{0} into \overline{ℚ}_{ℓ}. [We will now need to invert ℓ, so arguably the most efficient choice is to take for ℓ a divisor of d_{W}.] We We form the sheaves

F^{i} := R^{i}π_{⋆}\overline{ℚ}_{ℓ}

on 𝔸_{R_{0}[1/ℓ]}^{1}. They vanish unless 0 ≤ i ≤ 2(n − 2), and they are all lisse on U[1/ℓ]. By the weak Lefschetz Theorem and Poincaré duality, the sheaves F^{i}|U[1/ℓ] for i ≠ n − 2 are completely understood. They vanish for odd i; for even i = 2j ≤ 2(n − 2), i ≠ n − 2, they are the Tate twists

F^{2j}|U[1/ℓ] ≅ \overline{ℚ}_{ℓ}(−j).

We now turn to the lisse sheaf F^{n−2}|U[1/ℓ]. It is endowed with an autoduality pairing (cup product) toward \overline{ℚ}_{ℓ}(−(n − 2)) which is symplectic if n − 2 is odd, and orthogonal if n − 2 is even. If n − 2 is even, say n − 2 = 2m, then F^{n−2}|U[1/ℓ] contains \overline{ℚ}_{ℓ}(−m) as a direct summand (m’th power of the hyperplane class from the ambient ℙ) with nonzero self-intersection. We define P rim^{n−2} (as a sheaf on U[1/ℓ] only) to be the annihilator in F^{n−2}|U[1/ℓ] of this \overline{ℚ}_{ℓ}(−m) summand under the cup product pairing. So we have

F^{n−2}|U[1/ℓ] = P rim^{n−2} ⊕ \overline{ℚ}_{ℓ}(−m),

when n − 2 = 2m. When n − 2 is odd, we define P rim^{n−2} := F^{n−2}|U[1/ℓ], again as a sheaf on U[1/ℓ] only.

The group Γ_{W} /Δ acts on our family, so on all the sheaves above. For i ≠ n − 2, it acts trivially on F^{i}|U[1/ℓ]. For i = n − 2 = 2m even, it respects the decomposition

F^{n−2}|U[1/ℓ] = P rim ⊕ \overline{ℚ}_{ℓ}(−m),

and acts trivially on the second factor. We thus decompose P rim^{n−2} into eigensheaves P rim^{n−2}(V mod W). The basic information on the eigensheaves P rim^{n−2}(V mod W) is encoded in elementary combinatorics of the coset V mod W. An element V = (v_{1}, ..., v_{n}) ∈ (ℤ/dℤ)_{0}^{n} is said to be totally nonzero if v_{i} ≠ 0 for all i. Given a totally nonzero element V ∈ (ℤ/dℤ)_{0}^{n}, we define its degree, deg(V) as follows. For each i, denote by ṽ_{i} the unique integer 1 ≤ ṽ_{i} ≤ d − 1 which mod d gives v_{i}. Then ∑_{i} ṽ_{i} is 0 mod d, and we define

deg(V) := (1/d) ∑_{i} ṽ_{i}.

Thus deg(V) lies in the interval 1 ≤ deg(V) ≤ n − 1. The Hodge type of a totally nonzero V ∈ (ℤ/dℤ)_{0}^{n} is defined to be

HdgT ype(V) := (n − 1 − deg(V), deg(V) − 1).

We now compute the rank and the the Hodge numbers of eigensheaves P rim^{n−2}(V mod W). We have already chosen an embedding of R_{0} into \overline{ℚ}_{ℓ}. We now choose an embedding of \overline{ℚ}_{ℓ} into ℂ. The composite embedding R_{0} ⊂ ℂ allows us to extend scalars in our family π : 𝕏 → 𝔸_{R_{0}}^{1}, which is projective and smooth over the open set U_{R_{0}} = 𝔸_{R_{0}}^{1}[1/(λ^{d} − 1)], to get a complex family π_{ℂ} : 𝕏_{ℂ} → 𝔸_{ℂ}^{1}, which is projective and smooth over the open set U_{ℂ} = 𝔸_{ℂ}^{1}[1/(λ^{d} − 1)]. Working in the classical complex topology with the corresponding analytic spaces, we can form the higher direct image sheaves R^{i}π_{ℂ}^{an}ℚ on 𝔸_{ℂ}^{1,an}, whose restrictions to U_{ℂ}^{an} are locally constant sheaves. We can also form the locally constant sheaf P rim^{n−2,an}(ℚ) on U_{ℂ}^{an}. Extending scalars in the coefficients from ℚ to \overline{ℚ}_{ℓ}, we get the sheaf P rim^{n−2,an}(\overline{ℚ}_{ℓ}). On the other hand, we have the lisse \overline{ℚ}_{ℓ}-sheaf P rim^{n−2} on U_{R_{0}[1/ℓ]}, which we can pull back, first to U_{ℂ}, and then to U_{ℂ}^{an}. By the fundamental comparison theorem, we have

P rim^{n−2,an}(\overline{ℚ}_{ℓ}) ≅ P rim^{n−2}|U_{ℂ}^{an}.

Extending scalars from \overline{ℚ}_{ℓ} to ℂ, we find

P rim^{n−2,an}(ℂ) ≅ (P rim^{n−2}|U_{ℂ}^{an}) ⊗_{\overline{ℚ}_{ℓ}} ℂ.

This is all Γ_{W} /Δ-equivariant, so we have the same relation for individual eigensheaves:

P rim^{n−2,an}(ℂ)(V mod W) ≅ (P rim^{n−2}(V mod W)|U_{ℂ}^{an}) ⊗_{\overline{ℚ}_{ℓ}} ℂ.

If we extend scalars on U_{ℂ}^{an} from the constant sheaf ℂ to the sheaf O_{C^{∞}}, then the resulting C^{∞} vector bundle P rim^{n−2,an}(ℂ) ⊗_{ℂ} O_{C^{∞}} has a Hodge decomposition,

P rim^{n−2,an}(ℂ) ⊗_{ℂ} O_{C^{∞}} = ⨁_{a≥0,b≥0,a+b=n−2} P rim^{a,b}.

This decomposition is respected by the action of Γ_{W} /Δ, so we get a Hodge decomposition of each eigensheaf:

P rim^{n−2,an}(ℂ)(V mod W) ⊗_{ℂ} O_{C^{∞}} = ⨁_{a≥0,b≥0,a+b=n−2} P rim^{a,b}(V mod W).

Lemma 3.1. We have the following results.

(1) The rank of the lisse sheaf P rim^{n−2}(V mod W) on U_{R_{0}[1/ℓ]} is given by

rk(P rim^{n−2}(V mod W)) = #{r ∈ ℤ/dℤ | V + rW is totally nonzero}.

In particular, the eigensheaf P rim^{n−2}(V mod W) vanishes if none of the W -translates V + rW is totally nonzero.

(2) For each (a, b) with a ≥ 0, b ≥ 0, a + b = n − 2, the rank of the C^{∞} vector bundle P rim^{a,b}(V mod W) on U_{ℂ}^{an} is given by

rk(P rim^{a,b}(V mod W))

= #{r ∈ ℤ/dℤ | V + rW is totally nonzero and deg(V + rW) = b + 1}.

Proof. To compute the rank of a lisse sheaf on U_{R_{0}[1/ℓ]}, or the rank of a C^{∞} vector bundle on U_{ℂ}^{an}, it suffices to compute its rank at a single geometric point of the base. We take the ℂ-point λ = 0, where we have the Fermat hypersurface. Here the larger group (ℤ/dℤ)_{0}^{n} operates. It is well known that under the action of this larger group, the eigenspace P rim(V) vanishes unless V is totally nonzero, e.g., cf. [Ka-IMH, Section 6]. One knows further that if V is totally nonzero, this eigenspace is one-dimensional, and of Hodge type HdgT ype(V) := (n − 1 − deg(V), deg(V) − 1), cf. [Grif-PCRI, 5.1 and 10.8].

⊓ ⊔

The main result of this paper is to describe the eigensheaves

P rim^{n−2}(V mod W)

as lisse sheaves on U[1/ℓ], i.e., as representations of π_{1}(U[1/ℓ]), and to describe the direct image sheaves j_{U}_{⋆}(P rim^{n−2}(V mod W)) on 𝔸_{R_{0}[1/ℓ]}^{1}, for j_{U} : U[1/ℓ] ⊂ 𝔸_{R_{0}[1/ℓ]}^{1} the inclusion. The description will be in terms of hypergeometric sheaves in the sense of [Ka-ESDE, 8.7.11].

4 Interlude: Hypergeometric sheaves

We first recall the theory in its original context of finite fields, cf. [Ka-ESDE, Chapter 8]. Let k be an R_{0}[1/ℓ]-algebra which is a finite field, and

ψ : (k, +) → \overline{ℚ}_{ℓ}^{×}

a nontrivial additive character. Because k is an R_{0}[1/ℓ]- algebra, it contains d_{W} distinct d_{W} ’th roots of unity, and the structural map gives a group isomorphism μ_{d_{W}} (R_{0}) ≅ μ_{d_{W}} (k). So raising to the #k^{×}/d_{W} ’th power is a surjective group homomorphism

k^{×} → μ_{d_{W}} (k) ≅ μ_{d_{W}} (R_{0}).

So for any character χ : μ_{d_{W}} (R_{0}) → μ_{d_{W}} (R_{0}), we can and will view the composition of χ with the above surjection as defining a multiplicative character of k^{×}, still denoted χ. Every multiplicative character of k^{×} of order dividing d_{W} is of this form. Fix two non-negative integers a and b, at least one of which is nonzero. Let χ_{1}, ..., χ_{a} be an unordered list of a multiplicative characters of k^{×} of order dividing d_{W}, some possibly trivial, and not necessarily distinct. Let ρ_{1}, ..., ρ_{b} be another such list, but of length b. Assume that these two lists are disjoint, i.e., no χ_{i} is a ρ_{j}. Attached to this data is a geometrically irreducible middle extension \overline{ℚ}_{ℓ}-sheaf

H(ψ; χ_{i} ′s; ρ_{j} ′s)

on 𝔾_{m}/k, which is pure of weight a + b − 1. We call it a hypergeometric sheaf of type (a, b). If a ≠ b, this sheaf is lisse on 𝔾_{m}/k; if a = b it is lisse on 𝔾_{m} − {1}, with local monodromy around 1 a tame pseudoreflection of determinant (∏_{j} ρ_{j})/(∏_{i} χ_{i}).

The trace function of H(ψ; χ_{i} ′s; ρ_{j} ′s) is given as follows. For E/k a finite extension field, denote by ψ_{E} the nontrivial additive character of E obtained from ψ by composition with the trace map T race_{E/k}, and denote by χ_{i,E} (resp. ρ_{j,E}) the multiplicative character of E obtained from χ_{i} (resp. ρ_{j}) by composition with the norm map Norm_{E/k}. For t ∈ 𝔾_{m}(E) = E^{×}, denote by V (a, b, t) the hypersurface in (𝔾_{m})^{a} × (𝔾_{m})^{b}/E, with coordinates x_{1}, ..., x_{a}, y_{1}, ..., y_{b}, defined by the equation

∏_{i} x_{i} = t ∏_{j} y_{j}.

Then

T race(F rob_{t,E}|H(ψ; χ_{i} ′s; ρ_{j} ′s))

= (−1)^{a+b−1} ∑_{V (n,m,t)(E)} ψ_{E}(∑_{i} x_{i} − ∑_{j} y_{j}) ∏_{i} χ_{i,E}(x_{i}) ∏_{j} \overline{ρ}_{j,E}(y_{j}).

In studying these sheaves, we can always reduce to the case a ≥ b, because under multiplicative inversion we have

inv^{⋆}H(ψ; χ_{i} ′s; ρ_{j} ′s)) ≅ H(\overline{ψ}; \overline{ρ}_{j} ′s; \overline{χ}_{i} ′s)).

If a ≥ b, the local monodromy around 0 is tame, specified by the list of χ_{i}’s: the action of a generator γ_{0} of I_{0}^{tame} is the action of T on the \overline{ℚ}_{ℓ}[T]-module \overline{ℚ}_{ℓ}[T]/(P (T)), for P (T) the polynomial

P (T) := ∏_{i}(T − χ_{i}(γ_{0})).

In other words, for each of the distinct characters χ on the list of the χ′_{i}s, there is a single Jordan block, whose size is the multiplicity with which χ appears on the list. The local monodromy around ∞ is the direct sum of a tame part of dimension b, and, if a > b, a totally wild part of dimension a − b, all of whose upper numbering breaks are 1/(a − b). The b-dimensional tame part of the local monodromy around ∞ is analogously specified by the list of ρ’s: the action of a generator γ_{∞} of I_{∞}^{tame} is the action of T on the \overline{ℚ}_{ℓ}[T]-module \overline{ℚ}_{ℓ}[T]/(Q(T)), for Q(T) the polynomial

Q(T) := ∏_{j}(T − ρ_{j}(γ_{0})).

When a = b, there is a canonical constant field twist of the hypergeometric sheaf H = H(ψ; χ_{i} ′s; ρ_{j} ′s) which is independent of the auxiliary choice of ψ, which we will call H^{can}. We take for A ∈ \overline{ℚ}_{ℓ}^{×} the nonzero constant

A = (∏_{i}(−g(ψ, χ_{i}))(∏_{j}(−g(\overline{ψ}, \overline{ρ_{j}})),

and define

H^{can} := H ⊗ (1/A)^{deg}.

[That H^{can} is independent of the choice of ψ can be seen in two ways. By elementary inspection, its trace function is independent of the choice of ψ, and we appeal to Chebotarev. Or we can appeal to the rigidity of hypergeometric sheaves with given local monodromy, cf. [Ka-ESDE, 8.5.6], to infer that with given χ’s and ρ’s, the hypergeometric sheaves H_{ψ}^{can} with different choices of ψ are all geometrically isomorphic. Being geometrically irreducible as well, they must all be constant field twists of each other. We then use the fact that H^{1}(𝔾_{m} ⊗_{k} \overline{k}, H_{ψ}^{can}) is one dimensional, and that F rob_{k} acts on it by the scalar 1, to see that the constant field twist is trivial.]

Here is the simplest example. Take χ ≠ ρ, and form the hypergeometric sheaf H^{can}(ψ; χ; ρ). Then using the rigidity approach, we see that

H^{can}(ψ; χ; ρ) ≅ L_{χ(x)} ⊗ L_{(ρ/χ)(1−x)} ⊗ (1/A)^{deg},

with A (minus) the Jacobi sum over k,

A = −J(k; χ, ρ/χ) := − ∑_{x∈k^{×}} χ(x)(ρ/χ)(1 − x).

The object

H(χ, ρ) := L_{χ(x)} ⊗ L_{(ρ/χ)(1−x)}

makes perfect sense on 𝔾_{m}/R_{0}[1/ℓ], cf. [Ka-ESDE, 8.17.6]. By [We-JS], attaching to each maximal ideal P of R_{0} the Jacobi sum −J(R_{0}/P; χ, ρ/χ) over its residue field is a grossencharacter, and so by [Se-ALR, Chapter 2] a \overline{ℚ}_{ℓ}-valued character, call it Λ_{χ,ρ/χ}, of π_{1}(Spec(R_{0}[1/ℓ]). So we can form

H^{can}(χ, ρ) := H(χ, ρ) ⊗ (1/Λ_{χ,ρ/χ})

on 𝔾_{m}/R_{0}[1/ℓ]. For any R_{0}[1/ℓ]-algebra k which is a finite field, its pullback to 𝔾_{m}/k is H^{can}(ψ; χ; ρ).

This in turn allows us to perform the following global construction. Suppose we are given an integer a > 0, and two unordered disjoint lists of characters, χ_{1}, ..., χ_{a} and ρ_{1}, ..., ρ_{a}, of the group μ_{d_{W}} (R_{0}) with values in that same group. For a fixed choice of orderings of the lists, we can form the sheaves H^{can}(χ_{i}, ρ_{i}), i = 1, ..., a on 𝔾_{m}/R_{0}[1/ℓ]. We can then define, as in [Ka-ESDE, 8.17.11], the ! multiplicative convolution

H^{can}(χ_{1}, ρ_{1})[1] ⋆_{!} H^{can}(χ_{2}, ρ_{2})[1] ⋆_{!} ... ⋆_{!} H^{can}(χ_{a}, ρ_{a})[1],

which will be of the form F[1] for some sheaf F on 𝔾_{m}/R_{0}[1/ℓ] which is “tame and adapted to the unit section”. This sheaf F we call H^{can}(χ_{i} ′s; ρ_{j} ′s). For any R_{0}[1/ℓ]-algebra k which is a finite field, its pullback to 𝔾_{m}/k is H^{can}(ψ; χ_{i} ′s; ρ_{j} ′s). By Chebotarev, the sheaf H^{can}(ψ; χ_{i} ′s; ρ_{j} ′s) is, up to isomorphism, independent of the orderings that went into its definition as an interated convolution. This canonical choice (as opposed to, say, the ad hoc construction given in [Ka-ESDE, 8.17.11], which did depend on the orderings) has the property that, denoting by

f : 𝔾_{m}/R_{0}[1/ℓ] → Spec(R_{0}[1/ℓ])

the structural map, the sheaf R^{1}f_{!}H^{can}(χ_{i} ′s; ρ_{j} ′s) on Spec(R_{0}[1/ℓ]) is the constant sheaf, i.e., it is the trivial one-dimensional representation of π_{1}(Spec(R_{0}[1/ℓ])).

If the unordered lists χ_{1}, ..., χ_{a} and ρ_{1}, ..., ρ_{b} are not disjoint, but not identical, then we can “cancel” the terms in common, getting shorter disjoint lists. The hypergeometric sheaf we form with these shorter, disjoint “cancelled” lists we denote H(ψ; Cancel(χ_{i} ′s; ρ_{j} ′s)), cf. [Ka-ESDE, 9.3.1], where this was denoted CancelH(ψ; χ_{i} ′s; ρ_{j} ′s). If a = b, then after cancellation the shorter disjoint lists still have the same common length, and so we can form the constant field twist H^{can}(ψ; Cancel(χ_{i} ′s; ρ_{j} ′s)). And in the global setting, we can form the object H^{can}(Cancel(χ_{i} ′s; ρ_{j} ′s)) on 𝔾_{m}/R_{0}[1/ℓ].

5 Statement of the main theorem

We continue to work with the fixed data (n, d, W). Given an element V = (v_{1}, ..., v_{n}) ∈ (ℤ/dℤ)_{0}^{n}, we attach to it an unordered list List(V, W) of d = ∑_{i} w_{i} multiplicative characters of μ_{d_{W}} (R_{0}), by the following procedure. For each index i, denote by χ_{v_{i}} the character of μ_{d_{W}} (R_{0}) given by

ζ ↦ ζ^{(v_{i}/d)d_{W}} .

Because w_{i} divides d_{W} /d, this characterχ_{v_{i}} has w_{i} distinct w_{i}’th roots. We then define

List(V, W) = {all w′_{1}th roots of χ_{v_{1}}, ..., all w′_{n}th roots of χ_{v_{n}}}.

We will also need the same list, but for −V, and the list

List(all d) := {all characters of order dividing d}.

So long as the two lists List(−V, W) and List(all d) are not identical, we can apply the Cancel operation, and form the hypergeometric sheaf

H_{V,W} := H^{can}(Cancel(List(all d); List(−V, W)))

on 𝔾_{m}/R_{0}[1/ℓ].

Lemma 5.1. If P rim^{n−2}(V mod W) is nonzero, then the unordered lists List(−V, W) and List(all d) are not identical.

Proof. If P rim^{n−2}(V mod W) is nontrivial, then at least one choice of V in the coset V mod W is totally nonzero. For such a totally nonzero V, the trivial character is absent from List(−V, W). If we choose another representative of the same coset, say V − rW, then denoting by χ_{r} the character of order dividing d of μ_{d_{W}} (R_{0}) given by ζ ↦ ζ^{(r/d)d_{W}}, we see easily that

List(−(V − rW), W) = χ_{r}List(−V, W).

Hence the character χ_{r} is absent from List(−V + rW, W). ⊓ ⊔

Lemma 5.2. If P rim^{n−2}(V mod W) is nonzero, then P rim^{n−2}(V mod W) and [d]^{⋆}H_{V,W} have the same rank on U_{R_{0}[1/ℓ]}.

Proof. Choose V in the coset V mod W. The rank of P rim^{n−2}(V mod W) is the number of r ∈ ℤ/dℤ such that V + rW is totally nonzero. Equivalently, this rank is d − δ, for δ the number of r ∈ ℤ/dℤ such that V + rW fails to be totally nonzero. On the other hand, the rank of H_{V,W} is d − ϵ, for ϵ the number of elements in List(all d) which also appear in List(−V, W). Now a given character χ_{r} in List(all d) appears in List(−V, W) if and only if there exists an index i such that χ_{r} is a w_{i}’th root of χ_{−v_{i}}, i.e., such that χ_{r}^{w_{i}} = χ_{−v_{i}}, i.e., such that rw_{i} ≡ −v_{i} mod d. ⊓ ⊔

Theorem 5.3. Suppose that P rim^{n−2}(V mod W) is nonzero. Denote by j_{1} : U_{R_{0}[1/ℓ]} ⊂ 𝔸_{R_{0}[1/ℓ]}^{1} and j_{2} : 𝔾_{m,R_{0}[1/ℓ]} ⊂ 𝔸_{R_{0}[1/ℓ]}^{1} the inclusions, and by [d] : 𝔾_{m,R_{0}[1/ℓ]} → 𝔾_{m,R_{0}[1/ℓ]} the d’th power map. Then for any choice of V in the coset V mod W, there exists a continuous character Λ_{V,W} : π_{1}(Spec(R_{0}[1/ℓ])) → \overline{ℚ}_{ℓ}^{×} and an isomorphism of sheaves on 𝔸_{R_{0}[1/ℓ]}^{1},

j_{1⋆}P rim^{n−2}(V mod W) ≅ j_{2⋆}[d]^{⋆}H_{V,W} ⊗ Λ_{V,W} .

Remark 5.4. What happens if we change the choice of V in the coset V mod W, say to V − rW? As noted above,

List(−(V − rW), W) = χ_{r}List(−V, W).

As List(all d) = χ_{r}List(all d) is stable by multiplication by any character of order dividing d, we find [Ka-ESDE, 8.2.5] that H_{V −rW,W} ≅ L_{χ_{r}} ⊗H_{V,W} ⊗Λ, for some continuous character Λ : π_{1}(Spec(R_{0}[1/ℓ])) → \overline{ℚ}_{ℓ}^{×}. Therefore the pullback [d]^{⋆}H_{V,W} is, up to tensoring with a character Λ of π_{1}(Spec(R_{0}[1/ℓ])), independent of the particular choice of V in the coset V mod W. Thus the truth of the theorem is independent of the particular choice of V.

Question 5.5. There should be a universal recipe for the character Λ_{V,W} which occurs in Theorem 5.3. For example, if we look at the Γ_{W} /Δ-invariant part, both P rim^{n−2}(0 mod W) and H_{0,W} are pure of the same weight n − 2, and both have traces (on Frobenii) in ℚ. So the character Λ_{0,W} must take ℚ-values of weight zero on Frobenii in large characteristic. [To make this argument legitimate, we need to be sure that over every sufficiently large finite field k which is an R_{0}[1/ℓ]-algebra, the sheaf P rim^{n−2}(0 mod W) has nonzero trace at some k-point. This is in fact true, in virtue of Corollary 8.7 and a standard equidistribution argument.] But the only rational numbers of weight zero are ±1. So Λ_{0,W}^{2} trivial. Is Λ_{0,W} itself trivial?

6 Proof of the main theorem: the strategy

Let us admit for a moment the truth of the following characteristic p theorem, which will be proven in the next section.

Theorem 6.1. Let k be an R_{0}[1/ℓ]-algebra which is a finite field, and let ψ : (k, +) → \overline{ℚ}_{ℓ}^{×} be a nontrivial additive character of k. Suppose that P rim^{n−2}(V mod W) is nonzero. Let j_{1,k} : U_{k} ⊂ 𝔸_{k}^{1} and j_{2,k} : 𝔾_{m,k} ⊂ 𝔸_{k}^{1} be the inclusions. Choose V in the coset V mod W, and put

H_{V,W,k} := H^{can}(ψ; Cancel(; List(all d); List(−V, W))).

Then on 𝔸_{k}^{1} the sheaves j_{1,k⋆}P rim^{n−2}(V mod W) and j_{2,k⋆}[d]^{⋆}H_{V,W,k} are geometrically isomorphic, i.e., they become isomorphic on 𝔸1 ; \overline{k}.

We now explain how to deduce the main theorem. The restriction to

U_{R_{0}} − {0} = 𝔾_{m,R_{0}} − μ_{d}

of our family

X_{λ} : ∑_{i=1}^{n} w_{i}X_{i}^{d} = dλX^{W}

is the pullback, through the d’th power map, of a projective smooth family over 𝔾_{m} − {1}, in a number of ways. Here is one way to write down such a descent π_{desc} : 𝕐 → 𝔾_{m} − {1}. Use the fact that gcd(w_{1}, ..., w_{n}) = 1 to choose integers (b_{1}, ..., b_{n}) with ∑_{i} b_{i}w_{i} = 1. Then in the new variables

Y_{i} := λ^{b_{i}}X_{i}

the equation of X_{λ} becomes

∑_{i=1}^{n} w_{i}λ^{−db_{i}}Y_{i}^{d} = dY^{W} .

Then the family

Y_{λ} : ∑_{i=1}^{n} w_{i}λ^{−b_{i}}Y_{i}^{d} = dY^{W}

is such a descent. The same group Γ_{W} /Δ acts on this family. On the base 𝔾_{m} − {1}, we have the lisse sheaf P rim_{desc}^{n−2} for this family, and its eigensheaves P rim_{desc}^{n−2}(V mod W), whose pullbacks [d]^{⋆}P rim_{desc}^{n−2}(V mod W) are the sheaves P rim^{n−2}(V mod W)|(𝔾_{m,R_{0}} − μ_{d}).

Lemma 6.2. Let k be an R_{0}[1/ℓ]-algebra which is a finite field. Suppose P rim_{desc}^{n−2}(V mod W) is nonzero. Then there exists a choice of V in the coset V mod W such that the lisse sheaves P rim_{desc}^{n−2}(V mod W) and H_{V,W,k} on 𝔾_{m,k} − \{1\} are geometrically isomorphic, i.e., isomorphic on 𝔾_{m,\overline{k}} − \{1\}.

Proof. Choose a V in the coset V mod W. By Theorem 6.1, the lisse sheaves [d]^{⋆}P rim_{desc}^{n−2}(V mod W) and [d]^{⋆}H_{V,W,k} are isomorphic on 𝔾_{m,\overline{k}} −μ_{d}. Taking direct image by [d] and using the projection formula, we find an isomorphism

⨁_{χ with χ^{d} trivial} L_{χ} ⊗ P rim_{desc}^{n−2}(V mod W) ≅ ⨁_{χ with χ^{d} trivial} L_{χ} ⊗ H_{V,W,k}

of lisse sheaves 𝔾_{m,\overline{k}} − \{1\}. The right hand side is completely reducible, being the sum of d irreducibles. Therefore the left hand side is completely reducible, and each of its d nonzero summands L_{χ} ⊗ P rim_{desc}^{n−2}(V mod W) must be irreducible (otherwise the left hand side is the sum of more than d irreducibles). By Jordan-Hölder, the summand P rim_{desc}^{n−2}(V mod W) on the left is isomorphic to one of the summands L_{χ} ⊗ H_{V,W,k} on the right, say to the summand L_{χ_{r}} ⊗ H_{V,W,k}. As explained in Remark 5.3, this summand is geometrically isomorphic to H_{V −rW,W,k}. ⊓ ⊔

Lemma 6.3. Suppose that the sheaf P rim_{desc}^{n−2}(V mod W) is nonzero. Choose an R_{0}[1/ℓ]-algebra k which is a finite field, and choose V in the coset V mod W such that the lisse sheaves P rim_{desc}^{n−2}(V mod W) and H_{V,W,k} on 𝔾_{m,k} − {1} are geometrically isomorphic. Then there exists a continuous character

Λ_{V,W} : π_{1}(Spec(R_{0}[1/ℓ])) → \overline{ℚ}_{ℓ}^{×}

and an isomorphism of lisse sheaves on 𝔾_{m,R_{0}[1/ℓ]} − {1},

P rim_{desc}^{n−2}(V mod W) ≅ H_{V,W} ⊗ Λ_{V,W} .

This is an instance of the following general phenomenon, which is well known to the specialists. In our application, the S below is Spec(R_{0}[1/ℓ]), C is ℙ^{1}, and D is the union of the three everywhere disjoint sections 0, 1, ∞. We will also use it a bit later when D is the union of the d + 2 everywhere disjoint sections 0, μ_{d}, ∞.

Theorem 6.4. Let S be a reduced and irreducible normal noetherian ℤ[1/ℓ]- scheme whose generic point has characteristic zero. Let \overline{s} be a chosen geometric point of S. Let C/S be a proper smooth curve with geometrically connected fibres, and let D ⊂ C be a Cartier divisor which is finite étale over S. Let F and G be lisse \overline{ℚ}_{ℓ}-sheaves on C − D. Then we have the following results.

(1) Denote by j : C −D ⊂ C and i : D ⊂ C the inclusions. Then the formation of j_{⋆}F on C commutes with arbitrary change of base T → S, and i^{⋆}j_{⋆}F is a lisse sheaf on D.

(2) Denoting by f : C − D → S the structural map, the sheaves R^{i}f_{!}F on S are lisse.

(3) The sheaves R^{i}f_{⋆}F on S are lisse, and their formation commutes with arbitrary change of base T → S.

(4) Consider the pullbacks F\overline{s} and G\overline{s} of F and of G to C\overline{s} − D\overline{s}. Suppose that F\overline{s} ≅ G\overline{s}, and that G\overline{s} (and hence also F\overline{s}) are irreducible. Then there exists a continuous character Λ : π_{1}(S) → \overline{ℚ}_{ℓ}^{×} an isomorphism of lisse sheaves on C − D,

G ⊗ Λ ≅ F.

Proof. The key point is that because the base S has generic characteristic zero, any lisse sheaf on C − D is automatically tamely ramified along the divisor D; this results from Abhyankar’s Lemma. See [Ka-SE, 4.7] for assertions (1) and (2). Assertion (3) results from (2) by Poincaré duality, cf. [De-CEPD, Corollaire, p. 72].

To prove assertion (4), we argue as follows. By the Tame Specialization Theorem [Ka-ESDE, 8.17.13], the geometric monodromy group attached to the sheaf F\overline{s} is, up to conjugacy in the ambient GL(rk(F), \overline{ℚ}_{ℓ}), independent of the choice of geometric point \overline{s} of S. Since F\overline{s} is irreducible, it follows that F\overline{s_{1}} is irreducible, for every geometric point \overline{s_{1}} of S. Similarly, G\overline{s_{1}} is irreducible, for every geometric point \overline{s_{1}} of S. Now consider the lisse sheaf \underline{Hom}(G, F) ≅ F ⊗ G^{∨} on C − D. By assertion (3), the sheaf f_{⋆}\underline{Hom}(G, F) is lisse on S, and its stalk at a geometric point \overline{s_{1}} of S is the group Hom(G\overline{s_{1}}, F\overline{s_{1}}). At the chosen geometric point \overline{s}, this Hom group is one-dimensional, by hypothesis. Therefore the lisse sheaf f_{⋆}\underline{Hom}(G, F) on S has rank one. So at every geometric point \overline{s_{1}}, Hom(G\overline{s_{1}}, F\overline{s_{1}}) is one-dimensional. As source and target are irreducible, any nonzero element of this Hom group is an isomorphism, and the canonical map

G\overline{s_{1}} ⊗ Hom(G\overline{s_{1}}, F\overline{s_{1}}) → F\overline{s_{1}}

is an isomorphism. Therefore the canonical map of lisse sheaves on C − D

G ⊗ f^{⋆}f_{⋆}\underline{Hom}(G, F) → F

is an isomorphism, as we see looking stalkwise. Interpreting the lisse sheaf f_{⋆}\underline{Hom}(G, F) on S as a character Λ of π_{1}(S), we get the asserted isomorphism.

⊓ ⊔

Applying this result, we get Lemma 6.3. Now pull back the isomorphism of that lemma by the d’th power map, to get an isomorphism

P rim^{n−2}(V mod W) ≅ [d]^{⋆}H_{V,W} ⊗ Λ_{V,W}

of lisse sheaves on 𝔾_{m,R_{0}[1/ℓ]} − μ_{d}. Then extend by direct image to 𝔸_{R_{0}[1/ℓ]}^{1} to get the isomorphism asserted in Theorem 5.3.

7 Proof of Theorem 6.1

Let us recall the situation. Over the ground ring R_{0}[1/ℓ], we have the family π : 𝕏 → 𝔸^{1} given by

X_{λ} := X_{λ}(W, d) : ∑_{i=1}^{n} w_{i}X_{i}^{d} − dλX^{W} = 0,

which is projective and smooth over U = 𝔸^{1} − μ_{d}. We denote by V ⊂ 𝕏 the open set where X^{W} is invertible, and by Z ⊂ 𝕏 the complementary reduced closed set, defined by the vanishing of X^{W}. As scheme over 𝔸^{1}, Z/A^{1} is the constant scheme with fibre

(X^{W} = 0) ∩ (∑_{i} w_{i}X_{i}^{d} = 0).

The group Γ_{W} /Δ, acting as 𝔸^{1}-automorphisms of 𝕏, preserves both the open set V and its closed complement Z. In the following discussion, we will repeatedly invoke the following general principle, which we state here before proceeding with the analysis of our particular situation.

Lemma 7.1. Let S be a noetherian ℤ[1/ℓ]-scheme, and f : X → S a separated morphism of finite type. Suppose that a finite group G acts admissibly (:= every point lies in a G-stable affine open set) as S-automorphisms of X. Then in D_{c}^{b}(S, \overline{ℚ}_{ℓ}), we have a direct sum decomposition of Rf_{!}\overline{ℚ}_{ℓ} into G-isotypical components

Rf_{!}\overline{ℚ}_{ℓ} = ⨁_{irred. \overline{ℚ}_{ℓ} rep.′s ρ of G} Rf_{!}\overline{ℚ}_{ℓ}(ρ).

Proof. Denote by h : X → Y := X/G the projection onto the quotient, and denote by m : Y → S the structural morphism of Y/S. Then Rh_{!}\overline{ℚ}_{ℓ} = h_{⋆}\overline{ℚ}_{ℓ} is a constructible sheaf of \overline{ℚ}_{ℓ}[G] modules on Y, so has a G-isotypical decomposition

Rh_{!}\overline{ℚ}_{ℓ} = h_{⋆}\overline{ℚ}_{ℓ} = ⨁_{irred. \overline{ℚ}_{ℓ} rep.′s ρ of G} h_{⋆}\overline{ℚ}_{ℓ}(ρ).

Applying Rm_{!} to this decomposition gives the asserted decomposition of Rf_{!}\overline{ℚ}_{ℓ}. ⊓ ⊔

We now return to our particular situation. We are given a R_{0}[1/ℓ]-algebra k which is a finite field, and a nontrivial additive character ψ : (k, +) → \overline{ℚ}_{ℓ}^{×}. We denote by

π_{k} : 𝕏_{k} → 𝔸_{k}^{1}

the base change to k of our family. Recall that the Fourier Transform F T_{ψ} is the endomorphism of the derived category D_{c}^{b}(𝔸_{k}^{1}, \overline{ℚ}_{ℓ}) defined by looking at the two projections pr_{1}, pr_{2} of 𝔸_{k}^{2} onto 𝔸_{k}^{1}, and at the “kernel” L_{ψ(xy)} on 𝔸_{k}^{2}, and putting

F T_{ψ}(K) := R(pr_{2})_{!}(L_{ψ(xy)} ⊗ pr_{1}^{⋆}K[1]),

cf. [Lau-TFCEF, 1.2]. One knows that F T_{ψ} is essentially involutive,

F T_{ψ}(F T_{ψ}(K)) ≅ [x ↦ −x]^{⋆}K(−1),

or equivalently

F T\overline{ψ}(F T_{ψ}(K)) ≅ K(−1),

that F T_{ψ} maps perverse sheaves to perverse sheaves and induces an exact autoequivalence of the category of perverse sheaves with itself.

We denote by K(𝔸_{k}^{1}, \overline{ℚ}_{ℓ}) the Grothendieck group of D_{c}^{b}(𝔸_{k}^{1}, \overline{ℚ}_{ℓ}). One knows that K is the free abelian group on the isomorphism classes of irreducible perverse sheaves, cf. [Lau-TFCEF, 0.7, 0.8]. We also denote by F T_{ψ} the endomorphism of K(𝔸_{k}^{1}, \overline{ℚ}_{ℓ}) induced by F T_{ψ} on D_{c}^{b}(𝔸_{k}^{1}, \overline{ℚ}_{ℓ}).

The key fact for us is the following, proven in [Ka-ESDE, 9.3.2], cf. also [Ka-ESDE, 8.7.2 and line -4, p.327].

Theorem 7.2. Denote by ψ_{−1/d} the additive character x ↦ ψ(−x/d), and denote by j : 𝔾_{m,k} ⊂ 𝔸_{k}^{1} the inclusion. Denote by Λ_{1}, ..., Λ_{d} the list List(all d) of all the multiplicative characters of k^{×} of order dividing d. For any unordered list of d multiplicative characters ρ_{1}, ...ρ_{d} of k^{×} which is different from List(all d), the perverse sheaf

F T_{ψ}j_{⋆}[d]^{⋆}H(ψ_{−1/d}; ρ_{1}, ...ρ_{d}; ∅)[1]

on 𝔸_{k}^{1} is geometrically isomorphic to the perverse sheaf

j_{⋆}[d]^{⋆}H(ψ; Cancel(List(all d); \overline{ρ_{1}}, ..., \overline{ρ_{d}}))[1].

Before we can apply this result, we need some preliminaries. We first calculate the Fourier Transform of Rπ_{k,!}\overline{ℚ}_{ℓ}, or more precisely its restriction to 𝔾_{m,k}, in a Γ_{W} /Δ-equivariant way. Recall that V_{k} ⊂ 𝕏_{k} is the open set where X^{W} is invertible, and Z_{k} ⊂ 𝕏_{k} is its closed complement. We denote by

f := π_{k}|V_{k} : V_{k} → 𝔸_{k}^{1}

the restriction to V_{k} of π_{k}. Concretely, V_{k} is the open set ℙ_{k}^{n−1}[1/X^{W}] of ℙ_{k}^{n−1} (with homogeneous coordinates (X_{1}, ..., X_{n})) where X^{W} is invertible, and f is the map

(X_{1}, ..., X_{n}) ↦ ∑_{i}(w_{i}/d)X_{i}^{d}/X^{W} .

Lemma 7.3. For any character V mod W of Γ_{W} /Δ, the canonical map of ρ-isotypical components Rf_{!}\overline{ℚ}_{ℓ}(V mod W) → Rπ_{k,!}\overline{ℚ}_{ℓ}(V mod W) induced by the 𝔸_{k}^{1}-linear open immersion V_{k} ⊂ 𝕏_{k} induces an isomorphism in D_{c}^{b}(𝔾_{m,k}, \overline{ℚ}_{ℓ}),

(F T_{ψ}Rf_{!}\overline{ℚ}_{ℓ})(V mod W)|𝔾_{m,k} ≅ (F T_{ψ}Rπ_{k,!}\overline{ℚ}_{ℓ})(V mod W)|𝔾_{m,k}.

Proof. We have an “excision sequence” distinguished triangle

Rf_{!}\overline{ℚ}_{ℓ}(V mod W) → Rπ_{k,!}\overline{ℚ}_{ℓ}(V mod W) → R(π|Z)_{k,!}\overline{ℚ}_{ℓ}(V mod W) → .

The third term is constant, i.e., the pullback to 𝔸_{k}^{1} of a an object on Spec(k), so its F T_{ψ} is supported at the origin. Applying F T_{ψ} to this distinguished triangle gives a distinguished triange

F T_{ψ}Rf_{!}\overline{ℚ}_{ℓ}(V mod W) → F T_{ψ}Rπ_{k,!}\overline{ℚ}_{ℓ}(V mod W)
→ F T_{ψ}R(π|Z)_{k,!}\overline{ℚ}_{ℓ}(V mod W) → .

Restricting to 𝔾_{m,k}, the third term vanishes. ⊓ ⊔

We next compute (F T_{ψ}Rf_{!}\overline{ℚ}_{ℓ})|𝔾_{m,k} in a Γ_{W} /Δ-equivariant way. We do this by working upstairs, on V_{k} with its Γ_{W} /Δ-action.

Denote by T_{W} ⊂ 𝔾_{m,k}^{n} the connected (because gcd(w_{1}, ...w_{n}) = 1) torus of dimension n − 1 in 𝔾_{m,k}^{n}, with coordinates x_{i}, i = 1, ...., n, defined by the equation x^{W} = 1. Denote by ℙ_{k}^{n−1}[1/X^{W}] ⊂ ℙ_{k}^{n−1} the open set of ℙ_{k}^{n−1} (with homogeneous coordinates (X_{1}, ..., X_{n})) where X^{W} is invertible. Our group Γ_{W} is precisely the group T_{W} [d] of points of order dividing d in T_{W}. And the subgroup Δ ⊂ Γ_{W} is just the intersection of T_{W} with the diagonal in the ambient 𝔾_{m,k}^{n}. We have a surjective map

g : T_{W} → ℙ_{k}^{n−1}[1/X^{W}], (x_{1}, ..., x_{n}) ↦ (x_{1}, ..., x_{n}).

This map g makes T_{W} a finite étale galois covering of ℙ_{k}^{n−1}[1/X^{W}] with group Δ. The d’th power map [d] : T_{W} → T_{W} makes T_{W} into a finite étale galois covering of itself, with group Γ_{W}. We have a beautiful factorization of [d] as h ∘ g, for

h : ℙ_{k}^{n−1}[1/X^{W}] → T_{W} , (X_{1}, ..., X_{n}) ↦ (X_{1}^{d}/X^{W} , ..., X_{n}^{d}/X^{W}).

This map h makes ℙ_{k}^{n−1}[1/X^{W}] a finite étale galois covering of T_{W} with group Γ_{W} /Δ. Denote by m the map

m : T_{W} → A_{k}^{1}, (x_{1}, ..., x_{n}) ↦ ∑_{i}(w_{i}/d)x_{i}.

Let us state explicitly the tautology which underlies our computation.

Lemma 7.4. The map f : V_{k} = ℙ_{k}^{n−1}[1/X^{W}] → A_{k}^{1} is the composition

f = m ∘ h : ℙ_{k}^{n−1}[1/X^{W}] →^{h} T_{W} →^{m} 𝔸_{k}^{1}.

Because h is a a finite étale galois covering of T_{W} with group Γ_{W} /Δ, we have a direct sum decomposition on T_{W},

Rh_{!}\overline{ℚ}_{ℓ} = h_{⋆}\overline{ℚ}_{ℓ} = ⨁_{char′s V mod W of Γ_{W} /Δ} L_{V mod W} .

More precisely, any V in the coset V mod W is a character of Γ/Δ, hence of Γ, so we have the Kummer sheaf L_{V} on the ambient torus 𝔾_{m,k}^{n}. In the standard coordinates (x_{1}, ..., x_{n}) on 𝔾_{m,k}^{n}, this Kummer sheaf L_{V} is L_{∏_{i} χ_{v_{i}} (x_{i})}. The restriction of L_{V} to the subtorus T_{W} is independent of the choice of V in the coset V mod W; it is the sheaf denoted L_{V mod W} in the above decomposition.

Now apply Rm_{!} to the above decomposition. We get a direct sum decomposition

Rf_{!}\overline{ℚ}_{ℓ} = Rm_{!}h_{⋆}\overline{ℚ}_{ℓ} = ⨁_{char′s V mod W of Γ_{W} /Δ} Rm_{!}L_{V mod W}

into eigenobjects for the action of Γ_{W} /Δ.

Apply now F T\overline{ψ}. We get a direct sum decomposition

F T\overline{ψ}Rf_{!}\overline{ℚ}_{ℓ} = ⨁_{char′s V mod W of Γ_{W} /Δ} F T\overline{ψ}Rm_{!}L_{V mod W}

into eigenobjects for the action of Γ_{W} /Δ; we have

(F T\overline{ψ}Rf_{!}\overline{ℚ}_{ℓ})(V mod W) = F T\overline{ψ}Rm_{!}L_{V mod W}

for each character V mod W of Γ_{W} /Δ.

Theorem 7.5. Given a character V mod W of Γ_{W} /Δ, pick V in the coset V mod W. We have a geometric isomorphism

(F T\overline{ψ}Rf_{!}\overline{ℚ}_{ℓ})(V mod W)|𝔾_{m,k}
≅ [d]^{⋆}H(ψ_{−1/d}; List(V, W); ∅)[2 − n].

Proof. By the definition of F T\overline{ψ}, and proper base change for Rm_{!}, we see that F T\overline{ψ}Rm_{!}L_{V mod W} is obtained as follows. Choose V in the coset V mod W. Endow the product T_{W} × 𝔸_{k}^{1}, with coordinates (x = (x_{1}, ..., x_{n}); t) from the ambient 𝔾_{m,k}^{n} × 𝔸_{k}^{1}. The product has projections pr_{1}, pr_{2} onto T_{W} and 𝔸_{k}^{1} respectively. On the product we have the lisse sheaf L_{\overline{ψ}(t ∑_{i}(w_{i}/d)x_{i})} ⊗ pr_{1}^{⋆}L_{V}. By definition, we have

F T\overline{ψ}Rm_{!}L_{V mod W} = Rpr_{2,!}(L_{\overline{ψ}(t ∑_{i}(w_{i}/d)x_{i})} ⊗ pr_{1}^{⋆}L_{∏_{i} χ_{v_{i}} (x_{i})})[1].

If we pull back to 𝔾_{m,k} ⊂ 𝔸_{k}^{1}, then the source becomes T_{W} ×𝔾_{m,k}. This source is isomorphic to the subtorus Z of 𝔾_{m,k}^{n+1}, with coordinates (x = (x_{1}, ..., x_{n}); t), defined by

x^{W} = t^{d},

by the map

(x = (x_{1}, ..., x_{n}); t) ↦ (tx = (tx_{1}, ..., tx_{n}); t).

On this subtorus Z, our sheaf becomes L_{\overline{ψ}(∑_{i}(w_{i}/d)x_{i})} ⊗pr_{1}^{⋆}L_{∏_{i} χ_{v_{i}} (x_{i})}[1]. [Remember that V has ∑_{i} v_{i} = 0, so L_{∏_{i} χ_{v_{i}} (x_{i})} is invariant by x ↦ tx.] Thus we have

F T\overline{ψ}Rm_{!}L_{V mod W} |𝔾_{m,k} = Rpr_{n+1,!}(L_{\overline{ψ}(∑_{i}(w_{i}/d)x_{i})} ⊗ pr_{1}^{⋆}L_{∏_{i} χ_{v_{i}} (x_{i})}[1]).

This situation,

L_{\overline{ψ}(∑_{i}(w_{i}/d)x_{i})} ⊗ pr_{1}^{⋆}L_{∏_{i} χ_{v_{i}} (x_{i})}[1] on Z := (x^{W} = t^{d}) →^{pr_{n+1}} 𝔾_{m,k},

is the pullback by the d’th power map on the base of the situation

L_{\overline{ψ}(∑_{i}(w_{i}/d)x_{i})} ⊗ pr_{1}^{⋆}L_{∏_{i} χ_{v_{i}} (x_{i})}[1] on 𝔾_{m,k}^{n} →^{x^{W}} 𝔾_{m,k}.

Therefore we have

F T\overline{ψ}Rm_{!}L_{V mod W} |𝔾_{m,k} ≅ [d]^{⋆}R(x^{W})_{!}(L_{\overline{ψ}(∑_{i}(w_{i}/d)x_{i})} ⊗ pr_{1}^{⋆}L_{∏_{i} χ_{v_{i}} (x_{i})}[1]).

According to [Ka-GKM, 4.0,4.1, 5.5],

R^{a}(x^{W})_{!}(L_{\overline{ψ}(∑_{i}(w_{i}/d)x_{i})} ⊗ pr_{1}^{⋆}L_{∏_{i} χ_{v_{i}} (x_{i})})

vanishes for a ≠ n − 1, and for a = n − 1 is the multiple multiplicative ! convolution

Kl(ψ_{−w_{1}/d}; χ_{v_{1}}, w_{1}) ⋆_{!} Kl(ψ_{−w_{2}/d}; χ_{v_{2}}, w_{2}) ⋆_{!} ... ⋆_{!} Kl(ψ_{−w_{n}/d}; χ_{v_{n}}, w_{n}).

By [Ka-GKM, 4.3,5.6.2], for each convolvee we have geometric isomorphisms

Kl(ψ_{−w_{i}/d}; χ_{v_{i}}, w_{i}) = [w_{i}]_{⋆}Kl(ψ_{−w_{i}/d}; χ_{v_{i}}) ≅ Kl(ψ_{−1/d}; all w′_{i}th roots of χ_{v_{i}}).

So the above multiple convolution is the Kloosterman sheaf

Kl(ψ_{−1/d}; all w′_{1}th roots of χ_{v_{1}}, ..., all w′_{n}th roots of χ_{v_{n}})
:= H(ψ_{−1/d}; all w′_{1}th roots of χ_{v_{1}}, ..., all w′_{n}th roots of χ_{v_{n}}; ∅).

Recall that by definition

List(V, W) := (all w′_{1}th roots of χ_{v_{1}}, ..., all w′_{n}th roots of χ_{v_{n}}).

Putting this all together, we find the asserted geometric isomorphism

(F T\overline{ψ}Rf_{!}\overline{ℚ}_{ℓ})(V mod W)|𝔾_{m,k}
≅ [d]^{⋆}H(ψ_{−1/d}; List(V, W); ∅)[2 − n].
⊓ ⊔

We are now ready for the final step in the proof of Theorem 6.1. Recall that j_{1,k} : U_{k} := 𝔸_{k}^{1} − μ_{d} ⊂ 𝔸_{k}^{1}, and j_{2,k} : 𝔾_{m,k} ⊂ 𝔸_{k}^{1} are the inclusions. We must prove

Theorem 7.6. (Restatement of 6.1) Let V mod W be a character of Γ_{W} /Δ for which P rim^{n−2}(V mod W) is nonzero. Pick V in the coset V mod W. Then we have a geometric isomorphism of perverse sheaves on 𝔸_{k}^{1}

j_{1,k,⋆}P rim^{n−2}(V mod W)[1] ≅ j_{2,k,⋆}[d]^{⋆}H_{V,W,k}[1].

Proof. Over the open set U_{k}, we have seen that sheaves R^{i}π_{k,⋆}\overline{ℚ}_{ℓ}|U_{k} are geometrically constant for i ≠ n − 2, and that R^{n−2}π_{k,⋆}\overline{ℚ}_{ℓ}|U_{k} is the direct sum of P rim^{n−2} and a geometrically constant sheaf. The same is true for the Γ_{W} /Δ-isotypical components. Thus in K(U_{k}, \overline{ℚ}_{ℓ}), we have

Rπ_{k,⋆}\overline{ℚ}_{ℓ}(V mod W)|U_{k} := ∑_{i}(−1)^{i}R^{i}π_{k,⋆}\overline{ℚ}_{ℓ}(V mod W)|U_{k}

= (−1)^{n−2}P rim^{n−2}(V mod W) + (geom. const.).

Comparing this with the situation on all of 𝔸_{k}^{1}, we don’t know what happens at the d missing points of μ_{d}, but in any case we will have

Rπ_{k,⋆}\overline{ℚ}_{ℓ}(V mod W) = (−1)^{n−2}j_{1,k,⋆}P rim^{n−2}(V mod W)
+(geom. const.) + (punctual, supported in μ_{d})

in K(𝔸_{k}^{1}, \overline{ℚ}_{ℓ}).

Taking Fourier Transform, we get

F T\overline{ψ}j_{1,k,⋆}P rim^{n−2}(V mod W) =

(−1)^{n−2}F T\overline{ψ}Rπ_{k,⋆}\overline{ℚ}_{ℓ}(V mod W)+(punctual, supported at 0)+(sum of L_{ψ_{ζ}} ′s)

in K(𝔸_{k}^{1}, \overline{ℚ}_{ℓ}).

By Lemma 7.3 , we have

(F T_{ψ}Rπ_{k,!}\overline{ℚ}_{ℓ})(V mod W)|𝔾_{m,k} ≅ F T_{ψ}Rf_{!}\overline{ℚ}_{ℓ}(V mod W)|𝔾_{m,k},

so we have

F T\overline{ψ}j_{1,k,⋆}P rim^{n−2}(V mod W) =

(−1)^{n−2}F T_{ψ}Rf_{!}\overline{ℚ}_{ℓ}(V mod W)+(punctual, supported at 0)+(sum of L_{ψ_{ζ}} ′s)

in K(𝔸_{k}^{1}, \overline{ℚ}_{ℓ}).

By the previous theorem, we have

(F T\overline{ψ}Rf_{!}\overline{ℚ}_{ℓ})(V mod W)|𝔾_{m,k}
= (−1)^{n−2}[d]^{⋆}H(ψ_{−1/d}; List(V, W); ∅)

in K(𝔾_{m,\overline{k}}, \overline{ℚ}_{ℓ}). We don’t know what happens at the origin, but in any case we have

(F T\overline{ψ}Rf_{!}\overline{ℚ}_{ℓ})(V mod W) =

(−1)^{n−2}j_{2,k,⋆}[d]^{⋆}H(ψ_{−1/d}; List(V, W); ∅) + (punctual, supported at 0)

in K(𝔸1 ; \overline{k}, \overline{ℚ}_{ℓ}). So we find

F T\overline{ψ}j_{1,k,⋆}P rim^{n−2}(V mod W) =
j_{2,k,⋆}[d]^{⋆}H(ψ_{−1/d}; List(V, W); ∅)+
(punctual, supported at 0) + (sum of L_{ψ_{ζ}} ′s)

in K(𝔸1 ; \overline{k}, \overline{ℚ}_{ℓ}). Now apply the inverse Fourier Transform F T_{ψ}. By Theorem 7.2, we obtain an equality

j_{1,k,⋆}P rim^{n−2}(V mod W)[1] =
j_{2,k,⋆}[d]^{⋆}H_{V,W,k}[1] + (geom. constant) + (punctual)

in the group K(𝔸1 ; \overline{k}, \overline{ℚ}_{ℓ}). This is the free abelian group on isomorphism classes of irreducible perverse sheaves on 𝔸1 ; \overline{k}. So in any equality of elements in this group, we can delete all occurrences of any particular isomorphism class, and still have an equality.

On the open set U_{k}, the lisse sheaves P rim^{n−2}(V mod W) and [d]^{⋆}H_{V,W,k} are both pure, hence completely reducible on U\overline{k} by [De-Weil II, 3.4.1 (iii)]. So both perverse sheaves j_{1,k,⋆}P rim^{n−2}(V mod W)[1] and j_{2,k,⋆}[d]^{⋆}H_{V,W,k}[1] on 𝔸1 ; \overline{k} are direct sums of perverse irreducibles which are middle extensions from U\overline{k}, and hence have no punctual constituents. So we may cancel the punctual terms, and conclude that we have

j_{1,k,⋆}P rim^{n−2}(V mod W)[1] − j_{2,k,⋆}[d]^{⋆}H_{V,W,k}[1] = (geom. constant)

in the group K(𝔸1 ; \overline{k}, \overline{ℚ}_{ℓ}). By Lemma 5.2, the left hand side has generic rank zero, so there can be no geometrically constant virtual summand. Thus we have an equality of perverse sheaves

j_{1,k,⋆}P rim^{n−2}(V mod W)[1] = j_{2,k,⋆}[d]^{⋆}H_{V,W,k}[1]

in the group K(𝔸1 ; \overline{k}, \overline{ℚ}_{ℓ}). Therefore the two perverse sheaves have geometrically isomorphic semisimplifications. But by purity, both are geometrically semisimple. This concludes the proof of Theorem 6.1, and so also the proof of Theorem 5.3 ⊓ ⊔

8 Appendix I: The transcendental approach

In this appendix, we continue to work with the fixed data (n, d, W), but now over the groundring ℂ. We give a transcendental proof of Theorem 5.3, but only for the Γ_{W} /Δ-invariant part P rim^{n−2}(0 mod W). Our proof is essentially a slight simplification of an argument that Shepherd-Barron gave in a November, 2006 lecture at MSRI, where he presented a variant of [H-SB-T, pages 5-22]. We do not know how to treat the other eigensheaves P rim^{n−2}(V mod W), with V mod W a nontrivial character of Γ_{W} /Δ, in an analogous fashion.

First, let us recall the bare definition of hypergeometric D-modules. We work on 𝔾_{m} (always over ℂ), with coordinate λ. We write D := λd/dλ. We denote by D := ℂ[λ, 1/λ][D] the ring of differential operators on 𝔾_{m}. Fix nonnegative integers a and b, not both 0. Suppose we are given an unordered list of a complex numbers α_{1}, ..., α_{a} ,not necessarily distinct. Let β_{1}, ..., β_{b} be a second such list, but of length b. We denote by Hyp(α′_{i}s; β′_{j}s) the differential operator

Hyp(α′_{i}s; β′_{j}s) := ∏_{i}(D − α_{i}) − λ ∏_{j}(D − β_{j})

and by H(α′_{i}s; β′_{j}s) the holonomic left D-module

H(α′_{i}s; β′_{j}s) := D/DHyp(α′_{i}s; β′_{j}s).

We say that H(α′_{i}s; β′_{j}s) is a hypergeometric of type (a, b).

One knows [Ka-ESDE, 3.2.1] that this H is an irreducible D-module on 𝔾_{m}, and remains irreducible when restricted to any dense open set U ⊂ 𝔾_{m}, if and only if the two lists are disjoint “mod ℤ”, i.e., for all i, j, α_{i} − β_{j} is not an integer. [If we are given two lists List_{1} and List_{2} which are not identical mod ℤ, but possibly not disjoint mod ℤ, we can “cancel” the common (mod ℤ) entries, and get an irreducible hypergeometric H(Cancel(List_{1}, List_{2})).]

We will assume henceforth that this disjointness mod ℤ condition is satisfied, and that a = b. Then H(α′_{i}s; β′_{j}s) has regular singular points at 0, 1, ∞. If all the α_{i} and β_{j} all lie in ℚ, pick a common denominator N, and denote by χ_{α_{i}} the character of μ_{N}(ℂ) given by

χ_{α_{i}}(ζ) := ζ^{α_{i}N}.

Similarly for χ_{β_{j}}. For any prime number ℓ, the Riemann-Hilbert partner of H(α′_{i}s; β′_{j}s) is the \overline{ℚ}_{ℓ} perverse sheaf H^{can}(χ_{α_{i}} ′s; χ_{β_{j}} ′s)[1] on 𝔾_{m}, cf. [Ka-ESDE, 8.17.11].

We denote by D_{η} := ℂ(λ)[D] the ring of differential operators at the generic point. Although this ring is not quite commutative, it is near enough to being a one-variable polynomial ring over a field that it is left (and right) Euclidean, for the obvious notion of long division. So every nonzero left ideal in D_{η} is principal, generated by the monic (in D_{η}) operator in it of lowest order. Given a left D_{η}-module M, and an element m ∈ M, we denote by Ann(m, M) the left ideal in D_{η} defined as

Ann(m, M) := {operators L ∈ D_{η}|L(m) = 0 in M}.

If Ann(m, M) ≠ 0, we define L_{m,M} ∈ D_{η} to be the lowest order monic operator in Ann(m, M).

We have the following elementary lemma, whose proof is left to the reader.

Lemma 8.1. Let N and M be left D_{η}-modules, f : M → N a horizontal (:= D_{η}-linear) map, and m ∈ M. Suppose that Ann(m, M) ≠ 0. Then Ann(m, M) ⊂ Ann(f(m), N), and L_{m,M} is right-divisible by L_{f(m),N}.

We now turn to our complex family π : 𝕏 → 𝔸^{1}, given by

X_{λ} := X_{λ}(W, d) : ∑_{i=1}^{n} w_{i}X_{i}^{d} − dλX^{W} = 0.

We pull it back to U := 𝔾_{m} − μ_{d} ⊂ 𝔸^{1}, over which it is proper and smooth, and form the de Rham incarnation of P rim^{n−2}, which we denote P rim_{dR}^{n−2}. We also have the relative de Rham cohomolgy of (ℙ^{n−1} × U − 𝕏_{U})/U over the base U in degree n − 1, which we denote simply H_{dR}^{n−1}((ℙ − 𝕏)/U). Both are O-locally free D-modules (Gauss-Manin connection) on U, endowed with a horizontal action of Γ_{W} /Δ. The Poincaré residue map gives a horizontal, Γ_{W} /Δ-equivariant isomorphism

Res : H_{dR}^{n−1}((ℙ − 𝕏)/U) ≅ P rim_{dR}^{n−2}.

As in the discussion beginning Section 6, we write 1 = ∑_{i} b_{i}w_{i} to obtain a descent of our family through the d’power map: the family π_{desc} : 𝕐 → 𝔾_{m} given by

Y_{λ} : ∑_{i=1}^{n} w_{i}λ^{−b_{i}}Y_{i}^{d} = dX^{W} .

The same group Γ_{W} /Δ acts on this family, which is projective and smooth over 𝔾_{m} − {1}. So on 𝔾_{m} − {1}, we have P rim_{dR,desc}^{n−2} for this family, and its fixed part P rim_{dR,desc}^{n−2}(0 mod W), whose pullback [d]^{⋆}P rim_{dR,desc}^{n−2}(0 mod W) is the sheaf P rim_{dR}^{n−2}(0 mod W)|(𝔾_{m} − μ_{d}).

Our next step is to pull back further, to a small analytic disk. Choose a real constant C > 4. Pull back the descended family to a small disc U_{an,C} around C. We take the disc small enough that for λ ∈ U_{an,C}, we have |C/λ|^{b_{i}} < 2 for all i. The extension of scalars map

H_{dR}^{n−1}((ℙ − 𝕐)/(𝔾_{m} − {1})) → H_{dR}^{n−1}((ℙ − 𝕐)/(𝔾_{m} − {1})) ⊗_{O_{𝔾m−{1}}} O_{U_{an,C}}

is a horizontal map; we view both source and target as D -modules. Over this disc, the C^{∞} closed immersion

\begin{matrix}γ : (S^{1})^{n}/Diagonal & → & ℙ^{n−1}, \\ (z_{1}, ..., z_{n}) & ↦ & (C^{b_{1}/d}z_{1}, ..., C^{b_{n−1}/d}z_{n−1}, C^{b_{n}/d}z_{n}) \end{matrix}

lands entirely in ℙ − 𝕐: its image is an n − 1-torus Z ⊂ ℙ^{n−1} which is disjoint from Y_{λ} for λ ∈ U_{an,C}. Restricting to the Γ_{W} /Δ-invariant part

H_{dR}^{n−1}((ℙ − 𝕐)/(𝔾_{m} − {1}))(0 mod W),

we get a horizontal map

H_{dR}^{n−1}((ℙ − 𝕐)/(𝔾_{m} − {1}))(0 mod W) → H^{0}(U_{an,C}, O_{U_{an,C}}), ω ↦ ∫_{Z} ω.

Write y_{i} := Y_{i}/Y_{n} for i = 1, ..., n − 1. Denote by

ω ∈ H_{dR}^{n−1}((ℙ − 𝕐)/(𝔾_{m} − {1}))(0 mod W)

the (cohomology class of the) holomorphic n − 1-form

ω := (1/2πi)^{n−1}(\frac{dY^{W}}{dY^{W} − ∑_{i=1}^{n} w_{i}λ^{−b_{i}}Y_{i}^{d}}) ∏_{i=1}^{n−1} dy_{i}/y_{i}.

Our next task is to compute the integral

∫_{Z} ω.

The computation will involve the Pochammer symbol. For α ∈ ℂ, and k ≥ 1 a positive integer, the Pochammer symbol (α)_{k} is defined by

(α)_{k} := Γ (α + k)/Γ (α) = ∏_{i=0}^{k−1}(α + i).

We state for ease of later reference the following elementary identity.

Lemma 8.2. For integers k ≥ 1 and r ≥ 1, we have

(kr)!/r^{kr} = ∏_{i=1}^{r}(i/r)_{k}.

Lemma 8.3. We have the formula

∫_{Z} ω = 1 + ∑_{k≥1}(\frac{∏_{i=1}^{d}(i/d)_{k}}{∏_{i=1}^{n} ∏_{j=1}^{w_{i}}(j/w_{i})_{k}})(1/λ)^{k}.

Proof. Divide top and bottom by dY^{W}, expand the geometric series, and integrate term by term. This is legitimate because at a point z ∈ Z, the function ∑_{i=1}^{n}(w_{i}/d)λ^{−b_{i}}Y_{i}^{d}/Y^{W} has the value

∑_{i=1}^{n}(w_{i}/d)λ^{−b_{i}}C^{b_{i}}z_{i}^{d}/Cz^{W} = ∑_{i=1}^{n}(w_{i}/d)(C/λ)^{b_{i}}z_{i}^{d}/Cz^{W} ,

which has absolute value ≤ 2(∑_{i=1}^{n}(w_{i}/d))/C = 2/C ≤ 1/2. Because each term in the geometric series is homogeneous of degree zero, the integral of the k’th term in the geometric series is the coefficient of z^{kW} in (∑_{i=1}^{n}(w_{i}/d)(λ)^{−b_{i}}z_{i}^{d})^{k}. This coefficient vanishes unless k is a multiple of d (because gcd(w_{1}, ..., w_{n}) = 1). The integral of the dk’th term is the coefficient of z^{kdW} in (∑_{i=1}^{n}(w_{i}/d)(λ)^{−b_{i}}z_{i}^{d})^{dk}, i.e., the coefficient of z^{kW} in (∑_{i=1}^{n}(w_{i}/d)(λ)^{−b_{i}}z_{i})^{dk}. Expanding by the multinomial theorem, this coefficient is

(dk)! ∏_{i=1}^{n}(((w_{i}/d)λ^{−b_{i}})^{kw_{i}}/(kw_{i})!) = (λ)^{−k}((dk)!/d^{dk})/ ∏_{i=1}^{n}((kw_{i})!/w_{i}^{kw_{i}}),

which, by the previous lemma, is as asserted. ⊓ ⊔

This function

F (λ) : ∫_{Z} ω = 1 + ∑_{k≥1}(\frac{∏_{i=1}^{d}(i/d)_{k}}{∏_{i=1}^{n} ∏_{j=1}^{w_{i}}(j/w_{i})_{k}})(1/λ)^{k}

is annihilated by the following differential operator. Consider the two lists of length d.

List(all d) := {1/d, 2/d, ..., d/d},
List(0, W) := {1/w_{1}, 2/w_{1}, ..., w_{1}/w_{1}, ..., 1/w_{n}, 2/w_{n}, ..., w_{n}/w_{n}}.

These lists are certainly not identical mod ℤ; the second one contains 0 with multiplicity n, while the first contains only a single integer. Let us denote the cancelled lists, whose common length we call a,

Cancel(List(all d); List(0, W)) = (α_{1}, ..., α_{a}); (β_{1}, ..., β_{a}).

So we have

F (λ) : ∫_{Z} ω = 1 + ∑_{k≥1}(\frac{∏_{i=1}^{a}(α_{i})_{k}}{∏_{i=1}^{a}(β_{i})_{k}})(1/λ)^{k},

which one readily checks is annihilated by the differential operator

Hyp_{0,W} := Hyp(α′_{i}s; β_{i} − 1′s) := ∏_{i=1}^{a}(D − α_{i}) − λ ∏_{i=1}^{a}(D − (β_{i} − 1)).

Theorem 8.4. We have an isomorphism of D-modules on 𝔾_{m} − {1},

H_{dR}^{n−1}((ℙ−𝕐)/(𝔾_{m}−{1})) ≅ H_{0,W} |(𝔾_{m}−{1}) := H(α′_{i}s; β_{i}−1′s)|(𝔾_{m}−{1}).

Proof. Both sides of the alleged isomorphism are O-coherent D-modules on 𝔾_{m} − {1}, so each is the “middle extension” of its restriction to any Zariski dense open set in 𝔾_{m} − {1}. So it suffices to show that both sides become isomorphic over the function field of 𝔾_{m} − {1}, i.e., that they give rise to isomorphic D_{η}-modules. For this, we argue as follows. Denote by A the ring

A := H^{0}(U_{an,C}, O_{U_{an,C}}) ⊗_{O_{𝔾m−{1}}} ℂ(λ),

which we view as a D_{η}-module. We have the horizontal map

H_{dR}^{n−1}((ℙ − 𝕐)/(𝔾_{m} − \{1\}))(0 mod W) \underrightarrow{_{∫_{Z}}} H^{0}(U_{an,C}, O_{U_{an,C}}).

Tensoring over O_{𝔾_{m}−{1}} with ℂ(λ), we obtain a horizontal map

H_{dR}^{n−1}((ℙ − 𝕐)/ℂ(λ))(0 mod W) \underrightarrow{_{∫_{Z}}} A.

By (the Hyp analogue of) Lemma 5.2, we know that the source has ℂ(λ)- dimension a:= the order of Hyp(α′_{i}s; β_{i} − 1′s). So the element ω in the source is annihilated by some operator in D_{η} of order at most a, simply because ω and its first a derivatives must be linearly dependent over ℂ(λ). So the lowest order operator annihilating ω in H_{dR}^{n−1}((ℙ − 𝕐)/ℂ(λ))(0 mod W), call it L_{ω,H_{dR}}, has order at most a. On the other hand, the irreducible operator Hyp(α′_{i}s; β_{i} − 1′s) annihilates ∫_{Z} ω ∈ A. But ∫_{Z} ω ≠ 0, so Ann(∫_{Z} ω, A) is a proper left ideal in D_{η}, and hence is generated by the irreducible monic operator (1/(1 − λ))Hyp(α′_{i}s; β_{i} − 1′s). By Lemma 8.2, we know that L_{ω,H_{dR}} is divisible by (1/(1 − λ))Hyp(α′_{i}s; β_{i} − 1′s). But L_{ω,H_{dR}} has order at most a, the order of Hyp(α′_{i}s; β_{i} − 1′s), so we conclude that L_{ω,H_{dR}} = (1/(1 − λ))Hyp(α′_{i}s; β_{i} − 1′s). Thus the D_{η}-span of ω in the group H_{dR}^{n−1}((ℙ − 𝕐)/ℂ(λ))(0 mod W) is D_{η}/D_{η}Hyp(α′_{i}s; β_{i} −1′s). Comparing dimensions we see that this D_{η}-span is all of H_{dR}^{n−1}((ℙ − 𝕐)/ℂ(λ))(0 mod W).

⊓ ⊔

Corollary 8.5. For the family

X_{λ} := X_{λ}(W, d) : ∑_{i=1}^{n} w_{i}X_{i}^{d} − dλX^{W} = 0,

its P rim_{dR}^{n−2}(0 mod W) as D-module on 𝔸^{1} − μ_{d} is related to the D-module [d]^{⋆}(H_{0,W} |(𝔾_{m} − {1})) on 𝔾_{m} − μ_{d} as follows.

(1) We have an isomorphism of D-modules on 𝔾_{m} − μ_{d},

P rim_{dR}^{n−2}(0 mod W)|(𝔾_{m} − μ_{d}) ≅ [d]^{⋆}(H_{0,W} |(𝔾_{m} − {1})).

(2) Denote by j_{1} : 𝔸^{1} − μ_{d} ⊂ 𝔸^{1} and j_{2} : 𝔾_{m} − μ_{d} ⊂ 𝔸^{1} the inclusions. The we have an isomorphism of D-modules on 𝔸^{1} of the middle extensions

j_{1,!,⋆}(P rim_{dR}^{n−2}(0 mod W)) ≅ j_{2,!,⋆}([d]^{⋆}(H_{0,W} |(𝔾_{m} − {1}))).

Proof. The first isomorphism is the pullback by d’th power of the isomorphism of the theorem above. We obtain the second isomorphism as follows. Denote by j_{3} : 𝔾_{m} − μ_{d} ⊂ 𝔸^{1} − μ_{d} the inclusion. Because P rim_{dR}^{n−2}(0 mod W) is an O-coherent D-module on 𝔸^{1} − μ_{d}, it is the middle extension

j_{3,!,⋆}(P rim_{dR}^{n−2}(0 mod W)|(𝔾_{m} − μ_{d})).

Because j_{2} = j_{1} ∘ j_{3}, we obtain the second isomorphism by applying j_{2,!,⋆} to the first isomorphism. ⊓ ⊔

Theorem 8.6. Suppose n ≥ 3. For either the family

X_{λ} := X_{λ}(W, d) : ∑_{i=1}^{n} w_{i}X_{i}^{d} − dλX^{W} = 0,

over 𝔸^{1} − μ_{d}, or the descended family

Y_{λ} : ∑_{i=1}^{n} w_{i}λ^{−b_{i}}Y_{i}^{d} = dX^{W}

over 𝔾_{m}−{1}, consider its P rim_{dR}^{n−2}(0 mod W) (resp. P rim_{dR,desc}^{n−2}(0 mod W)) as a D-module, and denote by a its rank. For either family, its differential galois group G_{gal} (which here is the Zariski closure of its monodromy group) is the symplectic group Sp(a) if n − 2 is odd, and the orthogonal group O(a) if n − 2 is even.

Proof. Poincaré duality induces on P rim_{dR}^{n−2}(0 mod W) (resp. on the module P rim_{dR,desc}^{n−2}(0 mod W)) an autoduality which is symplectic if n − 2 is odd, and orthogonal if n − 2 is even. So we have a priori inclusions G_{gal} ⊂ Sp(a) if n − 2 is odd, G_{gal} ⊂ O(a) if n − 2 is even. It suffices to prove the theorem for the descended family. This is obvious in the Sp case, since the identity component of G_{gal} is invariant under finite pullback. In the O case, we must rule out the possibility that the pullback has group SO(a) rather than O(a). For this, we observe that an orthogonally autodual hypergeometric of type (a, a) has a true reflection as local monodromy around 1 (since in any case an irreducible hypergeometric of type (a, a) has as local monodromy around 1 a pseudoreflection, and the only pseudoreflection in an orthogonal group is a true reflection). As the d’th power map is finite étale over 1, the pullback has a true reflection as local monodromy around each ζ ∈ μ_{d}. So the group for the pullback contains true reflections, so must be O(a).

We now consider the descended family. So we are dealing with H_{0,W} := H(α′_{i}s; β_{i}−1′s). From the definition of H_{0,W}, we see that β = 1 mod ℤ occurs among the β_{i} precisely n − 1 times (n − 1 times and not n times because of a single cancellation with List(all d)). Because n − 1 ≥ 2 by hypothesis, local monodromy around ∞ is not semisimple [Ka-ESDE, 3.2.2] and hence H(α′_{i}s; β′_{j}s) is not Belyi induced or inverse Belyi induced, cf. [Ka-ESDE, 3.5], nor is its G^{0,der} trivial.

We next show that H_{0,W} is not Kummer induced of any degree r ≥ 2. Suppose not. As the α_{i} all have order dividing d in ℂ/ℤ, r must divide d, since 1/r mod ℤ is a difference of two α_{i}’s, cf. [Ka-ESDE, 3.5.6]. But the β_{j} mod ℤ are also stable by x ↦ x + 1/r, so we would find that 1/r mod ℤ occurs with the same multiplicity n − 1 as 0 mod ℤ among the β_{j} mod ℤ. So r must divide at least n − 1 of the w_{i}; it cannot divide all the w_{i} because gcd(w_{1}, ..., w_{n}) = 1. But this 1/r cannot cancel with List(all d), otherwise its multiplicity would be at most n − 2. This lack of cancellation means that r does not divide d, contradiction.

Now we appeal to [Ka-ESDE, 3.5.8]: let H(α′_{i}s; β′_{j}s) be an irreducible hypergeometric of type (a, a) which is neither Belyi induced nor inverse Belyi induced not Kummer induced. Denote by G its differential galois group G_{gal}, G^{0} its identity component, and G^{0,der} the derived group (:= commutator subgroup) of G^{0}. Then G^{0,der} is either trivial or it is one of SL(a) or SO(a) or, if a is even, possibly Sp(a).

In the case of H_{0,W}, we have already seen that G_{gal}^{0,der} is not trivial. Given that G_{gal} lies in either Sp(a) or O(a), depending on the parity of n−2, the only possibility is that G_{gal} = Sp(a) for n − 2 odd, and that G_{gal} = O(a) or SO(a) if n − 2 is even. In the even case, the presence of a true reflection in G_{gal} rules out the SO case. ⊓ ⊔

Corollary 8.7. In the context of Theorem 5.3, on each geometric fibre of U_{R_{0}[1/ℓ]}/Spec(R_{0}[1/ℓ]), the geometric monodromy group G_{geom} of the sheaf P rim^{n−2}(0 mod W) is the full symplectic group Sp(a) if n − 2 is odd, and is the full orthogonal group O(a) if n − 2 is even.

Proof. On a ℂ-fibre, this is just the translation through Riemann-Hilbert of the theorem above. The passage to other geometric fibres is done by the Tame Specialization Theorem [Ka-ESDE, 8.17.3]. ⊓ ⊔

When does it happen that P rim_{dR}^{n−2}(0 mod W) has rank n − 1 and all Hodge numbers 1?

Lemma 8.8. The following are equivalent.

(1) P rim_{dR}^{n−2}(0 mod W) has rank n − 1.

(2) Every w_{i} divides d, and for all i ≠ j, gcd(w_{i}, w_{j}) = 1.

(3) Local monodromy at ∞ is a single unipotent Jordan block.

(4) Local monodromy at ∞ is a single Jordan block.

(5) All the Hodge numbers P rim_{dR}^{a,b}(0 mod W)_{a+b=n−2} are 1.

Proof. (1)⇒(2) The rank is at least n − 1, as this is the multiplicity of 0 mod ℤ as a β in H_{0,W}. If the rank is no higher, then each w_{i} must divide d, so that the elements 1/w_{i}, ..., (w_{i} − 1)/w_{i} mod ℤ can cancel with List(all d). And the w_{i} must be pairwise relatively prime, for if a fraction 1/r mod ℤ with r ≥ 2 appeared among both 1/w_{i}, ..., (w_{i} −1)/w_{i} and 1/w_{j}, ..., (w_{j} −1)/w_{j}, only one of its occurrences at most can cancel with List(all d).

(2)⇒(1) If all w_{i} divide d, and if the w_{i} are pairwise relatively prime, then after cancellation we find that H_{0,W} has rank n − 1.

(1)⇒(3) If (1) holds, then the β_{i}’s are all 0 mod ℤ, and there are n − 1 of them. This forces H_{0,W} and also [d]^{⋆}H_{0,W} to have its local monodromy around ∞, call it T, unipotent, with a single Jordan block, cf. [Ka-ESDE, 3.2.2].

(3)⇒(4) is obvious.

(4)⇒(3) Although d’th power pullback may change the eigenvalues of local monodromy at ∞, it does not change the number of distinct Jordan blocks. But there is always one unipotent Jordan block of size n − 1, cf. the proof of (1)⇒(2).

(3)⇒(5) If not all the n−1 Hodge numbers are 1, then some Hodge number vanishes, and at most n − 2 Hodge numbers are nonzero. But by [Ka-NCMT, 14.1] [strictly speaking, by projecting its proof onto Γ_{W} /Δ-isotypical components] any local monodromy is quasiunipotent of exponent of nilpotence ≤ h:= the number of nonzero Hodge numbers. So our local monodromy T around ∞, already unipotent, would satisfy (T − 1)^{n−2} = 0. But as we have already remarked, H_{0,W} always has unipotent Jordan block of size n − 1. Therefore all the Hodge numbers are nonzero, and hence each is 1.

(5)⇒(1) is obvious. ⊓ ⊔

Remark 8.9. Four particular n = 5 cases where condition (2) is satisfied, namely W = (1, 1, 1, 1, 1), W = (1, 1, 1, 1, 2), W = (1, 1, 1, 1, 4), and W = (1, 1, 1, 2, 5), were looked at in detain in the early days of mirror symmetry, cf. [Mor, Section 4, Table 1].

Whatever the rank of P rim_{dR}^{n−2}(0 mod W), we have:

Lemma 8.10. All the Hodge numbers P rim_{dR}^{a,b}(0 mod W)_{a+b=n−2} are nonzero.

Proof. Repeat the proof of (3)⇒(5). ⊓ ⊔

9 Appendix II: The situation in characteristic p, when p divides some w_{i}

We continue to work with the fixed data (n, d, W). In this appendix, we indicate briefly what happens in a prime-to-d characteristic p which divides one of the w_{i}. For each i, we denote by w_{i}^{∘} the prime-to-p part of w_{i}, i.e.,

w_{i} = w_{i}^{∘} × (a power of p),

and we define

W^{∘} := (w_{1}^{∘}, ..., w_{n}^{∘}).

We denote by d_{W^{∘}} the integer

d_{W^{∘}} := lcm(w_{1}^{∘}, ..., w_{n}^{∘})d,

and define

d′ := ∑_{i} w_{i}^{∘}.

For each i, we have w_{i} ≡ w_{i}^{∘} mod p − 1, so we have the congruence, which will be used later,

d ≡ d′ mod p − 1.

We work over a finite field k of characteristic p prime to d which contains the d_{W^{∘}}’th roots of unity. We take for ψ a nontrivial additive character of k which is of the form ψ_{𝔽_{p}} ∘ T race_{k/𝔽_{p}}, for some nontrivial additive character ψ_{𝔽_{p}} of 𝔽_{p}. The signifigance of this choice of ψ is that for q = p^{e}, e ≥ 1, any power of p, under the q’th power map we have

[q]_{⋆}L_{ψ} = L_{ψ}, [q]^{⋆}L_{ψ} = L_{ψ}

on 𝔸_{k}^{1}.

The family we study in this situation is π : 𝕏 → 𝔸^{1},

X_{λ} := X_{λ}(W, d) : ∑_{i=1}^{n} w_{i}^{∘}X_{i}^{d} − dλX^{W} = 0.

The novelty is that, because p divides some w_{i}, this family is projective and smooth over all of 𝔸^{1}.

The group Γ_{W} /Δ operates on this family. Given a character V mod W of this group, the rank of the eigensheaf P rim^{n−2}(V mod W) is still given by the same recipe as in Lemma 3.1(1), because at λ = 0 we have a smooth Fermat hypersurface of degree d.

Given an element V = (v_{1}, ..., v_{n}) ∈ (ℤ/dℤ)_{0}^{n}, we attach to it an unordered list List(V, W) of d′ = ∑_{i} w_{i}^{∘} multiplicative characters of k^{×}, by the following procedure. For each index i, we denoted by χ_{v_{i}} the character of k^{×} given by

ζ ↦ ζ^{(v_{i}/d)#k^{×}}.

This characterχ_{v_{i}} has w_{i}^{∘} (as opposed to w_{i}) distinct w_{i}’th roots. We then define

List(V, W) = {all w_{1} ′th roots of χ_{v_{1}}, ..., all w_{n} ′th roots of χ_{v_{n}}}.

We will also need the same list, but for −V, and the list

List(all d) := {all characters of order dividing d}.

The two lists List(−V, W) and List(all d) are not identical, as they have different lengths d′ and d respectively, so we can apply the Cancel operation, and form the hypergeometric sheaf

H_{V,W} := H^{can}(Cancel(List(all d); List(−V, W)))

on 𝔾_{m,k}. Exactly as in Lemma 5.2, if P rim^{n−2}(V mod W) is nonzero, its rank is the rank of H_{V,W}.

An important technical fact in this situation is the following variant of Theorem 7.2, cf. [Ka-ESDE, 9.3.2], which “works” because 𝔽_{p}^{×} has order p−1.

Theorem 9.1. Denote by ψ_{−1/d} the additive character x ↦ ψ(−x/d), and denote by j : 𝔾_{m,k} ⊂ 𝔸_{k}^{1} the inclusion. Denote by Λ_{1}, ..., Λ_{d} the list List(all d) of all the multiplicative characters of k^{×} of order dividing d. Let d′ be a strictly positive integer with d′ ≡ d mod p − 1. For any unordered list of d′ multiplicative characters ρ_{1}, ...ρ_{d′} of k^{×} which is not identical to List(all d), the perverse sheaf

F T_{ψ}j_{⋆}[d]^{⋆}H(ψ_{−1/d}; ρ_{1}, ...ρ_{d′}; ∅)[1]

on 𝔸_{k}^{1} is geometrically isomorphic to the perverse sheaf

j_{⋆}[d]^{⋆}H(ψ; Cancel(List(all d); \overline{ρ_{1}}, ..., \overline{ρ_{d′}}))[1].

The main result is the following.

Theorem 9.2. Suppose that P rim^{n−2}(V mod W) is nonzero and denote by j : 𝔾_{m} ⊂ 𝔸^{1} the inclusion. Choose V in the coset V mod W. There exists a constant A_{V,W} ∈ \overline{ℚ}_{ℓ}^{×} and an isomorphism of lisse sheaves on 𝔸_{k}^{1},

P rim^{n−2}(V mod W) ≅ j_{⋆}[d]^{⋆}H_{V,W} ⊗ (A_{V,W})^{deg}.

Proof. Because our family is projective and smooth over all of 𝔸^{1}, Deligne’s degeneration theorem [De-TLCD, 2.4] gives a decomposition

Rπ_{⋆}\overline{ℚ}_{ℓ} ≅ P rim^{n−2}[2 − n] ⨁(geom. constant).

So applying Fourier Transform, we get

F T\overline{ψ}Rπ_{⋆}\overline{ℚ}_{ℓ}(V mod W)|𝔾_{m} ≅ F T\overline{ψ}P rim^{n−2}(V mod W)[2 − n]|𝔾_{m}.

On the open set V ⊂ 𝕏 where X^{W} is invertible, the restriction of π becomes the map f, now given by

(X_{1}, ..., X_{n}) ↦ ∑_{i}(w_{i}^{∘}/d)X_{i}^{d}/X^{W} .

Then the argument of Lemma 7.3 gives

F T\overline{ψ}P rim^{n−2}(V mod W)[2 − n]|𝔾_{m} ≅ F T\overline{ψ}Rf_{!}\overline{ℚ}_{ℓ}(V mod W)|𝔾_{m}.

Theorem 7.5 remains correct as stated. [In its proof, the only modification needed is the analysis now of the sheaves Kl(ψ_{−w_{i}^{∘}/d}; χ_{v_{i}}, w_{i}). Pick for each i a w_{i}’th root ρ_{i} of χ_{v_{i}}. We have geometric isomorphisms

Kl(ψ_{−w_{i}^{∘}/d}; χ_{v_{i}}, w_{i}) = [w_{i}]_{⋆}Kl(ψ_{−w_{i}^{∘}/d}; χ_{v_{i}}) = L_{ρ_{i}} ⊗ [w_{i}]_{⋆}L_{ψ_{−w^{∘}_{i}/d}}

= L_{ρ_{i}}⊗[w_{i}^{∘}]_{⋆}L_{ψ_{−w^{∘}_{i}/d}} ≅ L_{ρ_{i}}⊗Kl(ψ_{−1/d}; all the w_{i}^{∘} char′s of order dividing w_{i})

≅ Kl(ψ_{−1/d}; all the w_{i}^{∘} w′_{i}th roots of χ_{v_{i}}).]

At this point, we have a geometric isomorphism

F T\overline{ψ}P rim^{n−2}(V mod W)[2 − n]|𝔾_{m}
≅ [d]^{⋆}H(ψ_{−1/d}; List(V, W); ∅)[2 − n].

So in the Grothendieck group K(𝔸1 ; \overline{k}, \overline{ℚ}_{ℓ}), we have

F T\overline{ψ}P rim^{n−2}(V mod W)
= j_{⋆}[d]^{⋆}H(ψ_{−1/d}; List(V, W); ∅) + (punctual, supported at 0).

Applying the inverse Fourier Transform, we find that in K(𝔸1 ; \overline{k}, \overline{ℚ}_{ℓ}) we have

P rim^{n−2}(V mod W) = j_{⋆}[d]^{⋆}H_{V,W} + (geom. constant).

As before, the fact that P rim^{n−2}(V mod W) and j_{⋆}[d]^{⋆}H_{V,W} have the same generic rank shows that there is no geometically constant term, so we have an equality of perverse sheaves in K(𝔸1 ; \overline{k}, \overline{ℚ}_{ℓ}),

P rim^{n−2}(V mod W) = j_{⋆}[d]^{⋆}H_{V,W} .

So these two perverse sheaves have isomorphic semisimplifications. Again by purity, both are geometrically semisimple. So the two sides are geometrically isomorphic. To produce the constant field twist, we repeat the descent argument of Lemma 6.2 to reduce to the case when both descended sides are geometrically irreducible and geometrically isomorphic, hence constant field twists of each other. ⊓ ⊔

10 Appendix III: Interesting pieces in the original Dwork family

In this appendix, we consider the case n = d, W = (1, 1, ..., 1). We are interested in those eigensheaves P rim^{n−2}(V mod W) that have unipotent local monodromy at ∞ with a single Jordan block. In view of the explicit description of P rim^{n−2}(V mod W)|(𝔾_{m} − μ_{d}) as [d]^{⋆}H_{V,W}, and the known local monodromy of hypergeometric sheaves, as recalled in Section 4, we have the following characterization.

Lemma 10.1. In the case n = d, W = (1, 1, ..., 1), let V mod W be a character of Γ_{W} /Δ such that P rim^{n−2}(V mod W) is nonzero. The following are equivalent.

(1) Local monodromy at ∞ on P rim^{n−2}(V mod W) has a single Jordan block.

(2) Local monodromy at ∞ on P rim^{n−2}(V mod W) is unipotent with a single Jordan block.

(3) Every V = (v_{1}, ..., v_{n}) in the coset V mod W has the following property: there is at most one v_{i} which occurs more than once, i.e., there is at most one a ∈ ℤ/dZ for which the number of indices i with v_{i} = a exceeds 1.

(4) A unique V = (v_{1}, ..., v_{n}) in the coset V mod W has the following property: the value 0 ∈ ℤ/dℤ occurs more than once among the v_{i}, and no other value a ∈ ℤ/dZ does.

Proof. In order for P rim^{n−2}(V mod W) to be nonzero, the list List(−V, W) must differ from List(all d). In this n = d case, that means precisely that List(−V, W) must have at least one value repeated. Adding a suitable multiple of W = (1, 1, ..., 1), we may assume that the value 0 occurs at least twice among the v_{i}. So (3) ⇔ (4).

For a hypergeometric H^{can}(χ′_{i}s; ρ′_{j}s) of type (a, a), local monodromy at ∞ has a single Jordan block if and only if all the ρ_{j}’s coincide, in which case the common value of all the ρ_{j}’s is the eigenvalue in that Jordan block. And [d]^{⋆}H^{can}(χ′_{i}s; ρ′_{j}s)’s local monodromy at at ∞ has the same number of Jordan blocks (possibly with different eigenvalues) as that of H^{can}(χ′_{i}s; ρ′_{j}s). In our situation, if we denote by (χ_{1}, ..., χ_{d}) all the characters of order dividing d, and by (χ_{−v_{1}}, ..., χ_{−v_{d}}) the list List(−V, W), then

H_{V,W} = H^{can}(Cancel((χ_{1}, ..., χ_{d}); (χ_{−v_{1}}, ..., χ_{−v_{d}}))).

So in order for local monodromy at ∞ to have a single Jordan block, we need all but one of the characters that occur among the χ_{v_{i}} to cancel into List(all d). But those that cancel are precisely those which occur with multiplicity 1. So (1) ⇔ (3). Now (2) ⇒ (1) is trivial, and (2) ⇒ (4) by the explicit description of local monodromy at ∞ in terms of the ρ_{j}’s. ⊓ ⊔

Lemma 10.2. Suppose the equivalent conditions of Lemma 10.1 hold. Denote by a the rank of P rim^{n−2}(V mod W). Then on any geometric fibre of (𝔸^{1} − μ_{d})/Spec(ℤ[ζ_{d}][1/dℓ]), the geometric monodromy group G_{geom} attached to P rim^{n−2}(V mod W) has identity component either SL(a) or SO(a) or, if a is even, possibly Sp(a).

Proof. By the Tame Specialization Theorem [Ka-ESDE, 8.17.13], the group is the same on all geometric fibres. So it suffices to look in some characteristic p > a. Because on our geometric fibre H_{V,W} began life over a finite field, and is geometrically irreducible, G_{geom}^{0} is semisimple. The case a = 1 is trivial. Suppose a ≥ 2. Because its local monodromy at ∞ is a single unipotent block, the hypergeometric H_{V,W} is not Belyi induced, or inverse Belyi induced, or Kummer induced, and G_{geom}^{0,der} is nontrivial. The result now follows from [Ka-ESDE, 8.11.2]. ⊓ ⊔

Lemma 10.3. Suppose the equivalent conditions of Lemma 10.1 hold. Denote by a the rank of P rim^{n−2}(V mod W). Suppose a ≥ 2. Denote by V the unique element in the coset V mod W in which 0 ∈ ℤ/dZ occurs with multiplicity a+1, while no other value occurs more than once. Then we have the following results.

(1) Suppose that −V is not a permutation of V. Then

G_{geom} = SL(a)

if n − 2 is odd, and

G_{geom} = {A ∈ GL(a)|det(A) = ±1}

if n − 2 is even.

(2) If −V is a permutation of V and n − 2 is odd, then a is even and

G_{geom} = Sp(a).

(3) If −V is a permutation of V and n − 2 is even, then a is odd and

G_{geom} = O(a).

Proof. That these results hold for H_{V,W} results from [Ka-ESDE, 8.11.5, 8.8.1,8.8.2]. In applying those results, one must remember that ∑_{i} v_{i} = 0 ∈ ℤ/dℤ, which implies that (“even after cancellation”) local monodromy at ∞ has determinant one. Thus in turn implies that when d, or equivalently n−2, is even, then (“even after cancellation”) local monodromy at 0 has determinant the quadratic character, and hence local monodromy at 1 also has determinant the quadratic character. So in the cases where the group does not have determinant one, it is because local monodromy at 1 is a true reflection. After [d]^{⋆}, which is finite étale over 1, we get a true reflection at each point in μ_{d}. ⊓ ⊔

Lemma 10.4. If the equivalent conditions of the previous lemma hold, then over ℂ the Hodge numbers of P rim^{n−2}(V mod W) form an unbroken string of 1’s, i.e., the nonzero among the P rim^{b,n−2−b}(V mod W) are all 1, and the b for which P rim^{b,n−2−b}(V mod W) is nonzero form (the integers in) an interval [A, A − 1 + a] for some A.

Proof. From the explicit determination of G_{geom}, we see in particular that P rim^{n−2}(V mod W) is an irreducible local system. Looking in a ℂ-fibre of (𝔸^{1} − μ_{d})/Spec(ℤ[ζ_{d}][1/dℓ]) and applying Riemann-Hilbert, we get that the D-module P rim_{dR}^{n−2}(V mod W) is irreducible. By Griffiths transversality, this irreducibility implies that the b for which P rim^{b,n−2−b}(V mod W) is nonzero form (the integers in) an interval. The fact that local monodromy at ∞ is

unipotent with a single Jordan block implies that the number of nonzero Hodge groups P rim^{b,n−2−b}(V mod W) is at least a, cf. the proof of Lemma 8.8, (3) ⇔ (5). ⊓ ⊔

References

[Abh-GTL] Abhyankar, S., “Galois theory on the line in nonzero characteristic”,

Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 68-133. [Abh-PP] Abhyankar, S., “Projective polynomials”, Proc. Amer. Math. Soc. 125

(1997), no. 6, 1643-1650.

[Ber] Bernardara, M., “Calabi-Yau complete intersections with infinitely many lines”, preprint, math.AG/0402454

[Car] Carlitz, L., “Resolvents of certain linear groups in a finite field”, Canad. J. Math. 8 (1956), 568-579.

[C-dlO-RV] Candelas, P., de la Ossa, X., Rodriguez-Villegas, F., “Calabi-Yau manifolds over finite fields, II”, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001), 121-157, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003.

[C-dlO-G-P] Candelas, P., de la Ossa, X., Green, P., Parkes, L., “A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory”, Nuclear Phys. B 359 (1991), no. 1, 21-74.

[De-CEPD] Deligne, P., “Cohomologie étale: les points de départ”, redigé par J.F.

Boutot, pp. 6-75 in SGA 4 1/2, cited below. [De-ST] Deligne, P., “Applications de la formule des traces aux sommes

trigonométriques”, pp. 168-232 in SGA 4 1/2, cited below. [De-TLCD] Deligne, P., “Théorème de Lefschetz et critres de dégénérescence de

suites spectrales”, Publ. Math. IHES 35 (1968) 259-278. [De-Weil I] Deligne, P., “La conjecture de Weil”, Publ. Math. IHES 43 (1974),

273-307. [De-Weil II] Deligne, P., “La conjecture de Weil II”, Publ. Math. IHES 52 (1981),

313-428. [Dw-Def] Dwork, B., “A deformation theory for the zeta function of a hypersur-

face” Proc. Internat. Congr. Mathematicians (Stockholm, 1962), 247-259. [Dw-Rat] Dwork, B., “On the rationality of the zeta function of an algebraic va-

riety”, Amer. J. Math 82 (1960), 631-648. [Dw-Hyp1] Dwork, B., “On the Zeta Function of a Hypersurface”, Publ. Math.

IHES 12 (1962), 5-68. [Dw-HypII] Dwork, B., “On the zeta function of a hypersurface, II”, Ann. of Math.

(2) 80 (1964), 227-299. [Dw-HypIII] Dwork, B., “On the zeta function of a hypersurface, III”, Ann. of

Math. (2) 83 (1966), 457-519.

[Dw-PAA] Dwork, B., “On p-adic analysis”, Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966), 129-154.

[Dw-PC] Dwork, B., “p-adic cycles”, Publ. Math. IHES 37 (1969), 27-115. [Dw-NPI] Dwork, B., “Normalized period matrices, I, Plane curves”, Ann. of

Math. (2) 94 (1971), 337-388. [Dw-NPII] Dwork, B., “Normalized period matrices, II”, Ann. of Math. (2) 98

(1973), 1-57. [Dw-BP] Dwork, B., “On the Boyarsky principle”, Amer. J. Math. 105 (1983),

no. 1, 115-156. [Grif-PCRI] Griffiths, P., “On the periods of certain rational integrals, I, II”,

Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496-541. [Gr-Rat] Grothendieck, A., “Formule de Lefschetz et rationalité des fonctions L”,

Séminaire Bourbaki, Vol. 9, Exp. No. 279, 41-55, Soc. Math. France, 1995. [H-SB-T] Harrris, M., Shepherd-Barron, N., Taylor, R., “A family of Calabi-

Yau varieties and potential automorphy”, preprint, June 19, 2006. [Ka-ASDE] Katz, N., “Algebraic solutions of differential equations (p-curvature

and the Hodge filtration)”, Invent. Math. 18 (1972), 1-118. [Ka-ESES] Katz, N., “Estimates for “singular” exponential sums”, IMRN 16

(1999), 875-899. [Ka-ESDE] Katz, N., “Exponential sums and differential equations”, Annals of

Math. Study 124, Princeton Univ. Press, 1990. [Ka-GKM] Katz, N., “Gauss sums, Kloosterman sums, and monodromy groups”,

Annals of Math. Study 116, Princeton Univ. Press, 1988. [Ka-IMH] Katz, N., “On the intersection matrix of a hypersurface”, Ann. Sci.

École Norm. Sup. (4) 2 (1969), 583-598. [Ka-NCMT] Katz, N., “Nilpotent connections and the monodromy theorem: Ap-

plications of a result of Turrittin”, Publ. Math. IHES 39 (1970), 175-232. [Ka-SE] Katz, N., “Sommes Exponentielles”, Astérisque 79, Soc. Math. Fr., 1980.

[Kob] Koblitz, N., “The number of points on certain families of hypersurfaces over finite fields”, Compositio Math. 48 (1983), no. 1, 3-23.

[Ma-ACFD] Manin, Yu. I., “Algebraic curves over fields with differentiation”,

(Russian) Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958), 737-756. [Lau-TFCEF] Laumon, G., “Transformation de Fourier, constantes d’équations

fonctionnelles et conjecture de Weil”, Publ. Math. IHES 65 (1987), 131-210.

[Mor] Morrison, D. R., “Picard-Fuchs equations and mirror maps for hypersurfaces”, Essays on mirror manifolds, 241-264, Int. Press, Hong Kong, 1992. Also

available at http://arxiv.org/pdf/hep-th/9111025.

[Mus-CDPMQ] Mustata, A., “Degree 1 Curves in the Dwork Pencil and the Mirror

Quintic” preprint, math.AG/0311252 [Ogus-GTCC] Ogus, A., “Griffiths transversality in crystalline cohomology”, Ann.

of Math. (2) 108 (1978), no. 2, 395-419. [RL-Wan] Rojas-Leon, A., and Wan, D., “Moment zeta functions for toric calabi-

yau hypersurfaces”, preprint, 2007. [Se-ALR] Serre, J.-P., “Abelian l-adic representations and elliptic curves”, W. A.

Benjamin, Inc., New York-Amsterdam 1968 xvi+177 pp. [SGA 4 1/2] Cohomologie Etale. Séminaire de Géométrie Algébrique du Bois Marie

SGA 4 1/2. par P. Deligne, avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie, et J. L. Verdier. Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, 1977.

[SGA 1] Revêtements étales et groupe fondamental. Séminaire de Géométrie Algébrique du Bois Marie 1960-1961 (SGA 1). Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, 1971.

[SGA 4 Tome 3] Théorie des Topos et Cohomologie Etale des Schémas, Tome 3. Séminaire de Géométrie Algébrique du Bois Marie 1963-1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Lecture Notes in Mathematics, Vol.305, Springer-Verlag, 1973.

[SGA 7 II] Groupes de monodromie en géométrie algébrique. II. Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 II). Dirigé par P. Deligne et N. Katz. Lecture Notes in Mathematics, Vol. 340. Springer-Verlag, 1973.

[St] Stevenson, E., “Integral representations of algebraic cohomology classes on hypersurfaces” Pacific J. Math. 71 (1977), no. 1, 197-212.

[St-th] Stevenson, E., “Integral representations of algebraic cohomology classes

on hypersurfaces”, Princeton thesis, 1975. [We-JS] Weil, A., “Jacobi sums as Grössencharaktere”, Trans. Amer. Math. Soc

73, (1952), 487-495.