CHAPTER 29

The Situation over Z: Questions

There is another sense in which we might ask about “situations over Z,” namely we might try to mimic the setting of a theorem of Pink [Ka-ESDE, 8.18.2] about how “usual” (geometric) monodromy groups vary in a family. There the situation is that we are given a normal noetherian connected scheme S, a smooth X/S with geometrically connected fibres, and a lisse Q_ℓ-sheaf F on X of rank n ≥ 1. For each geometric point s in S, we have the restriction F_s of F to the fibre X_s. Pick a geometric point x_s in X_s. Then for each s in S we have the closed subgroup Γ(s) ⊂ GL(n, Q_ℓ) which is the image of π₁(X_s, x_s) in the representation corresponding to F_s. The assertion is that these groups Γ(s) are, up to GL(n)-conjugacy, constant on a dense open set of S, and that they decrease under specialization.

This result leads naturally to the following question. Suppose we are given a normal noetherian connected scheme S which is of finite type over Z[1/ℓ], an object N in the derived category D^b_c((G_m)_S, Q_ℓ), and an integer n ≥ 1. We make the following assumptions on this data.

(1) For every geometric point s of S, the restriction, N_s, of N to the fibre over s is perverse and lies in P.

(2) The formation of the Verdier dual, D_(Gm)S/S(N), commutes with arbitrary change of base on S to a good scheme S′ (a condition which is always satisfied after we shrink S to a dense open set of itself [Ka-Lau, 1.1.7]).

(3) For every finite field k and for every k-valued point s ∈ S(k), the restriction N_k,s of N to the G_m/k which is the fibre of (G_m)_S over s is perverse, lies in P_arith, has “dimension” n, and is geometrically semisimple. [This last condition holds if, for example, each N_k,s is pure of some integer weight w.]

The first two conditions say that N is “perverse relative to S”inthe sense of [Ka-Lau, 1.2.1], and in addition satisfies P on each geometric fibre. For η the generic point of S, and η a geometric point lying over it, we have the perverse object N_η on G_m over an algebraically closed field κ(η), which lies in P. The results of Gabber-Loeser developed in their seminal paper, especially [Ga-Loe, 3.7.2, 3.7.5], are stated when the ground field has strictly positive characteristic, but remain valid in the case of characteristic zero, cf. [Ga-Loe, lines 22-25 on page 505]. So whatever the characteristic of κ(η), we may speak of the Tannakian group G_geom,Nη. Using the construcibility of the Euler characteristic on fibres, the third condition implies that N_η has “dimension” n. Hence we have an inclusion G_geom,Nη ⊂ GL(n), well-defined up to GL(n)conjugacy.

On the other hand, for every finite field k and for every k-valued point s ∈ S(k), we have the Tannakian group G_geom,Nk,s ⊂ GL(n). The first natural question is whether this group is always conjugate, in GL(n), to a subgroup of G_geom,Nη. We cannot expect equality of these groups, even if we are willing to shrink S to an open dense subset of itself, as the following example shows. [See [Ka-ESDE, 2.4.1, 2.4.4] for a discussion of the analogous phenomenon for differential galois groups.]

Take for S the spectrum of Z[1/ℓ]. Then ℓ is a global section of (G_m)_S, so we can speak of the delta sheaf δ_ℓ. The finite fields k for which S(k) is nonempty are precisely those of characteristic p ≠ ℓ, and for each such k there is a unique point s in S(k). For each such point s, the Tannakian group G_{geom,(δℓ)k, s} is the finite group μ_{N(ℓ, p)} ⊂ GL(1) of roots of unity of order N(ℓ, p) := the multiplicative order of ℓ mod p. As this order, for fixed ℓ and variable p, is unbounded (otherwise ℓ would itself be a root of unity in Q), these Tannakian groups do not become constant, no matter how large the finite set of primes p we invert. Nor do they ever become the entire group GL(1), which is the value of G_geom,Nη (exactly because ℓ has infinite multiplicative order in Z[1/ℓ]^×).

So two plausible questions in this context are the following.

(Q1) Is it true that for every finite field k and for every k-valued point s ∈ S(k), the Tannakian group G_geom,Nk,s is conjugate in GL(n) to a subgroup of G_geom,Nη ?

(Q2) After possibly shrinking S to an open dense subset U ⊂ S, is it true that for every finite field k and for every k-valued point u ∈ U(k), the derived group (commutator subgroup) of the identity component of G_geom,Nk,u, ((G_geom,Nk,u)⁰)^der, is conjugate in GL(n) to the derived group of the identity component of G_geom,Nη, ((G_geom,Nη)⁰)^der?

Another natural question, involving only finite field fibres, is this.

(Q3) Suppose given an object N as above, and a reductive group G ⊂ GL(n). Suppose there is a dense open set U ⊂ S such that for every finite field k and every k-valued point u ∈ U(k), G_geom, N_k,u is conjugate in GL(n) to the given group G. Is it then true that for every finite field k and for every k-valued point s ∈ (S U)(k), G_geom, N_k,s is conjugate in GL(n) to a subgroup of the given group G?

Here is an example, based on Theorem 18.7, where this last question has an affirmative answer.

Example 29.1. Fix an even integer 2g ≥ 4, and consider the one parameter (“a”) family of palindromic polynomials

f_a(x) := x^2g + ax^2g–1 + ax + 1.

Denote by ∆(a) ∈ Z[a] the discriminant of f_a. Its constant term is invertible in Z[1/2g] (because f₀(x) = x^2g + 1 has 2g distinct roots in any field in which 2g in invertible). Take for S the spectrum of the ring Z[1/2gℓ][a][1/∆(a)]. On (G_m)_S, we have the polynomial function f_a(x), and the dense open set j : (G_m)_S[1/f_a(x)] ⊂ (G_m)_S. We take for N the object

N := j_!L_χ2(fa(x))[1].

Then on each geometric fibre of (G_m)_S/S, the object N(1/2) is perverse, has P, is pure of weight zero and is symplectically self-dual of “dimension” 2g (“dimension” 2g because we inverted ∆). On any fibre over the open set S[1/a] where a is invertible, f_a has an x-term, so is not a polynomial in x^d for any d ≥ 2. So by Theorem 18.7, over U := S[1/a], we are in the situation of this last question, with G = Sp(2g).

For points in S U = Spec(Z[1/2gℓ]), i.e., points where a = 0, we are looking at

N₀ := j_!L_{χ2(x^2g + 1)}[1].

Over any field in which 2gℓ is invertible, N₀(1/2) remains symplectic, but it is no longer Lie-irreducible, cf. Corollary 8.3. So its G_geom is a subgroup of Sp(2g), but it is no longer the entire symplectic group.

We can be more precise about what its G_geom is. The key observation is that L_χ2(x^2g + 1) is the Kummer pullback [2g]^? (L_{χ2(x+ 1)}) of Lχ₂(x+ 1). Its direct image by [2g] is the direct sum

[2g]_?(L_{χ2(x^2g + 1)}) = [2g]_?[2g]^?(L_{χ2(x+ 1)}) = ⊕_iL_{χ2(x+ 1)} ⊗L_ρi

over the 2g multiplicative characters ρ_i of order dividing 2g. The dual of j_!L_{χ2(x+ 1)}[1](1/2) is j_!L_{χ2((1/x)+ 1)}[1](1/2) = j_!L_{χ2(x+ 1)} ⊗L_χ2[1](1/2). So for each ρ_i, the dual of j_!L_{χ2(x+ 1)}⊗L_ρi[1](1/2) is j_!L_{χ2(x+ 1)}⊗L_ρiχ2[1](1/2). Now apply Corollary 20.3, with a := –1, Λ := χ₂, and a choice of g among the ρ_i which picks one out of each pair (ρ_i, χ₂ρ_i). Then the partial direct sum

⊕_chosen ρ_i j_!L_{χ2(x+ 1)} ⊗L_ρi[1](1/2)

has its G_geom a g-dimensional torus, with the unchosen ρ_i terms being the inverse characters. In other words, the G_geom of the entire direct sum

[2g]_?N₀(1/2) = ⊕_all ρ_i j_!L_{χ2(x+ 1)} ⊗L_ρi[1](1/2)

is the maximal torus of Sp(2g). So by Theorems 8.1 and 8.2, we see that for N₀ := j_!L_χ2(x^2g + 1)[1] itself, the identity component of its G_geom is the maximal torus of Sp(2g).

Here is another example, exhibiting more extensive and interesting specialization behavior.

Example 29.2. Again fix an even integer 2g ≥ 4, and consider now the two parameter family of polynomials

f_a,b(x) = x^2g + ax^2g–1 + bx + 1.

We denote by ∆(a, b) ∈ Z[a, b] its discriminant. Just as in the previous example, its constant term is invertible in Z[1/2g]. We now take for S the spectrum of the ring Z[1/2gℓ][a, b][1/∆(a, b)], and for N the object

N := j_!L_{χ2(fa, b(x))}[1].

Using now Theorem 23.2, elementary computation shows that over the locus where a^2g – b^2g is invertible, we have G_geom = SL(2g).

The locus where ab is invertible and a^2g – b^2g is not invertible is, set theoretically, the disjoint union over divisors d of 2g, of the sets where ζ := a/b is a primitive d’th root of unity. As we will see below, the question of whether or not d divides g, i.e., whether ζ^g = 1 or ζ^g = –1, has a huge effect on what G_geom turns out to be.

Suppose first that ζ^g = 1, i.e., that d|g. We claim that on the locus a = ζb, b invertible, G_geom is the group GSp_d(2g) := μ_2dSp(2g) of symplectic similitudes with multiplicator of order dividing d. Indeed, if we choose a square root η of ζ of order 2d (both choices have order 2d if d is even, just one choice does if d is odd), then f_a,b(ηx) is palindromic, and we are dealing with its multiplicative translate by η. Then δ_1/η “is” a character of order 2d. By Theorem 18.7, the direct sum

j_!L_{χ2(fa,b(ηx))}[1] ⊕ δ_η

has its G_geom a subgroup of the product group Sp(2g)×μ_2d, which maps onto each factor. As Sp(2g) has no nontrivial quotients, Goursat’s lemma shows that this direct sum has its G_geom the product group, and hence our N has its G_geom equal to μ_2dSp(2g). [If d is odd and we chose η of order d, we would find that G_geom is μ_dSp(2g), which for d odd is also group μ_2dSp(2g).]

Suppose next that ζ^g = –1, i.e., that d does not divide g. We claim that on the locus a = ζb, b invertible, G_geom is SL(2g) ∩ (μ_2dO(2g)), the group of orthogonal similitudes with multiplicator of order d and determinant 1. Indeed, for either choice of η with η² = ζ (here d is even, and so both choices of η have order 2d), f_a,b(ηx) is antipalindromic, and we are again dealing with its multiplicative translate by η. Here, however, there is an additional subtlety. By Theorem 24.1, the direct sum

j_!L_{χ2(fa,b(ηx))}[1] ⊕ δ_η

has its G_geom a subgroup of the product group O(2g) × μ_2d, which maps onto each factor. By Goursat’s lemma, this G_geom is the (inverse image in the product of the) graph of an isomorphism of a quotient of O(2g) with a quotient of μ_2d. The only nontrivial abelian quotient of O(2g) is ±1, via the determinant, since SO(2g) is, for 2g ≥ 4, its own commutator subgroup. The only ±1 quotient of μ_2d is given by the d’th power map. Because d divides 2g but does not divide g, the ratio 2g/d is odd. So we may also describe the only ±1 quotient of μ_2d as given by the 2g’th power map. So for the direct sum above, its G_geom is either the full product μ_2d × O(2g) or it is the subgroup consisting of those elements (° ∈ μ_2d, A ∈ O(2g)) satisfying det(A) = °^2g. So for the object N, its G_geom is either μ_2dO(2g) or it is the intersection of that group with SL(2g). From Theorem 23.2, we know that G_geom, N lies in SL(2g), and this rules out the μ_2dO(2g) possibility.

On the locus a = b = 0, the identity component of G_geom is, as we have seen in Example 29.1, the maximal torus of Sp(2g).