SHIODA CYCLES IN FAMILIES OF SURFACES

AISE JOHAN DE JONG

Contents

Section 1. Introduction 1 Section 2. Setup and Notation 1 Section 3. Extra cycles 2 Section 4. The basic example 3 Section 5. A construction of collections of cycles 4 Section 6. A wild guess 5 Section 7. Change of surface 7 Section 8. Algebra 7 Section 9. Another example 8 Section 10. Related questions 9 References 10

Section 1. Introduction

We recall how the Tate conjecture forces jumps of Picard numbers (over finite fields) in arithmetic families of surfaces. The goal of this note is to describe a mechanism that could explain this behaviour, see 6.1 below. This mechanism is shown to be valid for the universal Weierstrass curve squared (Section 4) and for “general” families of abelian surfaces (Section 9).

This document describes work in progress.

Section 2. Setup and Notation

Let f : X → S be a morphism of schemes. We are going to assume the following conditions.

(1) The scheme S is an irreducible normal scheme of finite type over Spec(Z) of dimension ≥ 1.

(2) The morphism X → S is proper and flat.

(3) All fibres of X → S are projective, smooth, geometrically irreducible sur-

faces. Let K be the algebraic closure of Z in the function field of S. Then either K is a finite field (the positive characteristic case), or K is a number field (the characteristic zero case).

This work was supported in part by the National Science Foundation under Award 0600425.

Any Opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect those of the National Science Foundation.

Let β_{2} be the second Betti number of the geometric fibres of the family f. For every prime number ℓ we obtain a continuous representation

τ_{ℓ} : π_{1}(S ⊗ Z[1/ℓ]) ⟶ GL_{β_{2}}(Q_{ℓ})

by looking at the monodromy of the ℓ-adic lisse sheaf R^{2}f_{∗}Q_{ℓ}(1) over S ⊗ Z[1/ℓ]. Let us agree to only consider τ_{ℓ} when S ⊗Z[1/ℓ] is not empty. These representations form a compatible system of Galois representations, if we think of π_{1}(S) as a Galois group with a dense set of Frobenii coming from closed points of S. Note that our choice of twist garantees that this is a representation pure of weight zero.

We let ρ_{geom,gen} be the Picard number of the geometric generic fibre of X → S. Standard conjectures imply that ρ_{geom,gen} should equal the dimension of the largest subspace of Q_{ℓ}^{β_{2}} on which τ_{ℓ} acts through a finite quotient.

Assumption 2.1. We assume that ρ_{geom,gen} agrees with the prediction mentioned above, and we assume that all divisor classes on the geometric generic fibre Xη̄ are already defined on the generic fibre X_{η}.

Remarks 2.2. (1) Note that this implies that the Picard number of every fibre is at least ρ_{geom,gen}.

(2) The assumption implies that

R^{2}f_{∗}Q_{ℓ}(1) = Q_{ℓ}^{⊕ρ_{geom,gen}} ⊕ F_{ℓ}

is a direct sum of a constant ℓ-adic sheaf of rank ρ_{geom,gen} and another summand F_{ℓ} which has no nontrivial subsheaf on which the monodromy is finite. Namely, by the Hodge index theorem the subspace spanned by cycles in H^{2}(Xη̄, Q_{ℓ}(1)) is nondegenerate, by the assumption it is Galois stable. Hence the other summand corresponds to its orthogonal complement.

(3) Standard conjectures predict the monodromy on F_{ℓ} is semi-simple.

(4) The second part of the assumption can always be satisfied by replacing S by a finite etale cover, allthough taking a finite covering changes the content of what we do below drastically in some cases.

(5) There are many examples of families where we can prove the assumption holds. For example the universal family of surfaces of degree d in P^{3} when d ≥ 4, or the universal family of elliptic surfaces of height d when d ≥ 3.

The last ingredient in our discussion below is the sign of the representation τ_{ℓ}, namely det(τ_{ℓ}) is independent of ℓ and defines a character

ϵ : π_{1}(S) ⟶ {±1}

of order 2. If s ∈ S is a closed point we set ϵ(s) equal to ϵ(F_{s}), where F_{s} is a Frobenius element corresponding to s.

Section 3. Extra cycles

This section is a short summary of a part of [KdJ00]. Let f : X → S be as above. We are going to look at the behaviour of the ranks of the Picard groups of X_{s} as s various over the closed points of S.

Definition 3.1. We say that X_{s} has an extra cycle if the Picard number ρ_{s} = ρ(X_{s}) is bigger than ρ_{geom,gen}.

Now it turns out that, assuming the Tate conjecture, we can sometimes predict the existence of an extra cycle in terms of the parity of the rank of F and the sign. Namely, assuming the Tate conjecture and assuming 2.1, there is an extra cycle on X_{s} in each of the following cases:

(1) The rank of F_{ℓ} is even and ϵ(s) = −1.

(2) The rank of F_{ℓ} is odd and ϵ(s) = 1. For an explanation, see [KdJ00]. It is clear that if this happens for one s then it will happen for a set of closed points s of density 1/2 or 1.

Definition 3.2. We say that parity forces an extra cycle on X_{s} for a closed point s ∈ S if one of the two cases enumerated above occurs.

The point of view in this note is that the extra cycles forced for parity reasons should fit a pattern. We guess a shape for this pattern, but it is perhaps already interseting to point out that a pattern might exist.

Remark 3.3. Let S′ → S be the 2 : 1 covering corresponding to ϵ, and let X′ be the pullback of the family X to S′. In the case that F_{ℓ} has odd rank we see that we get a cycle forced for parity reasons in all closed points of S′. In the case that the rank is even there are no points of S′ where this is so.

Section 4. The basic example

The basic example of the construction in the next section is that the graph of Frobenius of an elliptic curve over a finite field of q elements should be thought of as a “family of cycles varying with q”. More precisely, let

W = Spec(Z[a_{1}, a_{2}, a_{3}, a_{4}, a_{6}][1/Δ])

be the Weierstrass scheme. Here Δ = Δ(a_{1}, a_{2}, a_{3}, a_{4}, a_{6}) is the disciminant of the Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

Let E → W be the universal family of elliptic curves over W (defined as the projective closure of the scheme defined by the Weierstrass equation displayed above). Finally, let X → W be the family of surfaces with

X = E ×_{W} E ⟶ W.

This family of surfaces satisfies all the assumptions mentioned in the first section. Furthermore, ρ_{geom,gen} = 3 and the rank of the sheaf F_{ℓ} is three as well. Furthermore the determinant of sheaf F_{ℓ} is trivial in this case. Thus we see that on each X_{w} parity forces an extra cycle.

Of course, in this case an extra cycle is known: the graph of the geometric Frobenius morphism of E_{w} is typically linearly independent of the 3 cycle classes coming from the generic fibre. Namely, this fails only if q_{w} is a square, E_{w} is supersingular, and π_{w} = ±[q_{w}^{1/2}]. It is easy to see, for prime powers q, that the number of F_{q}-points w of W where this happens make up a vanishingly small fraction as q → ∞.

Let us write the equations of these cycles in the following way. Let w ∈ W be a closed point with residue field κ(w) having q_{w} elements. Let us denote the images of the elements a_{i} in κ(w) by α_{i}. Consider the affine open part

Spec(κ(w)[x_{1}, y_{1}, x_{2}, y_{2}]/I)

of X_{w}, where I is generated by the Weierstrass equations

\begin{aligned}I = & (y_{1}^{2} + α_{1}x_{1}y_{1} + α_{3}y_{1} − x_{1}^{3} − α_{2}x_{1}^{2} − α_{4}x_{1} − α_{6}, \\ & y_{2}^{2} + α_{1}x_{2}y_{2} + α_{3}y_{2} − x_{2}^{3} − α_{2}x_{2}^{2} − α_{4}x_{2} − α_{6}). \end{aligned}

The graph of the geometric Frobenius is the cycle Z_{w} ⊂ X_{w} cut out by the equations

x_{1} − x_{2}^{q_{w}}, and, y_{1} − y_{2}^{q_{w}}

on the affine part above. So in this basic example the additional cycles can be cut out by a bunch of equations that are somehow universal polynomials in the coordinates and the coordinates raised to the q_{w}th power.

We can formalize this “uniformity” as follows. Consider the scheme X ×_{W} X = E_{W}^{4}. It has an affine open with coordinates x_{1}, x_{2}, x_{3}, x_{4}, y_{1}, y_{2}, y_{3}, y_{4} and a_{1}, a_{2}, a_{3}, a_{4}, a_{6}. Consider the closed subset of this defined by the equations x_{1} − x_{4} and y_{1} − y_{4}, and let Z ⊂ X ×_{W} X be its closure. Let π_{w} : X_{w} → X_{w} be the geometric Frobenius of X_{w}. Then the pullback of the fibre Z_{w} via the morphism (id, π_{w}) : X_{w} → X_{w} ×_{w} X_{w} gives us the cycle Z_{w}.

Remark 4.1. A more general cycle we could allow is one where there is not only an exponent q but also an exponent p in the equations.

Section 5. A construction of collections of cycles

Let f : X → S be a morphism of schemes of finite type over Z. Suppose we have an integer r ≥ 0. Our notation will be

X^{r+1} = X ×_{S} X ×_{S} … ×_{S} X

Here there are r + 1 factors in the fibre product. We will associate to a closed subscheme

Z ⊂ X^{r+1}

a “collection” of cycles {Z_{s}}_{s∈|S|}. Here |S| denotes the set of closed points of S.

Example 5.1. Suppose that S = Spec(R), with R = Z[a_{1}, …, a_{n}] and that X = Spec(A), with A = Z[a_{1}, …, a_{n}, x_{1}, …, x_{m}]. In this case X^{r+1} is Spec(A^{r+1}) where the ring

A^{r+1} = A ⊗_{R} A ⊗ … ⊗_{R} A

is simply a polynomial algebra over Z. Let us label the variables a_{i}, 1 ≤ i ≤ n, and x_{i,j}, 1 ≤ i ≤ m, 0 ≤ j ≤ r. Here x_{i,j} is the pullback of x_{i} to X^{r+1} via the (j + 1)st projection morphism. To understand what follows it is a good idea to think x_{i,j} = (x_{i})^{q^{j}}. We will say that an element G ∈ A^{r+1} is a q-polynomial in x_{1}, …, x_{m} over Z[a_{1}, …, a_{n}].

Remark 5.2. More generally, we could look at a ring with variables (a_{i})^{p^{j}}, (x_{i})^{p^{j}q^{k}} whose elements could be called (q, p)-polynomials, see Remark 4.1.

Any closed point s ∈ |S| has a finite residue field κ(s) having q_{s} elements and characteristic p_{s}. The scheme X_{s} is of finite type over s = Spec(κ(s)) and has an absolute Frobenius morphism

F : X_{s} ⟶ X_{s},

as well as a geometric Frobenius morphism

π_{s} = F^{log_{ps} (q_{s})} : X_{s} ⟶ X_{s}.

The geometric Frobenius is a morphism over s, whereas the absolute Frobenius morphism is Frobenius semi-linear.

Now suppose we have a closed subscheme

Z ⊂ X^{r+1}.

For our closed point s the fibre Z_{s} is a closed subscheme of X_{s}^{r+1} (product of schemes over s). We define a closed subscheme Z_{s} ⊂ X_{s} as follows:

Z_{s} = {x ∈ X_{s} | (x, π_{s}(x), π_{s}^{2}(x), …, π_{s}^{r}(x)) ∈ Z_{s}}

In other words it is the scheme theoretic inverse image of Z_{s} under the morphism

Φ = (id, π_{s}, …, π_{s}^{r}) : X_{s} ⟶ X_{s}^{r+1}

Remark 5.3. Of course it is possible that Z_{s} = ∅ or Z_{s} = X_{s}. If Z is an ample divisor for example, then Z_{s} = ∅ won’t happen. In Lemma 8.1 we show that usually Z_{s} ≠ X_{s} for “most s in a Zariski open of S”.

Definition 5.4. Any collection of closed subschemes Z_{s}, s ∈ |S| obtained from Z ⊂ X^{r+1} in the manner constructed above is called a q-algebraic family of cycles.

Example 5.5. We continue with the notation of Example 5.1. Suppose that Z ⊂ X^{r+1} is a divisor, and hence the zero scheme of a q-polynomial G in x_{i} with coefficients which are polynomials in a_{i}. Let s ∈ S be a closed point. It correspond to a surjective ring map Z[a_{1}, …, a_{n}] → κ(s). Let α_{i} be the image of a_{i} in κ(s). Let q = q_{s} be the order of κ(s) as before. Then the scheme Z_{s} is cut out by the equation

G|_{a_{i}=α_{i},x_{i,j}=x_{i}^{q^{j}}} = G(…, α_{i}, …, x_{i}^{q^{j}}, …).

in the ring κ(s)[x_{1}, …, x_{n}]. In the general case Z_{s} is cut out by a finite list of such polynomials.

Example 5.6. So for example with analogous notation if you have Z[a, x] and G = a + ax_{2} + x^{4} then the schemes Z_{s} are cut out by the equation

α + αx^{q^{2}} + x^{4} = 0.

Remark 5.7. There is no elimination theory for q-polynomials. For example, consider the system of q-polynomial equations in x, y, z over Z

\begin{cases}x^{2} = y \\ x^{q+1} = z \end{cases}

and try to eliminate x. As long as q is not a power of 2 the solution is clearly z = y^{(q+1)/2}. However, this family of cycles in A_{Z}^{2} over Z cannot be cut out by a system of q-polynomials in y, z (exercise).

Section 6. A wild guess

Suppose that X → S is a morphism as in Section 2 satisfying assumption 2.1. Choose an integer r and a closed subscheme Z ⊂ X^{r+1}. Consider the associated q-algebraic family of cycles Z_{s} constructed in Section 5. Let q be a power of a prime p. We define

(1) N_{q} = the number closed points s ∈ S with residue field of order q, (2) P_{q} = the number of closed points s ∈ S with residue field of order q, such

that parity forces an extra cycle on X_{s}, and

(3) E_{q} = the number of closed points s ∈ S with residue field of order q, such that parity forces an extra cycle on X_{s} and there is an irreducible component^{1} of Z_{s} whose divisor class is not in the group of divisor classes coming from the generic fibre.

In other words, E_{q}/P_{q} is the fraction of those F_{q}-fibres of the family, where there should be an extra cycle (by the Tate conjecture) and where an irreducible component of Z_{s} gives one.

For simplicity we define the fraction E_{q}/P_{q} to be one if N_{q} = 0 or if P_{q} = 0.

Wild Guess 6.1. For any family X → S as above, after possibly replacing S by a nonempty Zariski open, the following holds: there exists an integer r ≥ 0 and a closed subscheme Z ⊂ X^{r+1} such that the fraction E_{q}/P_{q} tends to 1 as q → ∞.

Note that if S lives in characteristic p there is no need for the phrase “after possibly replacing S by a nonempty Zariski open”. This is because the number of F_{q} points lying in the thrown out Zariski closed is vanishingly small compared to P_{q} (see the remark directly above Definition 3.2).

Note also that even in the basic example (Section 4) we do not know how to construct a q-algebraic family of cycles such that E_{q} = P_{q} for all sufficiently large q. Of course, in stead of the limit in 6.1 we could postulate/guess/speculate on a suitable behaviour of the “bad points” s ∈ |S|.

Remark 6.2. We note that 6.1 changes meaning under pull back to a ramified cover S′ → S. For example, if S′ → S is the double covering from Remark 3.3, then in the odd even rank case 6.1 becomes vacuous for the family X′/S′. (Whereas in the odd rank case it kind of becomes a “stronger” property.)

Example 6.3. The universal surface of even degree in P^{3}. Here is a completely “explicit” case of 6.1 above. Fix an even integer d ≥ 4. Our base ring will be R = Z[a_{I}, |I| = d][1/Δ] where I = (i_{0}, i_{1}, i_{2}, i_{3}). Here Δ is the universal polynomial in a_{I} expressing that the surface defined by the equation

∑_{I} a_{I}X_{0}^{i_{0}}X_{1}^{i_{1}}X_{2}^{i_{2}}X_{3}^{i_{3}} = 0

is singular. We set S = Spec(R) and we let X ⊂ P_{S}^{3} be the universal nonsingular degree d hypersurface. In this case the sheaf F_{ℓ} has odd rank and its determinant ϵ is nontrivial. Then 6.1 predicts for example the existence of a q-polynomial G = G(a_{I}, X_{i,0}, X_{i,1}, …) in X_{0}, X_{1}, X_{2}, X_{3} over R with the following property: For almost all points s ∈ S(F_{p}) with ϵ(s) = 1 there exists some irreducible component of

X_{s} ∩ (G(a_{I}(s), X_{i}, X_{i}^{p}, …) = 0)

whose divisor class is Q-independent of c_{1}(O(1)) on X_{s}.

1This is probably necessary since there is no elimination theory for q-polynomals.

Section 7. Change of surface

The only good property of our cycles Z_{s} we can esthablish right now is that it behave reasonably well under morphisms between families of surfaces. First we discuss pull back.

Lemma 7.1. Suppose we are given two families f : X → S and f′ : X′ → S as in Section 2. Let γ : X′ → X be a morphism over S. Suppose we are given Z ⊂ X^{r+1} giving rise to cycles Z_{s} as described in Section 5. The cycles γ^{∗}Z_{s} are equal to the cycles Z′_{s} constructed from the closed subscheme Z′ = (γ^{r+1})^{∗}Z.

Proof. This is just unwinding the definitions. □

This does not imply that we can deduce 6.1 for the family X′ from that for the family X. The reason is that the ranks of F_{ℓ} and F′_{ℓ} may differ as well as the determinants. A more precisely formulated version of our wild guess would predict the existence of one q-algebraic family of cycles for each orthogonally selfdual simple summand of F_{ℓ} and the lemma above would imply the prediction per simple factor is stable under pullback by a morphism between families of surfaces.

Next we come to pushforward, which is a bit more interesting.

Proposition 7.2. Let X , X′, S, f, f′, r, Z be as in the lemma above, but now suppose γ : X → X′. Assume that Z ⊂ X^{r+1} is a proper closed subscheme. Then there exists a proper closed subscheme Z′ ⊂ (X′)^{r+1} such that for all closed points s ∈ S we have γ_{s}(Z_{s}) ⊂ Z′_{s}.

Note that the way this proposition is stated it allows us to replace S by a Zariski open. If the morphism γ is not dominant then the proposition is trivial. We may also replace Z by an (ample) divisor containing it in X. By unwinding definitions in this way, the proposition boils down to the algebra problem stated in Lemma 8.2 below.

The relevance of proposition is that 6.1 for X /S implies X′/S, at least if F_{ℓ} = F′_{ℓ}. As above, a more precise version of 6.1 would allow for such a conclusion per simple factor that F_{ℓ} and F′_{ℓ} have in commmon.

Section 8. Algebra

Let R be a Z-algebra of finite type. Let A be an R-algebras of finite type. Let G ∈ A^{r+1} = A ⊗_{R} … ⊗_{R} A (r + 1 factors), compare Example 5.1. Let us call this a q-polynomial in A over R.

Consider a closed point s of the spectrum of R, in other words a surjection R ⟶ κ(s) where κ(s) is a finite field with q_{s} elements. We denote G_{s} ∈ A ⊗_{R} κ(s) the image of G in A ⊗_{R} κ(s) under the map

A^{r+1} ⟶ A ⊗_{R} κ(s), a_{0} ⊗ … ⊗ a_{r} ↦ a_{0}a_{1}^{q_{s}} … a_{r}^{q_{s}^{r}} ⊗ 1.

Lemma 8.1. Assume that G is nonzero, R is a domain, and R → A is flat with geometrically integral fibres. After possibly replacing R → A by R_{f} → A_{f} for some nonzero f ∈ R, the element G_{s} is nonzero for all points s.

Proof. The assumptions imply that A and A^{r+1} are domains. Namely, let K = f.f.(R), then A ⊂ A_{K} by flatness and A_{K} is a domain by assumption. Similarly, A^{r+1} ⊂ A_{K}^{r+1} = A_{K} ⊗_{K} … ⊗_{K} A_{K}. To see that A_{K}^{r+1} is a domain use that AK̄ is a domain, and that the product of varieties is a variety. We claim may localize on Spec(A). Namely, let 0 ≠ h ∈ A and let H ∈ (A_{h})^{r+1} denote the image of G. Then H ≠ 0 because (A_{h})^{r+1} is a localization of the domain A^{r+1}. Also, H_{s} is the image of G_{s} in the algebra A_{h} ⊗_{R} κ(s). Next, observe that A ⊗_{R} κ(s) → A_{h} ⊗_{R} κ(s) is injecitve if and only if A_{h} ⊗_{R} κ(s) ≠ 0. Thus after localizing R injectivity will hold for all s. We prove the lemma by induction on the number of generators of A over R. Write A = A′[x] for some subring A′ ⊂ A that can be generated by fewer elements. There are two cases to consider. Case I: The element x is algebraically independent over A′. In this case A = A′[x] is a polynomial algebra over A′. It follows immediately that A′ is flat over R with geometrically irreducible fibres, so the induction hypothesis applies to A′. We write symbolically A^{r+1} = (A′)^{r+1}[x, x^{q}, …, x^{q^{r}}]. Now the result for A follows directly from the result for A′ by looking at the largest power of x in the expansion of G_{s}. Case II: The element x is algebraically dependent on A′. In this case, we may localize on A′ (see above) and assume that A is finite free over A′ (in particular A′ → A is split as an R-module map). Then A′ is flat over R and for any R-algebra k̄ we have A′ ⊗_{R} k̄ ⊂ A ⊗_{r} k̄. So A′ has geometrically integral fibres over R too, and the induction hypothesis applies to A′. Now (A′)^{r+1} ⊂ A^{r+1} is a finite free extension of domains and thus there exists a nonzero element G̃ ∈ A^{r+1} such that G′ := G · G̃ ∈ (A′)^{r+1}. Clearly G′ is not zero and G′_{s} = G_{s} · G̃_{s}, and we deduce the result for G from that for G′ which we know by induction hypothesis. □

Lemma 8.2. Assume that R is a domain, and R → A and R → A′ are flat with geometrically integral fibres. Assume further that A′ → A is a finite flat map of R-algebras. Let G ∈ A^{r+1} be nonzero. After possibly replacing R by R_{f} for some nonzero f ∈ R there exists a G′ ∈ (A′)^{r+1} such that G_{s} divides G′_{s} for all closed points s.

Proof. The argument is the same as in the proof of Lemma 8.1. □

Section 9. Another example

Let S be a scheme of finite type over Z and let

f : X → S

be a family of abelian surfaces. Let λ : X → X^{t} be a polarization, and let P be the Poincaré sheaf on X ×_{S} X^{t}. Let ℓ be invertible on S. We will assume that the monodromy on R^{1}f_{∗}Q_{ℓ} is large in this family: For example let’s assume that the Zariski closure of the image of π_{1}(S) in GL_{4}(Q_{ℓ}) equals the group of simplectic similitudes CSP_{4} (largest possible). This garantees that

R^{2}f_{∗}Q_{ℓ}(1) = Q_{ℓ} ⊕ F_{ℓ}

with F_{ℓ} absloutely irreducible. For example, X /S could be some “universal family” of abelian surfaces with fixed polarization degree and fixed level structure.

In this case the rank of F_{ℓ} is 6 − 1 = 5. Thus we are looking for closed points s ∈ S with ϵ(s) = 1. But since R^{2}f_{∗}Q_{ℓ}(1) = ( ∧^{2} R^{1}f_{∗}Q_{ℓ})(1) we see that ϵ(s) = 1 for all closed points s. As in Section 4 we can construct a suitable q-algebraic family of cycles directly.

Namely, consider the invertible sheaf N = (id × λ)^{∗}L on X ×_{S} X. The fact that λ is a polarization means that A := Δ_{X /S}^{∗}N is a relatively ample sheaf on X /S. Let Z ⊂ X ×_{S} X be an effective Cartier divisor such that N is trivial over the complement. We claim that the q-algebraic family of cycles {Z_{s}}_{s∈|S|} associated to Z is a solution to 6.1 in this case.

The reason this works is that the pullback of the invertible sheaf N_{s} on X_{s} × X_{s} via the morphism id × π_{s} has a chern class that is Q-linearly independent of the chern class of the ample generator A_{s}, as long as [1] and π_{s} are Q linearly independent as endomorphisms of X_{s}. To see this one uses the exact sequence

0 → Pic^{0}(X_{s}) → Pic(X_{s}) → Hom(X_{s}, X_{s}^{t})

which embeds NS(X_{s}) into the group of homomorphisms of X_{s} into X_{s}^{t}, see [Mum70]. Again there is going to be a “small” set of exceptions, namely those s ∈ |S| corresponding to abelian surfaces whose geometric frobenius is equal to ±[q_{s}^{1/2}].

Section 10. Related questions

Set A = Z[x] and let A ⊂ B be a finite ring extension, with B a normal domain. Let d be the rank of B as an A-module. Then for all large enough primes p the induced map A_{p} = F_{p}[x] → B_{p} = B/pB, is finite locally free of rank d. For f ∈ A^{2} = Z[x, u] we define f_{p}(x) = f(x, x^{p}) mod p ∈ F_{p}[x]. For g ∈ B^{2} = B ⊗_{Z} B we define g_{p} to be the image of g mod p under the map B_{p} ⊗ B_{p} → B_{p}, b_{1} ⊗ b_{2} ↦ b_{1}b_{2}^{p}. In Section 8 we showed for any (B, g), if g is not zero and B ⊗_{Z} \overline{Q} is a domain, then there exists a nonzero f such that for all p >> 0 we have

Norm_{B_{p}/A_{p}}(g_{p}) divides f_{p}.

We can take f ∈ Z[x, u] to be

f = NNorm_{B_{Q}^{2}/A_{Q}^{2}}(g)

for a suitably divisible integer N (where we use the map A^{2} = Z[x, u] → B^{2}, x ↦ x ⊗ 1, u ↦ 1 ⊗ x). It is also an easy excercise to show this directly in this case; the assumption “B ⊗_{Z} \overline{Q} is a domain” is used only to garantee that this norm is nonzero.

Questions. They may be really hard or they may be easy. I don’t know.

(1) Is there any nonlinear f ∈ Z[x, u] such that f_{p} is irreducible for an inifinite number of p?

(2) Let f ∈ Z[x, u] nonzero, and suppose deg_{x}(f_{p}) = dp + d′ for all large p. (In other words deg_{u}(f) = d and the coefficient c of u^{d} in f has deg_{x}(c) = d′.) Suppose we can write d = d_{1} + d_{2} and d′ = d′_{1} + d′_{2} with d_{1}, d_{2} > 0 and d′_{1}, d′_{2} ≥ 0 such that for all sufficiently large p there exists a factorization

f_{p} = f_{1,p}f_{2,p}

with deg_{x}(f_{1,p}) = d_{1}p + d′_{1}. Is it true that f comes about via a norm construction as above? (More precisely, is there an extension A ⊂ B and a g ∈ B^{2} such that Norm_{B_{p}/A_{p}}(g_{p}) nontrivially divides f_{p} for all p >> 0?)

(3) Suppose that f ∈ Z[x_{1}, x_{2}, u_{1}, u_{2}] irreducible. Assume that f_{p}(x_{1}, x_{2}) := f(x_{1}, x_{2}, x_{1}^{p}, x_{2}^{p}) mod p factors for all p >> 0. Is it true that the degrees of irreducible factors of f_{p} are of the form d_{1}p + d′_{1} for suitable d_{1}, d′_{1} ∈ Z?

(More precisely: Can one find a finite list of pairs (d_{1}, d′_{1}) such that the degrees of irreducible factors are among the integers d_{1}p + d′_{1}?)

The motivation behind these questions is that a nontrivial example of 6.1 gives rise to a nontrivial factorizations of specializations of q-polynomials.

References

[KdJ00] N.M. Katz and A.J. de Jong. Monodromy and the tate conjecture: Picard numbers and mordell-weil ranks in families. Israel journal of mathematics, 120:47–79, 2000.

[Mum70] David Mumford. Abelian varieties, volume 5 of Tata Institute of Fundamental Research Studies in Mathematics. Oxford University Press, 1970.