Local Monodromy for Deformations of One-Dimensional Formal Groups

Ching-Li Chai^{1}

University of Pennsylvania

Version 3.1, November 12, 1999

§1. Introduction

Let k be an algebraically closed field of characteristic p > 0. Let R = k[[t]] be the formal power series ring over k, and let K = k((t)) be the fraction field of R. Let G_{R} be a smooth one-dimensional formal group over R, such that its fiber G_{k} has finite height h and its generic fiber G_{K} of G has height one. Let G[p^{∞}] be the Barsotti-Tate group of height h over R attached to G; see 2.5 for the definition of G[p^{∞}]. Let T_{p}(G) be the p-adic Tate module attached to the maximal étale quotient G[p^{∞}]_{K}^{ét} of G[p^{∞}]_{K}. In Katz’s terminology T_{p}(G) is the “physical” p-adic Tate module, and can be identified with the inverse limit \underleftarrow{lim}_{n} G[p^{n}](K^{sep}). So T_{p}(G) is a free ℤ_{p}-module of rank d = h − 1. Let

ρ_{G} : Gal(K^{sep}/K) → GL(T_{p}(G))

be the natural representation of the Galois group Gal(K^{sep}/K) on T_{p}(G).

When h = 2, a famous result of Igusa implies that the image of ρ_{G} is an open subgroup of ℤ_{p}^{×}. In this article we show that Katz’ proof of Igusa’s theorem in [Ka, Thm. 4.3], coupled with a limit argument, can be used to prove that the image of the Galois representation ρ_{G} is quite big.

The main result of this paper says that ρ_{G} is an irreducible linear representation of Gal(K^{sep}/K) over ℚ_{p}, and the image of its determinant is an open subgroup of ℤ_{p}^{×}. Moreover the orbit of every non-zero vector in T_{p}(G)⊗_{ℤ_{p}}ℚ_{p} is open in the p-adic topology. The upper breaks of the determinant of ρ_{G} are computed after passing to a finite extension; they are quite similar to the results in [G]. The precise statements can be found in theorems 3.3, 3.5, 4.1, 4.6, 5.6.

In [G] Gross studied the monodromy of one-dimensional formal groups over k[[t]] when the height of the generic fiber is one less than the height of the special fiber, and obtained a very complete answer in that case. But the case where the generic fiber has height one has attracted little attention in the past and does not seem to have been studied in the literature.

In [E] Ekedahl proved that the local monodromy of the maximal étale quotient of the universal deformation of a product of supersingular elliptic curves in the moduli space of g-dimensional principally polarized abelian varieties of characteristic p is equal to GL_{g}(ℤ_{p}). A question analogous to the problem treated in this paper, in the context of Ekedahl’s theorem, is the following:

Suppose that A_{R} is a principally polarized abelian scheme over R, such that its special fiber is isomorphic to a product of supersingular elliptic curves, and its generic fiber is an ordinary abelian variety. Show that the monodromy of the maximal étale quotient of A[p^{∞}]_{K} is “big”.

One knows that the image of the monodromy representation is not finite, using the main theorem of de Jong in [dJ]. Otherwise not much is known.

1partially supported by grants DMS95-02186 AND DMS 9800609 from the National Science Foundation

For one-dimensional formal groups, there are many open questions about the monodromy representation ρ_{G}. The most important one is the following “wishful thinking”, suggested by the philosophy of motives: The image of ρ_{G} “should be” an open subgroup of GL(T_{p}(G)). Unfortunately we have not been able to supply any solid evidence in its support, even in the first nontrivial case when h = 3.

This paper was motivated by the Hecke orbit problem for some U(d, 1) Shimura varieties, and also by the related problem about Tate-linear subvarieties in these Shimura varieties. The results in this paper can be used to show that for a good reduction X of a U(d, 1) Shimura variety of PEL-type attached to an imaginary quadratic field, and for any point x_{0} of X corresponding to an ordinary abelian variety, the Hecke orbit of x_{0} is Zariski dense in X.

It is a pleasure to thank J. de Jong for discussions on applications of his theorem in [dJ] to p-adic representations, and on local monodromy resulting from change of slopes. I would like to thank the referee for suggestions which improved the exposition of this paper, and for the second, and better, proof of Theorem 4.1. I would also like to thank T. Chinburg, R. Donagi, B. Gross, D. Harbater, A. Kirillov, S. Shatz, J.-K. Yu, A. Yukie, and W. Ziller for discussions on topics ranging from group theory, prehomogenous vector spaces and computer-aided computation of Galois groups.

§2. Notations and Conventions

(2.1) Let k be an algebraically closed field of characteristic p > 0. Let R = k[[t]] be the formal power series ring over k, and let K = k((t)) be the fraction field of R.

(2.2) Let G_{R} be a smooth one-dimensional formal group over R such that its fiber G_{k} has finite height h ≥ 2. We assume that the generic fiber G_{K} of G has height one.

(2.3) Let G[p^{∞}] be the Barsotti-Tate group over R attached to G as explained in 2.5. Let T_{p}(G) be the p-adic Tate module attached to the maximal étale quotient G[p^{∞}]_{K}^{ét} of the generic fiber G[p^{∞}]_{K} of G[p^{∞}].

(2.4) Let ρ_{G} : Gal(K^{sep}/K) → GL(T_{p}(G)) be the natural representation of the Galois group Gal(K^{sep}/K) on the free ℤ_{p}-module T_{p}(G) of rank d = h − 1. The image of ρ_{G} is a p-adic analytic subgroup H of GL(T_{p}(G)) ≅ GL_{d}(ℤ_{p}). Let 𝔥 be the Lie algebra of H.

(2.5) The relation between the smooth formal group G_{R} and the Barsotti-Tate group G[p^{∞}] is as follows. Let G[p^{n}]_{R/𝔪^{m}} be the p^{n}-torsion subgroup of the base extension of G_{R} to R/𝔪^{m}, where 𝔪 is the maximal ideal of R. Taking the limit as m → ∞, one obtains a finite flat group scheme G[p^{n}]_{R} over R. The inductive limit of G[p^{n}]_{R} is a Barsotti-Tate group over R, which we call G[p^{∞}]_{R} by abuse of notation. Conversely, from this Barsotti-Tate group G[p^{∞}]_{R} one gets the finite flat group schemes G[p^{n}]_{R/𝔪^{m}}. Taking the limit as n → ∞, one gets a formal group G_{R/𝔪^{m}} over R/𝔪^{m}. The limit of G_{R/𝔪^{m}} is the smooth formal group G_{R}. So G_{R} and G[p^{n}]_{R} determines each other. This kind of going back and forth already appeared in [Ta] and works for all p-divisible formal groups over complete Noetherian local rings with residue characteristic p.

(2.6) Choosing a suitable parameter for G_{R}, we may and do assume that the group law for G_{R} is such that [p]_{G} is represented by a polynomial [p]_{G}(x) with coefficients in R, of the form

[p]_{G}(x) = A_{1}(t) · x^{p} + A_{2}(t) · x^{p^{2}} + ⋯ + A_{h−1}(t) · x^{p^{h−1}} + x^{p^{h}}

with each A_{j}(t) ∈ t · k[[t]], and A_{1}(t) ≠ 0. This fact is well-known, and can be justified by [Dr, §4].

The Verschiebung for G is given by the polynomial

V (x) = A_{1}(t) · x + A_{2}(t) · x^{p} + ⋯ + A_{h−1}(t) · x^{p^{h−2}} + x^{p^{h−1}} .

Its p-iterates V^{(p^{i})} are

V^{(p^{i})}(x) = A_{1}(t)^{p^{i}} · x + A_{2}(t)^{p^{i}} · x^{p} + ⋯ + A_{h−1}^{p^{i}}(t) · x^{p^{h−2}} + x^{p^{h−1}} .

§3. Ramification

(3.1) We proceed as in [Ka, Thm. 4.3]. To find a point of G[p^{N}]^{ét}, we need to solve the following system of equations

{ V (y_{0})  = 0 ; V^{(p)}(y_{1})  = y_{0} ; V^{(p^{2})}(y_{2})  = y_{1} ; ⋮ ; V^{(p^{N−1})}(y_{N−1}) = y_{N−2}

Moreover if one requires that y_{0} ≠ 0, then y_{N−1} has order p^{N}.

(3.2) We would like to estimate the valuation of elements y_{0}, y_{1}, …, y_{i}, … as above, with y_{0} ≠ 0 and V^{(p^{i})}(y_{i}) = y_{i−1} for all i ≥ 1, using the method of Newton polygons. Let v = v_{K} denote the normalized valuation for K, extended to the algebraic closure of K. We want to show that v(y_{i}) = p^{−(h−1)} v(y_{i−1}) for all sufficiently large i. Here is the simple strategy: once one has a lower bound for v(y_{i}), the Newton polygon of the equation V^{(p^{i})}(x) = y_{i−1} will have to be a straight line segment for i large, since the valuation of the coefficients of all but the leading term of the monic polynomial V^{(p^{i})}(x) grow with i. This estimate itself is a simple consequence of the convexity of the Newton polygon.

(3.3) Theorem Let G_{R} be a one-dimensional formal group over R = k[[t]], where k is an algebraically closed field of characteristic p > 0. Suppose that the closed fiber G_{0} of G has finite height h and the generic fiber G_{K} of G over K = k((t)) has height one. Then there exists a natural number n_{1} such that for any sequence 0 ≠ y_{0}, y_{1}, y_{2}, … in K^{sep} with V (y_{0}) = 0, V^{(p^{i})}(y_{i}) = y_{i−1} for all i ≥ 1, we have v_{K}(y_{n}) = p^{−(n−n_{1})(h−1)} · v_{K}(y_{n_{1}}) for all n ≥ n_{1}. Furthermore, for any sequence 0 ≠ y_{0}, y_{1}, y_{2}, … as above, there exists a natural number N_{0} ≥ n_{1}, such that the extension K(y_{n})/K(y_{N_{0}}) is totally ramified of degree p^{(n−N_{0})(h−1)} for all n ≥ N_{0}.

Proof. First we make a trivial estimate. Let c_{1} = v(A_{1}(t)). Looking at the Newton polygon of V (x), clearly we have v(y_{0}) ≤ c_{1}. For the same reason, we have v(y_{i}) ≤ c_{1} for all i ≥ 0.

Now we look at the equation V^{(p^{i})}(x) = y_{i−1} for y_{i} when i ≫ 0. Since the v(y_{i})’s are bounded, and v(A_{j}(t)^{(p^{i})}) = p^{i}v(A_{j}(t)) goes to infinity as i → ∞, the Newton polygon for the equation

−y_{i−1} + A_{1}(t)^{p^{i}} · x + A_{2}(t)^{p^{i}} · x^{p} + ⋯ + A_{h−1}^{p^{i}}(t) · x^{p^{h−2}} + x^{p^{h−1}} = 0

is the line segment connecting (0, v(y_{i−1})) and (p^{h−1}, 0) for i ≫ 0. Hence all solutions of the equation V^{(p^{i})}(x) = y_{i−1} have the same valuation, equal to p^{−(h−1)}v(y_{i−1}), again for i ≫ 0. This argument shows that there exists a constant n_{1}, independent of the sequence y_{0}, y_{1}, … such that v(y_{n}) = p^{−(h−1)}v(y_{n−1}) for all n ≥ n_{1}. This will be the case for instance if p^{n_{1}} · v(A_{j}(t)) ≥ c_{1} for all j = 1, …, h − 1. This proves the first part of the theorem.

The growth behavior of the denominator of the valuation of the y_{i}’s as i increases clearly implies that the ramification index e(K(y_{n})/K) is at least p^{(n−n_{1})(h−1)} for all n ≥ n_{1}. On the other hand we have the trivial bound [K(y_{i}) : K(y_{i−1}] ≤ p^{h−1} for all i ≥ 1. Therefore [K(y_{i}) : K(y_{i−1}] = e(K(y_{i})/K(y_{i−1})) = p^{h−1} for all sufficiently large i’s.

(3.4) Remark As pointed out by the referee, the same argument actually proves more: Let ϵ = min{v_{K}(A_{1}(t), …, v_{K}(A_{h−1}(t)}. Then the convexity of the Newton polygon implies that

0 < v(y_{i}) ≤ max \{\frac{v(y_{i−1})}{p^{h−1}}, v(y_{i−1}) − p^{i}ϵ\} ,

and v(y_{i}) = \frac{v(y_{i−1})}{p^{h−1}} if v(y_{i−1}) ≤ p^{i}ϵ. In particular, the number N_{0} in the statement of Theorem 3.3 is independent of the sequence 0 ≠ y_{0}, y_{1}, y_{2}, … one considers. This also follows from a compactness argument below; see Corollary 3.6 (ii).

The image H of ρ_{G} operates linearly on T_{p}(G), and its orbits are p-adic analytic subvarieties of T_{p}(G). Denote T_{p}(G) − p · T_{p}(G) by U; clearly U is stable under the action of H.

(3.5) Theorem Every non-zero orbit of H on T_{p}(G) is open. Especially the linear representation ρ of H on the ℚ_{p}-vector space T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p} is irreducible.

Proof. Let {y_{i}} be a family of elements in K^{sep} as in Theorem 3.3. From Theorem 3.3 we know that for any i ≥ N_{0}, the orbit of y_{n} under Gal(K^{sep}/K(y_{n−1})) has p^{h−1} elements. The family {y_{i}} above corresponds to an element z ∈ U, and the Gal(K^{sep}/K(y_{n−1}))-orbit of y_{n} corresponds to the coset

Gal(K^{sep}/K(y_{n−1})) · z + p^{n+1}T_{p}(G)/p^{n+1}T_{p}(G)

So for each orbit Z of H contained in U and for each point z of Z, there exists a natural number N_{1} such that

((z + p^{n}T_{p}(G)) ∩ Z) + p^{n+1}T_{p}(G) = z + p^{n}T_{p}(G)

for all n ≥ N_{1}. Since Z is a p-adic analytic subvariety of T_{p}(G), this is possible only when dim(Z) = dim(T_{p}(G)), that is Z is an open subvariety. So every H-orbit in U is open. Clearly this implies that every H-orbit in T_{p}(G) − {0} is open. The last statement of the theorem follows immediately.

(3.6) Corollary (i) There are only a finite number of H-orbits in U.

(ii) The constant N_{0} in the last part of Theorem 3.3 can be chosen uniformly for all sequences 0 ≠ y_{0}, y_{1}, y_{2}, . . .. In other words there exists a natural number N_{0} such that any sequence 0 ≠ y_{0}, y_{1}, y_{2}, … as in Theorem 3.3, the extension K(y_{n})/K(y_{N_{0}}) is totally ramified of degree p^{(n−N_{0})(h−1)} for all n ≥ N_{0}.

Proof. Statement (i) follows from the compactness of U. As for (ii), the families {y_{i}} considered in Theorem 3.3 correspond to points of U. Since the one can use the same number N_{0} on each Galois orbit in U, statement (ii) follows from statement (i).

§4. The determinant of ρ_{G} and its breaks

(4.1) Theorem The image of the one-dimensional Galois representation det ∘ρ_{G} is an open subgroup of ℤ_{p}^{×}. In other words the image is infinite.

The First Proof. According to [G, Thm. 2.7], det ∘ρ_{G} is equal to the inverse of the action of Gal(K^{sep}/K) on the character group of G_{K}. So it suffices to show that the formal group G_{L} is not isomorphic to 𝔾̂_{m} over any finite separable extension L of K.

As before, choose a parameter x for G_{R} such that

[p]_{G}(x) = A_{1}(t) · x^{p} + ⋯ + A_{h−1}(t) · x^{p^{h−1}} + x^{p} .

Suppose that there exists an isomorphism ϕ from G to 𝔾̂_{m} over a finite extension L of K. This isomorphism ϕ is given by a power series

ϕ(x) = a_{1} · x + a_{2} · x^{2} + ⋯, a_{i} ∈ L, a_{1} ≠ 0 ,

such that ϕ(x)^{p} = ϕ([p]_{G}(x)). From this one quickly deduces that a_{i} = 0 unless i is equal to a power of p, and that all the a_{i}’s are integral over L. To be more explicit, let ϕ_{i} = a_{p^{i}}, then the condition ϕ(x)^{p} = ϕ([p]_{G}(x)) becomes the following system of equations.

ϕ_{0}^{p−1} − A_{1}  = 0
ϕ_{1}^{p} − ϕ_{1}A_{1}^{p} − ϕ_{0}A_{2}  = 0
ϕ_{2}^{p} − ϕ_{2}A_{1}^{p^{2}} − ϕ_{1}A_{2}^{p} − ϕ_{0}A_{3}  = 0
⋮
ϕ_{h−2}^{p} − ϕ_{h−2}A_{1}^{p^{h−2}} − ϕ_{h−3}A_{2}^{p^{h−3}} − ⋯ − ϕ_{1}A_{h−2}^{p} − ϕ_{0}A_{h−1}  = 0
ϕ_{h−1}^{p} − ϕ_{h−1}A_{1}^{p^{h−1}} − ϕ_{h−2}A_{2}^{p^{h−2}} − ⋯ − ϕ_{1}A_{h−1}^{p} − ϕ_{0}  = 0
ϕ_{h}^{p} − ϕ_{h}A_{1}^{p^{h}} − ϕ_{h−1}A_{2}^{p^{h−1}} − ⋯ − ϕ_{2}A_{h−1}^{p^{2}} − ϕ_{1}  = 0
ϕ_{h+1}^{p} − ϕ_{h+1}A_{1}^{p^{h+1}} − ϕ_{h}A_{2}^{p^{h}} − ⋯ − ϕ_{3}A_{h−1}^{p^{3}} − ϕ_{2}  = 0
⋮
ϕ_{m}^{p} − ϕ_{m}A_{1}^{p^{m}} − ϕ_{m−1}A_{2}^{p^{m−1}} − ⋯ − ϕ_{m−h+2}A_{h−1}^{p^{m−h+2}} − ϕ_{m−h+1} = 0
∀m ≥ h − 1

Let F (x_{1}, x_{2}) ∈ R[[x_{1}, x_{2}]] be the group law for G_{R} with respect to the chosen parameter x. We have ϕ(F (x_{1}, x_{2})) = ϕ(x_{1}) · ϕ(x_{2}) in L[[x_{1}, x_{2}]]. Since ϕ(x) ∈ 𝒪_{L}[[x]], ϕ extends to a homomorphism ϕ_{R′} from G to 𝔾̂_{m} over R′ = 𝒪_{L}. Over the residue field κ′ ≅ k of R′, the reduction of ϕ_{R′} gives a homomorphism from G_{k} to 𝔾̂_{m}, which must be trivial since G_{k} has height h ≥ 2. (This also follows from the system of equations in last paragraph.) On the other hand, ϕ_{R′} defines a homomorphism ϕ[p^{∞}]_{R′} from the Barsotti-Tate group G[p^{∞}]_{R} ×_{Spec R} Spec 𝒪_{L} to the Barsotti-Tate group 𝔾_{m}[p^{∞}]_{𝒪_{L}} , which induces the trivial homomorphism between the closed fibers. This implies that ϕ[p^{∞}]_{R′} induces the trivial homomorphism over 𝒪_{L}/π_{L}^{n}𝒪_{L} for every n ≥ 1. Therefore ϕ_{R′} is trivial, a contradiction.

The Second Proof. The previous proof uses a rigidity property of homomorphisms between Barsotti-Tate groups. Here is a more elementary alternative to the last paragraph of the first proof. Take any sequence 0 ≠ y_{0}, y_{1}, y_{2}, … of elements in K^{sep} such that V (y_{0}) = 0 and V^{(p^{i})}(y_{i}) = y_{i−1} for all i ≥ 1. Let z_{i} = y_{i}^{i+1}, i = 0, 1, . . .. Then [p]_{G}(z_{i}) = z_{i−1} for all i ≥ 1. Hence ϕ(z_{i}) = ϕ(z_{i−1})^{p} for all i ≥ 1. We want to show that ϕ(z_{0}) = 0, i.e. ϕ(z_{i}) = 0 for all i. This will force ϕ to be trivial and we would have proved the theorem.

Suppose that ϕ(z_{0}) ≠ 0. Then v_{K}(ϕ(z_{i})) = p^{−i}δ_{1}, where δ_{1} = v_{K}(ϕ(z_{0})). On the other hand, Theorem 3.3 tells us that there exists δ_{2} ∈ ℚ_{>0} such that v_{K}(z_{i}) = p^{−ih} δ_{2} for i ≫ 0. Write

ϕ(x) = ϕ_{0} x + ϕ_{1} x^{p} + ϕ_{2} x^{p^{2}} + ⋯ ϕ_{i} ∈ 𝒪_{L} .

Suppose that ϕ_{m} is such that v_{K}(ϕ_{m}) < v_{K}(ϕ_{0}) …, v_{K}(ϕ_{m−1}), and v_{K}(ϕ_{m}) ≤ v_{K}(ϕ_{i}) for all i > m. Then we have

v_{K}(ϕ(z_{i})) = v(ϕ_{m} z_{i}^{p^{m}}) = v_{K}(ϕ_{m}) + δ_{2} p^{m−ih}

for all i ≫ 0. This is a contradiction.

(4.2) Remark The first proof can be interpreted as a direct verification of a special case of the theorem of de Jong in [dJ].

(4.3) Once one knows that the image of det ∘ρ_{G} : Gal(K^{sep}/K) → ℤ_{p}^{×} is open, one can determine the breaks of the corresponding abelian extension of K asymptotically. We shall need a modified version of Tate’s lemma in [G, Lemma 1.5]. The proof is essentially the same as in [G], hence omitted. A systematic exposition of the upper-number filtration in the generality we need can be found in the appendix of [De], see also [G].

(4.4) Lemma Let K be the fraction field of a complete discrete valuation ring O_{K} with perfect residue field. Let

f(Y) = Y^{e} + a_{e−1} · Y^{e−1} + ⋯ + a_{1} · Y + a_{0}

be an irreducible separable polynomial in K[Y] such that for any root β of f(Y), the field K(β) is totally ramified. Pick a root β, let E = K(β), and let π_{E} be a uniformizer of E. Let Γ_{E/K} = Hom(E, K^{sep}) be the set of embeddings of E into K^{sep}, endowed with the upper-numbering filtration. Consider the polynomial

h(Y) = (\frac{1}{π_{E}^{e}} ) · f(π_{E} Y + β) ∈ E[Y]

and the Newton polygon N(h) of h(Y) on the (x, y)-plane constructed using the normalized valuation v_{K} of K. We have (a) The upper breaks of Γ_{E/K} occur at the y-intercepts of the non-trivial sides of N(h).

(b) For each non-trivial side of N(h) with slope λ, let y_{λ} be the corresponding upper break. Then the horizontal length of this non-trivial side is equal to

Card (Γ_{E/K}^{y_{λ}} − Γ_{E/K}^{y_{λ}^{+}}) .

(4.5) Let

ϕ(x) = ∑_{i=0}^{∞} ϕ_{i}(t) · x^{p^{i}} , ϕ_{i}(t) ∈ K^{sep} ∀i, ϕ_{0}(t) ≠ 0

be an isomorphism from the toric part of G[p^{∞}]_{K} to 𝔾̂_{m}[p^{∞}] over K^{sep}, regarded as a generator of the character group of the toric part of G[p^{∞}]_{K}. The openness of det(H) in ℤ_{p}^{×} implies that there exists a natural number n_{2} ≥ h−1 such that for all n ≥ n_{2}, the extension K_{n} = K(ϕ_{0}, …, ϕ_{n−1}, ϕ_{n}) over K_{n_{2}} = K(ϕ_{0}, …, ϕ_{n_{2}}) is totally ramified and cyclic of degree p^{n−n_{2}}. Especially the polynomial

f_{n}(Y) = Y^{p} − A_{1}^{p^{n}}Y − (ϕ_{n−1}A_{2}^{p^{n−1}} + ϕ_{n−2}A_{3}^{p^{n−2}} + ⋯ + ϕ_{n−h+2}A_{h−1}^{p^{n−h+2}} + ϕ_{n−h+1})

is the minimal polynomial of ϕ_{n} over K_{n−1} = K(ϕ_{0}, …, ϕ_{n−1}). Let π_{n} be a uniformizer for K_{n}, and let v_{n−1} be the normalized valuation for K_{n−1}. The Newton polygon for

h_{n}(Y) = \frac{1}{π_{n}^{p}} · f_{n}(π_{n}Y + ϕ_{n}) = \frac{1}{π_{n}^{p}} (π_{n}^{p}Y^{p} − A_{1}^{p^{n}}π_{n}Y )

using v_{n−1} is the line segment connecting (1, p^{n}v_{n−1}(A) − \frac{p−1}{p}) to (p, 0). Hence the only upper

break for K_{n}/K_{n−1} is \frac{p^{n+1}}{p−1} v_{n−1}(A) − 1 according to the modified Tate’s lemma.

It is convenient to write the positive integer \frac{v_{n−1}(A)}{p−1} as c · p^{n−1}, i.e. we let c = \frac{v_{n−1}(A)}{(p−1)p^{n−1}} . From the above information, the upper breaks for K_{n}/K_{n−2} works out to be c · p^{2n−2} − 1 and c · p^{2n−3}(p^{2} − 1) + c · p^{2n−2} − 1 when n − 2 ≥ n_{2}. Continuing this way, one finds that whenever n > m ≥ n_{3}, the upper breaks of the totally ramified cyclic extension K_{n}/K_{m} are

c · (p + 1)(p^{n−m−1} − 1)p^{2m+1} + c · p^{2m+2} − 1
c · (p + 1)(p^{n−m−2} − 1)p^{2m+1} + c · p^{2m+2} − 1
⋮
c · (p + 1)(p − 1)p^{2m+1} + c · p^{2m+2} − 1
c · p^{2m+2} − 1

In other words, the n − m upper breaks for K_{n}/K_{m} occur at

c · p^{2m+1}(p^{i+1} + p^{i} − 1) , i = 0, 1, …, n − m − 1 .

We summarize our calculations as

(4.6) Theorem (a) The image of the determinant det ∘ρ_{G} of the p-adic Galois representation ρ_{G} attached to the maximal étale quotient of G[p^{∞}]_{K} is an open subgroup of ℤ_{p}^{×}.

(b) Pick an integer n_{2} ≥ h − 1 such that Im(det ∘ρ_{G}) ⊇ 1 + p^{n_{2}+1}ℤ_{p}. For each n ≥ n_{2}, let K_{n} be the fixed field corresponding to the subgroup 1 + p^{n+1}ℤ_{p} of Im(det ∘ρ_{G}). Then for every n > n_{2}, K_{n} is a totally ramified cyclic extension of K_{n_{2}} with Galois group

Gal(K_{n}/K_{n_{2}}) ≅ 1 + p^{n_{2}+1}ℤ_{p}/1 + p^{n+1}ℤ_{p} ≅ ℤ/p^{n−n_{2}}ℤ ,

and the upper breaks occur at

c · p^{2n_{2}+1}(p^{i+1} + p^{i} − 1) , i = 0, 1, …, n − n_{2} − 1 ,

where c := \frac{v_{m−1}(A)}{(p−1)p^{m−1}} for m ≫ 0.

§5. A weak classification

(5.1) In this section we use Theorem 3.5 and Theorem 4.1 to classify the possible p-adic representations ρ_{G}’s when h = 3, 4. Before doing so, it may be helpful to consider the general case first. As usual one first passes to the Zariski closure M of the image of ρ in GL(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}). Let M^{∘} be the neutral component of M. The group M^{∘} is a connected reductive algebraic group over ℚ_{p} since it has a faithful irreducible representation on T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}. Let 𝔪 be the Lie algebra of M in this section. By [B, §7, Cor. 7. 9], we know that

[𝔪, 𝔪] ⊆ [𝔥, 𝔥] ⊊ 𝔥 .

Since 𝔥 ⊆ 𝔪, [𝔥, 𝔥] is equal to [𝔪, 𝔪]. In particular, H contains an open subgroup of the ℚ_{p}-points of the derived group M^{∘,der} of M^{∘}.

(5.2) Prehomogeneous vector spaces

Recall that a prehomogeneous vector space is a algebraic linear action of an algebraic group on a vector space with a Zariski dense open orbit. Clearly the algebraic linear representation ρ : M^{∘} → T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p} contains a Zariski dense open orbit. Therefore (M^{∘}, ρ, T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) is a prehomogeneous vector space over ℚ_{p}. Moreover it is defined over ℚ_{p}, and has the additional property that every non-zero orbit which contains a ℚ_{p}-rational point is open in the Zariski topology.

There is an extensive literature on the classification of prehomogeneous vector spaces. The fundamental results are due to Sato and Kimura, and there are important contributions by Chen, Igusa, Kac, Rubenthaler and Vinberg. The standard reference for prehomogenous vector spaces seems to be [SK]; the survey article [Ki] is a convenient place to look up the definitions and results. This theory provides a classification, over algebraically closed field of characteristic 0, of prehomogenous vector spaces whose underlying representation is absolutely irreducible, under the so called castling equivalence. Combining this classification with the results of [Ti], in principle one can work out a weak classification of possible triples (M^{∘}, ρ, T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) which satisfy the restrictions imposed by Theorem 3.5 and Theorem 4.6. The interested readers are invited to consult [Ki, §3] and [Ti, §7].

For the convenience of the reader, we reproduce the definition of castling equivalence here; see [Ki, p. 227] for details. For any representation (H, ρ) of a linear algebraic group H and any two integers m, n ≥ 1, the pair (H × GL_{n}, ρ ⊗ std) is a prehomogeneous vector space if and only if the pair (H × GL_{m}, ρ^{∗} ⊗ std_{n}) is a prehomogeneous vector space, where ρ^{∗} denotes the contragradient representation of ρ and std_{n} denotes the standard representation for GL_{n}. Going from (H×GL_{n}, ρ⊗std) to (H×GL_{m}, ρ^{∗}⊗std_{m}) is called a castling transform. Two prehomogeneous vector spaces are castling equivalent if one can be obtained from the other by a finite number of castling transforms.

Here is a list of simple factors (allowing a central 𝔾_{m}) that may occur in absolutely irreducible prehomogeneous vector spaces. See [Ki, p. 228–229]

• the classical groups and their standard representations;

• the second symmetric product of the standard representation of GL_{n};

• the second exterior product of the standard representation of GL_{n} with n even;

• the second exterior product of the standard representation of GL_{5};

• the 14-dimensional fundamental representation for GSp_{6} which corresponds to the long root;

• the third exterior product of GL_{6}, GL_{7}, GL_{8};

• the spin representation for Spin_{7}, Spin_{9}, Spin_{11};

• a half-spin representation for Spin_{10}, Spin_{12}, Spin_{14};

• the 7-dimensional irreducible representation of G_{2};

• the 27-dimensional irreducible representation of E_{6};

• the 56-dimensional irreducible representation of E_{7}.

(5.3) It is instructive to look first at the case when M^{∘} is commutative. In other words M^{∘} is an algebraic torus. The ℚ_{p}-subalgebra of End(T_{p}(G)⊗_{ℤ_{p}}ℚ_{p}) generated by the image of its Lie algebra 𝔪 is a commutative semisimple algebra contained in End(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}), therefore its dimension is at most d = h − 1. On the other hand, 𝔥 ⊆ 𝔪 has dimension at least d, since H has at least one open orbit in T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}. So 𝔥 = 𝔪, and they are equal to a commutative semi-simple subalgebra F of End(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) of dimension d over ℚ_{p}. The fact that every non-zero orbit of H in T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p} is open implies that this commutative semi-simple subalgebra F is a field extension of ℚ_{p} with [F : ℚ_{p}] = d. Clearly this is the smallest size 𝔥 can possibly be under the restrictions we have imposed on it.

(5.4) Next we consider the first non-trivial case h = 3. If 𝔪 is not commutative, [𝔪, 𝔪] is a nontrivial semi-simple Lie algebra over ℚ_{p} contained in the traceless subalgebra of End(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) ≅ M_{2}(ℚ_{p}). Hence [𝔪, 𝔪] = 𝔰𝔩(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}). Therefore 𝔥 is equal to End(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) ≅ M_{2}(ℚ_{p}) since it is not traceless. On the other hand if 𝔪 is commutative, then 𝔥 is equal to some quadratic extension field of ℚ_{p} embedded in End(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) ≅ M_{2}(ℚ_{p}).

(5.5) Suppose now that h = 4. Of course we have two possibilities at the two extremes, that is 𝔥 can be either End(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) ≅ M_{3}(ℚ_{p}), or a cubic extension field of ℚ_{p} embedded in End(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) ≅ M_{3}(ℚ_{p}). But there is now a third possibility, namely that [𝔪, 𝔪] is a nontrivial semisimple proper subalgebra of 𝔰𝔩(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) ≅ 𝔰𝔩_{3}(ℚ_{p}). The dimension of the irreducible representation involved is so low that the only possibility left is that there exists a quadratic form Q on T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p} such that [𝔪, 𝔪] = 𝔰𝔬(Q), consequently 𝔥 = 𝔤𝔬(Q). Since all non-zero orbits of H are open, the quadratic form Q on T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p} has to be anisotropic.

We summarize the above discussions in the following

(5.6) Theorem Let ρ_{G} be the Galois representation on the p-adic Tate module T_{p}(G) of the maximal étale quotient of G[p^{∞}]_{K}, let H be the image of Gal(K^{sep}/K) in GL(T_{p}(G)), and let M^{∘} be the neutral component of the Zariski closure of the image of H in GL(T_{p}(G)). Then

(1) The connected algebraic group M^{∘} over ℚ_{p} is reductive, and its natural action ρ on T_{p}(G)⊗_{ℤ_{p}} ℚ_{p} is an irreducible representation of M^{∘} over ℚ_{p} with nontrivial determinant.

(2) The p-adic analytic group H contains an open subgroup of the derived group M^{∘,der} of M^{der}.

(3) The triple (M^{∘}, ρ, T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) is a prehomogeneous vectors space over ℚ_{p}, such that the every non-zero M^{∘}-orbit which contains a ℚ_{p}-rational point is open in the Zariski topology.

(4) The Lie algebra 𝔪 of M^{∘} is commutative if and only if the Lie algebra 𝔥 of H is commutative. If this is the case, then 𝔪 = 𝔥, and they are equal to a field extension F of ℚ_{p} of degree d = h − 1 embedded in End_{ℚ_{p}}(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) ≅ M_{d}(ℚ_{p}).

(5) If h = 3, then there are two possibilities for 𝔥. It can be either

– End_{ℚ_{p}}(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) ≅ M_{2}(ℚ_{p}), or

– a quadratic extension field F of ℚ_{p}, embedded in the matrix algebra End_{ℚ_{p}}(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}).

(6) If h = 4, there are three possibilities for 𝔥. It can be equal to

– End_{ℚ_{p}}(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) ≅ M_{3}(ℚ_{p}), or

– a cubic extension F of ℚ_{p} embedded in End_{ℚ_{p}}(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}), or

– the Lie algebra of the group of orthogonal similitudes GO(Q) attached to an anisotropic quadratic form Q on the three-dimension vector space T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p} over ℚ_{p}.

(5.7) Remark (a) If we knew that M^{∘} is the generic fiber of a reductive group scheme 𝔐^{∘} over ℤ_{p}, and that the action of M^{∘} on T_{p}(G)⊗_{ℤ_{p}} ℚ_{p} is the generic fiber of a linear action of 𝔐^{∘} on a free ℤ_{p}-module such that every non-zero orbit of the closed fiber of 𝔐^{∘} is Zariski open, then the classification would be a lot cleaner and shorter. Namely we would be able to conclude that (M^{∘}, ρ, T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}) is the restriction of scalar from an unramified extension of ℚ_{p} to ℚ_{p} of an unramified twist of the standard representation of a classical group, just by looking up the classification list of [SK] and use Lang’s theorem. Unfortunately such information does not seem to be available.

(b) The author does not know any example where h ≥ 3 and 𝔥 is commutative. As mentioned in the introduction, a “wishful thinking” is that under the general assumptions of this paper, we have 𝔥 = End_{ℚ_{p}}(T_{p}(G) ⊗_{ℤ_{p}} ℚ_{p}). In other words, H is an open subgroup of GL(T_{p}(G)).

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Ching-Li Chai

Department of Mathematics University of Pennsylvania Philadelphia, PA 19104 U.S.A.

                     Email: chai@math.upenn.edu