THE p-ADIC CLOSURE OF A SUBGROUP OF RATIONAL POINTS ON A COMMUTATIVE ALGEBRAIC GROUP

BJORN POONEN

Abstract. Let G be a commutative algebraic group over ℚ. Let Γ be a subgroup of G(ℚ) contained in the union of the compact subgroups of G(ℚ_{p}). We formulate a guess for the dimension of the closure of Γ in G(ℚ_{p}), and show that its correctness for certain tori is equivalent to Leopoldt’s conjecture.

1. Introduction

1.1. Notation. Let ℚ be the field of rational numbers. Let p be a prime, and let ℚ_{p} be the corresponding completion of ℚ. Let ℤ_{p} be the completion of ℤ at p. If K is a number field, then O_{K} is the ring of the integers, and for any finite set S of places, O_{K,S} is the ring of S-integers. If G is a group (or group scheme), then H ≤ G means that H is a subgroup (or subgroup scheme) of G. If A is an integral domain with fraction field K, and M is an A-module, then rk_{A} M is the dimension of the K-vector space M ⊗_{A} K; we write rk for rk_{ℤ}.

1.2. The logarithm map for a p-adic Lie group. Let G be a finite-dimensional commutative Lie group over ℚ_{p} (see [Bou98, III.§1] for terminology). The Lie algebra Lie G is the tangent space of G at the identity. So Lie G is a ℚ_{p}-vector space of dimension dim G. Let G_{f} be the union of the compact subgroups of G. By [Bou98, III.§7.6], G_{f} is an open subgroup of G, and there is a canonical homomorphism

log: G_{f} → Lie G,

defined first on a sufficiently small compact open subgroup by formally integrating translationinvariant 1-forms, and then extended by linearity. Moreover, log is a local diffeomorphism, and its kernel is the torsion subgroup of G_{f}. It behaves functorially in G.

Examples 1.1.

(i) If G = ℚ_{p} (the additive group), then G_{f} = ℚ_{p}, and log is an isomorphism. In this example, G_{f} is not compact.

(ii) If G = ℚ_{p}^{×}, then G_{f} = ℤ_{p}^{×}.

(iii) If G = A(ℚ_{p}) for an abelian variety A over ℚ_{p}, then G_{f} = G.

Date: November 30, 2007.

2000 Mathematics Subject Classification. Primary 14L10; Secondary 11E95, 14G05, 14G20, 22E35.

Key words and phrases. p-adic logarithm, rational points, algebraic group, Chabauty’s method, Leopoldt’s conjecture.

This research was supported by NSF grant DMS-0301280.

1.3. Dimension of an analytic subgroup. Let Γ be a finitely generated subgroup of G_{f}. Then log Γ ⊆ Lie G is a finitely generated abelian group of the same rank. The closure \overline{log Γ} = log \overline{Γ} with respect to the p-adic topology equals the ℤ_{p}-submodule of Lie G spanned by log Γ, so it is a finitely generated ℤ_{p}-module. Define

dim \overline{Γ} := rk_{ℤ_{p}} \overline{log Γ}.

This agrees with the dimension of \overline{Γ} viewed as a Lie group over ℚ_{p}.

1.4. Rational points. Now let G be a commutative group scheme of finite type over ℚ. Fix a prime p. Define G(ℚ)_{f} := G(ℚ) ∩ G(ℚ_{p})_{f}. We specialize the previous sections to the Lie group G(ℚ_{p}) and to a finitely generated subgroup Γ of G(ℚ)_{f}. Our goal is to predict the value of dim \overline{Γ}.

1.5. Applications. The value of dim \overline{Γ} is important for a few reasons:

(i) If C is a curve of genus g ≥ 2 over ℚ embedded in its Jacobian J, then the condition dim \overline{J(ℚ)} < g is necessary for the application of Chabauty’s method, which attempts to calculate C(k) or at least bound its size [Cha41,Col85].

(ii) Leopoldt’s conjecture on p-adic independence of units in a number field predicts dim \overline{Γ} in a special case: see Corollary 5.3 in Section 5. Leopoldt’s conjecture is important because it governs the abelian extensions of K of p-power degree.

1.6. Outline of the paper. Section 2 axiomatizes some of the properties of dim \overline{Γ} in order to identify possible candidates for its value. Section 3 defines a “maximal” function d(Γ) satisfying the same axioms and Question 3.3 asks whether it always equals dim \overline{Γ}. Section 4 shows that dim \overline{Γ} and d(Γ) share many other properties. Section 5 computes d(Γ) for subgroups of integer points on tori, and shows that a positive answer to Question 3.3 for certain tori would imply Leopoldt’s conjectures. We end with further open questions.

2. Dimension functions

Let G be the set of pairs (G, Γ) where G is a commutative group scheme of finite type over ℚ and Γ is a finitely generated subgroup of G(ℚ)_{f}.

Definition 2.1. A dimension function is a function ∂: G → ℤ_{≥0} satisfying

(1) If Γ ≤ H(ℚ)_{f} for some subgroup scheme H ≤ G, then ∂(H, Γ) = ∂(G, Γ). (Because of this, we generally write ∂(Γ) instead of ∂(G, Γ).)

(2) ∂(Γ) ≤ rk Γ.

(3) If H ≤ G and Γ′′ is the image of Γ in (G/H)(ℚ)_{f}, then ∂(Γ) ≤ dim H + ∂(Γ′′).

Proposition 2.2. The expression dim \overline{Γ} is a dimension function.

Proof. Since H(ℚ_{p}) is closed in G(ℚ_{p}), the closure of Γ in H(ℚ_{p}) equals the closure of Γ in G(ℚ_{p}); therefore (1) holds. The fact that \overline{log Γ} is the ℤ_{p}-submodule spanned by log Γ gives the middle step in

dim \overline{Γ} = rk_{ℤ_{p}} \overline{log Γ} ≤ rk(log Γ) ≤ rk Γ,

so (2) holds. Finally, by continuity, the image of \overline{Γ} in (G/H)(ℚ_{p}) equals \overline{Γ′′}, so we have an exact sequence

0 → \overline{Γ} ∩ H → \overline{Γ} → \overline{Γ′′} → 0.

Taking dimensions of p-adic Lie groups and observing that the group on the left has dimension at most dim H yields (3). □

3. The guess

With notation as before, define

d(G, Γ) := inf_{H≤G} (dim H + rk Γ − rk(Γ ∩ H)) ,

where the infimum is over all subgroup schemes H ≤ G.

Proposition 3.1. The function d is a dimension function, and any dimension function ∂ satisfies ∂ ≤ d.

Proof. First we check that d is a dimension function:

(1) Suppose G′ ≤ G and Γ ≤ G′(ℚ)_{f}. If H ≤ G′ then the subgroup H′ := H ∩G′ satisfies dim H′ ≤ dim H and Γ ∩ H′ = Γ ∩ H, so

dim H′ + rk Γ − rk(Γ ∩ H′) ≤ dim H + rk Γ − rk(Γ ∩ H).

Therefore the infimum in the definition of d(G, Γ) is attained for some H ≤ G′, so d(G, Γ) = d(G′, Γ).

(2) The H = {0} term in the infimum is 0 + rk Γ − 0, so d(Γ) ≤ rk Γ.

(3) Let K′′ be the subgroup of G/H realizing the infimum defining d(Γ′′). Let K be the inverse image of K′′ under G → G/H. Then Γ′′ ∩ K′′ is a homomorphic image of Γ ∩ K, so rk(Γ′′ ∩ K′′) ≤ rk(Γ ∩ K) and

d(Γ) ≤ dim K + rk Γ − rk(Γ ∩ K)
= dim H + dim K′′ + rk Γ − rk(Γ ∩ K)
≤ dim H + dim K′′ + rk Γ − rk(Γ′′ ∩ K′′)
= dim H + d(Γ′′).

Now we check that any dimension function ∂ satisfies ∂ ≤ d. If H ≤ G and Γ′′ is the image of Γ in (G/H)(ℚ), then properties (3) and (2) for ∂ and the isomorphism Γ′′ ≃ Γ/(Γ ∩ H) yield

∂(Γ) ≤ dim H + ∂(Γ′′) ≤ dim H + rk Γ′′ = dim H + rk Γ − rk(Γ ∩ H).

This holds for all H, so ∂(Γ) ≤ d(Γ). □

Corollary 3.2. We have dim \overline{Γ} ≤ d(Γ).

Proposition 3.1 shows that the function d(Γ) gives the largest guess for dim \overline{Γ} compatible with the elementary inequalities based on rank and the dimension of the group. Therefore we ask:

Question 3.3. Does dim \overline{Γ} = d(Γ) always hold?

In other words, are rational points p-adically independent whenever dependencies are not forced by having a subgroup of too high rank inside an algebraic subgroup?

4. Further properties shared by both dimension functions

Proposition 4.1. Let ∂(Γ) denote either dim \overline{Γ} or d(Γ). Then

(i) If Γ′ ≤ Γ ≤ G(ℚ)_{f}, then ∂(Γ′) ≤ ∂(Γ). (ii) If G → G′′ is a homomorphism and Γ ≤ G(ℚ)_{f}, then the image Γ′′ in G′′(ℚ) is

contained in G′′(ℚ)_{f} and ∂(Γ′′) ≤ ∂(Γ). (iii) If G → G′′ has finite kernel and Γ ≤ G(ℚ), then Γ is contained in G(ℚ)_{f} if and only

if its image Γ′′ in G′′(ℚ) belongs to G′′(ℚ)_{f}; in this case, ∂(Γ) = ∂(Γ′′). (iv) Suppose that Γ_{1}, Γ_{2} are commensurable subgroups of G(ℚ); i.e., Γ_{1} ∩ Γ_{2} has finite

index in both Γ_{1} and Γ_{2}. Then Γ_{1} ≤ G(ℚ)_{f} if and only if Γ_{2} ≤ G(ℚ)_{f}; in this case, ∂(Γ_{1}) = ∂(Γ_{2}).

(v) If Γ_{i} ≤ G_{i}(ℚ)_{f} for i = 1, 2, then Γ_{1} × Γ_{2} ≤ (G_{1} × G_{2})(ℚ)_{f} and ∂(Γ_{1} × Γ_{2}) = ∂(Γ_{1}) + ∂(Γ_{2}).

(vi) If Γ_{1}, Γ_{2} ≤ G(ℚ)_{f}, then ∂(Γ_{1} + Γ_{2}) ≤ ∂(Γ_{1}) + ∂(Γ_{2}). (vii) If rk Γ = 1, then ∂(Γ) = 1. (viii) If G ≃ 𝔾_{a}^{n}, then ∂(Γ) = rk Γ.

Proof.

(i) For ∂(Γ) := dim \overline{Γ} the result is obvious. For d(Γ) it follows since rk Γ − rk(Γ ∩ H) equals the rank of the image of Γ in G/H.

(ii) We have Γ′′ ≤ G′′(ℚ)_{f} by functoriality. For dim \overline{Γ}, the inequality follows since \overline{log Γ} surjects onto \overline{log Γ′′}. For d(Γ), if H ≤ G and H′′ is its image in G′′, then the subgroup Γ/(Γ∩H) of G/H surjects onto the subgroup Γ′′/(Γ′′∩H′′) of G′′/H′′, and this implies the second inequality in

d(Γ′′) ≤ dim H′′ + rk Γ′′ − rk(Γ′′ ∩ H′′)
≤ dim H′′ + rk Γ − rk(Γ ∩ H)
≤ dim H + rk Γ − rk(Γ ∩ H).

This holds for all H ≤ G, so d(Γ′′) ≤ d(Γ). (iii) The map of topological spaces G(ℚ_{p}) → G′′(ℚ_{p}) is proper, so the inverse image of

G′′(ℚ_{p})_{f} is contained in G(ℚ_{p})_{f}; this gives the first statement. To prove ∂(Γ) = ∂(Γ′′), first use (1) to assume that G → G′′ is surjective, so G′′ = G/H for some finite H ≤ G. By (ii), ∂(Γ′′) ≤ ∂(Γ). By (3), ∂(Γ) ≤ dim H + ∂(Γ′′) = ∂(Γ′′). Thus ∂(Γ) = ∂(Γ′′).

(iv) We may reduce to the case in which Γ_{1} is a finite-index subgroup of Γ_{2}. Let n = (Γ_{2} : Γ_{1}), so nΓ_{1} ≤ Γ_{2} ≤ Γ_{1}. If Γ_{1} ≤ G(ℚ)_{f}, then Γ_{2} ≤ G(ℚ)_{f}. Conversely, if Γ_{2} ≤ G(ℚ)_{f}, then nΓ_{1} ≤ G(ℚ)_{f}, so Γ_{1} ≤ G(ℚ)_{f} by (iii) applied to G →^{n} G. In this case, (iii) gives ∂(nΓ_{1}) = ∂(Γ_{1}), and (ii) implies that both equal ∂(Γ_{2}).

(v) Let Γ := Γ_{1} × Γ_{2}. Since a product of compact open subgroups is a compact open subgroup, we have G_{1}(ℚ_{p})_{f} × G_{2}(ℚ_{p})_{f} ≤ (G_{1} × G_{2})(ℚ_{p})_{f}. (In fact, equality holds.) Thus Γ ≤ (G_{1} × G_{2})(ℚ)_{f}. The equality dim \overline{Γ} = dim \overline{Γ_{1}} × dim \overline{Γ_{2}} follows from the definitions. To prove the corresponding equality for d, we must show that the infimum in the definition of d(Γ) is realized for an H of the form H_{1} × H_{2} with H_{i} ≤ G_{i}. Suppose instead that K ≤ G_{1} × G_{2} realizes the infimum. Let π_{1} : G_{1} × G_{2} → G_{1} be the first projection. Let H_{1} = π_{1}(K). Let H_{2} = ker(π_{1}|_{K}); view H_{2} as a subgroup scheme of G_{2}. Let H = H_{1} × H_{2}. Thus dim K = dim H. The exact sequence

0 → Γ_{2} ∩ H_{2} → Γ ∩ K →^{π_{1}} Γ_{1} ∩ H_{1}

shows that rk(Γ ∩ K) ≤ rk(Γ ∩ H), so

dim H + rk Γ − rk(Γ ∩ H) ≤ dim K + rk Γ − rk(Γ ∩ K).

Thus H too realizes the infimum in the definition of d(Γ), as desired. (vi) Apply (ii) to the addition homomorphism G × G → G and Γ := Γ_{1} × Γ_{2}, and use (v). (vii) We have ∂(Γ) ≤ 1 by (2). If dim \overline{Γ} = 0, then the finitely generated torsion-free ℤ_{p}-

module \overline{log Γ} is of rank 0, so it is 0; therefore Γ ⊆ ker log, so Γ is torsion, contradicting the hypothesis rk Γ = 1. If d(Γ) = 0, then there exists H ≤ G with dim H = 0 and rk Γ = rk(Γ ∩ H); then H is finite, so rk(Γ ∩ H) = 0 and rk(Γ) = 0, contradicting the hypothesis.

(viii) By applying an element of GL_{n}(ℚ) = Aut 𝔾_{a}^{n}, we may assume that Γ = ℤ^{r}×{0}^{n−r} ≤ ℚ^{n} = 𝔾_{a}^{n}(ℚ), where r := rk Γ. Using (v), we reduce to the case n = 1. If r = 0, then the result is trivial. If r = 1, use (vii).

□

5. Tori

Lemma 5.1. Let K be a Galois extension of ℚ. Let G := Gal(K/ℚ). Then the representation O_{K}^{×} ⊗ ℂ of G is a subquotient of the regular representation.

Proof. Define the G-set E := Hom_{ℚ-algebras}(K, ℂ) of embeddings and the G-set P of archimedean places of K, the difference being that conjugate complex embeddings are identified in P. Then E is a principal homogeneous space of G, and there is a natural surjection E → P. Therefore ℂ^{E} is the regular representation and the permutation representation ℂ^{P} is a quotient of ℂ^{E}. The proof of the Dirichlet unit theorem gives a G-equivariant exact sequence

0 → O_{K}^{×} ⊗ ℝ →^{log} ℝ^{P} → ℝ → 0

so O_{K}^{×} ⊗ ℂ is a subrepresentation of ℂ^{P}. □

Proposition 5.2. Let T be a group scheme of finite type over ℤ whose generic fiber T := T × ℚ is a torus. Then

(a) T (ℤ) is a finitely generated abelian group.

(b) rk T (ℤ) ≤ dim T.

(c) If Γ ≤ T (ℤ), then d(T, Γ) = rk Γ.

Proof.

(a) For some number field K and set S of places of K, we have T × O_{K,S} ≃ (𝔾_{m})_{O_{K,S}}^{n}, so T (O_{K,S}) is finitely generated by the Dirichlet S-unit theorem. Therefore the subgroup T (ℤ) is finitely generated.

(b) We may assume that K is Galois over ℚ. Let G = Gal(K/ℚ). Let X be the character group Hom(T_{K}, (𝔾_{m})_{K}) of T. Let χ_{X} be the character of the representation X ⊗ ℂ of G. Let χ_{K} be the character of the representation O_{K}^{×} ⊗ ℂ of G. By Theorem 6.7 and Corollary 6.9 of [Eis03], rk T (ℤ) = (χ_{X}, χ_{K}). On the other hand, dim T = rk X = (χ_{X}, χ_{reg}), where χ_{reg} is the character of the regular representation of G. The result now follows from Lemma 5.1.

(c) First, T (ℤ) is contained in the compact open subgroup T (ℤ_{p}) of T (ℚ_{p}), so d(T, Γ) is defined. By (2), d(T, Γ) ≤ rk Γ. To prove the opposite inequality, we must show that for every subgroup scheme H ≤ T, we have rk(Γ ∩ H) ≤ dim H. By replacing H by its connected component of the identity, we may assume that H is a subtorus of T. Let H be the Zariski closure of H in T . Then Γ ∩ H ≤ H(ℤ), so rk(Γ ∩ H) ≤ rk H(ℤ) ≤ dim H by (b).

□

Corollary 5.3. Let K be a number field. Let T be the restriction of scalars Res_{K/ℚ} 𝔾_{m}. Let Γ ≤ T (ℚ) ≃ K^{×} correspond to O_{K}^{×}. Then Leopoldt’s conjecture is equivalent to a positive answer to Question 3.3 for Γ.

Proof. Leopoldt’s conjecture is the statement dim \overline{Γ} = rk Γ. Let T = Res_{O_{K}/ℤ} 𝔾_{m}. By Proposition 5.2(c) applied to T , d(T, Γ) = rk Γ. So Leopoldt’s conjecture is equivalent to dim \overline{Γ} = d(T, Γ). □

Remark 5.4. In effect, we have shown that Leopoldt’s conjecture cannot be disproved simply by finding a subtorus H of Res_{K/ℚ} 𝔾_{m} containing a subgroup of integer points of rank greater than dim H. This seems to have been known to experts, but we could not find a published proof.

6. Further questions

Question 6.1. Is d(Γ) computable in terms of G and generators for Γ?

Question 6.2. If the answer to Question 6.1 is positive, can dim \overline{Γ} = d(Γ) be verified in each instance where it is true?

Question 6.3. Can one define a plausible generalization of d(Γ) for the analogous situation where ℚ and ℚ_{p} are replaced a number field k and some nonarchimedean completion k_{v}?

Remark 6.4. Applying restriction of scalars from k to ℚ and then applying d does not answer Question 6.3: it would instead predict the dimension of the closure of Γ in the product ∏_{v|p} G(k_{v}) instead of in a single G(k_{v}).

Remark 6.5. If G be a commutative group scheme of finite type over ℚ, we can consider also G(ℝ), and define G(ℝ)_{f} and G(ℚ)_{f}. The closure \overline{Γ} of any subgroup Γ ≤ G(ℚ)_{f} in G(ℝ) is a real Lie group. The natural guess for dim \overline{Γ} seems now to be that it equals the dimension of the Zariski closure H of Γ in G; in other words, \overline{Γ} should be open in H(ℝ). See [Maz92, §7] for a discussion of the abelian variety case.

Acknowledgements

I thank Robert Coleman for suggesting the reference [Bou98, III.§7.6].

References

[Bou98] Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. Translated from the French; Reprint of the 1989 English translation.MR1728312 (2001g:17006) ↑1.2, 6

[Cha41] Claude Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885 (French).MR0004484 (3,14d) ↑i [Col85] Robert F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770.MR808103

(87f:11043) ↑i

[Eis03] Anne Kirsten Eisenträger, Hilbert’s tenth problem and arithmetic geometry, May 2003. Ph.D. thesis, University of California at Berkeley. ↑5

[Maz92] Barry Mazur, The topology of rational points, Experiment. Math. 1 (1992), no. 1, 35–45.MR1181085 (93j:14020) ↑6.5

Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

  E-mail address: poonen@math.berkeley.edu
  URL: http://math.berkeley.edu/~poonen