Stability of Arakelov Bundles

and Tensor Products without Global Sections

Norbert Hoffmann

Received: June 12, 2003

Communicated by Ulf Rehmann

Abstract. This paper deals with Arakelov vector bundles over an arithmetic curve, i.e. over the set of places of a number field. The main result is that for each semistable bundle E, there is a bundle F such that E ⊗F has at least a certain slope, but no global sections. It is motivated by an analogous theorem of Faltings for vector bundles over algebraic curves and contains the Minkowski-Hlawka theorem on sphere packings as a special case. The proof uses an adelic version of Siegel’s mean value formula.

2000 Mathematics Subject Classification: Primary 14G40; Secondary 11H31, 11R56. Keywords and Phrases: Arakelov bundle, arithmetic curve, tensor product, lattice sphere packing, mean value formula, Minkowski-Hlawka theorem

Introduction

G. Faltings has proved that for each semistable vector bundle E over an algebraic curve of genus g, there is another vector bundle F such that E ⊗ F has slope g − 1 and no global sections. (Note that any vector bundle of slope > g −1 has global sections by Riemann–Roch.) See [3] and [4] where this result is interpreted in terms of theta functions and used for a new construction of moduli schemes of vector bundles. In the present paper, an arithmetic analogue of that theorem is proposed. The algebraic curve is replaced by the set X of all places of a number field K; we call X an arithmetic curve. Vector bundles are replaced by so-called Arakelov bundles, cf. section 3. In the special case K = ℚ, Arakelov bundles without global sections are lattice sphere packings, and the slope μ measures the packing density. We will see at the end of section 4 that the maximal slope of Arakelov bundles of rank n without global sections is d(log n + O(1))/2 + (log 𝔡)/2 where d is the degree and 𝔡 is the discriminant of K. Now the main result is:

Theorem 0.1 Let E be a semistable Arakelov bundle over the arithmetic curve X. For each n ≫ 0 there is an Arakelov bundle F of rank n satisfying

μ(E ⊗ F) > \frac{d}{2}(log n − log π − 1 − log 2) + \frac{log 𝔡}{2}

such that E ⊗ F has no nonzero global sections.

The proof is inspired by (and generalises) the Minkowski-Hlawka existence theorem for sphere packings; in particular, it is not constructive. The principal ingredients are integration over a space of Arakelov bundles (with respect to some Tamagawa measure) and an adelic version of Siegel’s mean value formula. Section 2 explains the latter, section 3 contains all we need about Arakelov bundles, and the main results are proved and discussed in section 4. This paper is a condensed and slightly improved part of the author’s Ph. D. thesis [6]. I would like to thank my adviser G. Faltings for his suggestions; the work is based on his ideas. It was supported by a grant of the Max-Planck-Institut in Bonn.

1 Notation

Let K be a number field of degree d over ℚ and with ring of integers O_{K}. Let X = Spec(O_{K}) ∪ X_{∞} be the set of places of K; this might be called an ‘arithmetic curve’ in the sense of Arakelov geometry. X_{∞} consists of r_{1} real and r_{2} complex places with r_{1} + 2r_{2} = d. w(K) is the number of roots of unity in K. For every place v ∈ X, we endow the corresponding completion K_{v} of K with the map | . |_{v} : K_{v} → ℝ_{≥0} defined by μ(a·S) = |a|_{v} ·μ(S) for a Haar measure μ on K_{v}. This is the normalised valuation if v is finite, the usual absolute value if v is real and its square if v is complex. The well known product formula ∏_{v∈X} |a|_{v} = 1 holds for every 0 ≠ a ∈ K. On the adele ring 𝔸, we have the divisor map div : 𝔸 → ℝ_{≥0}^{X} that maps each adele a = (a_{v})_{v∈X} to the collection (|a_{v}|_{v})_{v∈X} of its valuations. Let O_{v} be the set of those a ∈ K_{v} which satisfy |a|_{v} ≤ 1; this is the ring of integers in K_{v} for finite v and the unit disc for infinite v. Let O_{𝔸} denote the product ∏_{v∈X} O_{v}; this is the set of all adeles a with div(a) ≤ 1. By D ≤ 1 for an element D = (D_{v})_{v∈X} of ℝ_{≥0}^{X}, we always mean D_{v} ≤ 1 for all v. We fix a canonical Haar measure λ_{v} on K_{v} as follows:

• If v is finite, we normalise by λ_{v}(O_{v}) = 1.

• If v is real, we take for λ_{v} the usual Lebesgue measure on ℝ.

• If v is complex, we let λ_{v} come from the real volume form idz ∧ dz̄ on ℂ. In other words, we take twice the usual Lebesgue measure.

This gives us a canonical Haar measure λ := ∏_{v∈X} λ_{v} on 𝔸. We have λ(𝔸/K) = \sqrt{𝔡} where 𝔡 = 𝔡_{K/ℚ} denotes (the absolute value of) the discriminant. More details on this measure can be found in [12], section 2.1. Let V_{n} = \frac{π^{n/2}}{(n/2)!} be the volume of the unit ball in ℝ^{n}. For v ∈ X_{∞}, we denote by O_{v}^{n} the unit ball with respect to the standard scalar product on K_{v}^{n}. Observe that this is not the n-fold Cartesian product of O_{v} ⊆ K_{v}. Similarly, O_{𝔸}^{n} := ∏_{v∈X} O_{v}^{n} is not the n-fold product of O_{𝔸} ⊆ 𝔸. Its volume λ^{n}(O_{𝔸}^{n}) is V_{n}^{r_{1}}(2^{n}V_{2n})^{r_{2}}.

2 A mean value formula

The following proposition is a generalisation of Siegel’s mean value formula to an adelic setting: With real numbers and integers instead of adeles and elements of K, Siegel has already stated it in [10], and an elementary proof is given in [7]. (In the special case l = 1, a similar question is studied in [11].)

Proposition 2.1 Let 1 ≤ l < n, and let f be a nonnegative measurable function on the space Mat_{n×l}(𝔸) of n × l adele matrices. Then

∫_{Sl_{n}(𝔸)/Sl_{n}(K)} ∑_{{M∈Mat_{n×l}(K) \atop rk(M)=l}} f(g · M) dτ(g) = 𝔡^{−nl/2} ∫_{Mat_{n×l}(𝔸)} f dλ^{n×l} (1)

where τ is the unique Sl_{n}(𝔸)-invariant probability measure on Sl_{n}(𝔸)/Sl_{n}(K).

Proof: The case l = 1 is done in section 3.4 of [12], and the general case can be deduced along the same lines from earlier sections of this book. We sketch the main arguments here; more details are given in [6], section 3.2. Let G be the algebraic group Sl_{n} over the ground field K, and denote by τ_{G} the Tamagawa measure on G(𝔸) or any quotient by a discrete subgroup. The two measures τ and τ_{G} on Sl_{n}(𝔸)/Sl_{n}(K) coincide because the Tamagawa number of G is one. G acts on the affine space Mat_{n×l} by left multiplication. Denote the first l columns of the n × n identity matrix by E ∈ Mat_{n×l}(K), and let H ⊆ G be the stabiliser of E. This algebraic group H is a semi-direct product of Sl_{n−l} and Mat_{l×(n−l)}. Hence section 2.4 of [12] gives us a Tamagawa measure τ_{H} on H(𝔸), and the Tamagawa number of H is also one. Again by section 2.4 of [12], we have a Tamagawa measure τ_{G/H} on G(𝔸)/H(𝔸) as well, and it satisfies τ_{G} = τ_{G/H} · τ_{H} in the sense defined there. In particular, this implies

∫_{G(𝔸)/H(K)} f(g · E) dτ_{G}(g) = ∫_{G(𝔸)/H(𝔸)} f(g · E) dτ_{G/H}(g).

It is easy to see that the left hand sides of this equation and of (1) coincide. According to lemma 3.4.1 of [12], the right hand sides coincide, too. □

3 Arakelov vector bundles

Recall that a (Euclidean) lattice is a free ℤ-module Λ of finite rank together with a scalar product on Λ⊗ℝ. This is the special case K = ℚ of the following notion:

Definition 3.1 An Arakelov (vector) bundle E over our arithmetic curve X = Spec(O_{K})∪X_{∞} is a finitely generated projective O_{K}-module E_{O_{K}} endowed with

• a Euclidean scalar product ⟨ , ⟩_{E,v} on the real vector space E_{K_{v}} for every real place v ∈ X_{∞} and

• a Hermitian scalar product ⟨ , ⟩_{E,v} on the complex vector space E_{K_{v}} for every complex place v ∈ X_{∞}

where E_{A} := E_{O_{K}} ⊗ A for every O_{K}-algebra A.

A first example is the trivial Arakelov line bundle O. More generally, the trivial Arakelov vector bundle O^{n} consists of the free module O_{K}^{n} together with the standard scalar products at the infinite places. We say that E′ is a subbundle of E and write E′ ⊆ E if E′_{O_{K}} is a direct summand in E_{O_{K}} and the scalar product on E′_{K_{v}} is the restriction of the one on E_{K_{v}} for every infinite place v. Hence every vector subspace of E_{K} is the generic fibre of one and only one subbundle of E. From the data belonging to an Arakelov bundle E, we can define a map

∥ . ∥_{E,v} : E_{K_{v}} ⟶ ℝ_{≥0}

for every place v ∈ X:

• If v is finite, let ∥e∥_{E,v} be the minimum of the valuations |a|_{v} of those elements a ∈ K_{v} for which e lies in the subset a · E_{O_{v}} of E_{K_{v}}. This is the nonarchimedean norm corresponding to E_{O_{v}}.

• If v is real, we put ∥e∥_{E,v} := \sqrt{⟨e, e⟩_{v}}, so we just take the norm coming from the given scalar product.

• If v is complex, we put ∥e∥_{E,v} := ⟨e, e⟩_{v} which is the square of the norm coming from our Hermitian scalar product.

Taken together, they yield a divisor map

div_{E} : E_{𝔸} → ℝ_{≥0}^{X} e = (e_{v}) ↦ (∥e_{v}∥_{E,v}).

Although O_{𝔸} is not an O_{K}-algebra, we will use the notation E_{O_{𝔸}}, namely for the compact set defined by

E_{O_{𝔸}} := {e ∈ E_{𝔸} : div_{E}(e) ≤ 1}.

Recall that these norms are used in the definition of the Arakelov degree: If L is an Arakelov line bundle and 0 ≠ l ∈ L_{K} a nonzero generic section, then

deg(L) := − log ∏_{v∈X} ∥l∥_{L,v}

and the degree of an Arakelov vector bundle E is by definition the degree of the Arakelov line bundle det(E). μ(E) := deg(E)/rk(E) is called the slope of E. One can form the tensor product of two Arakelov bundles in a natural manner, and it has the property μ(E ⊗ F) = μ(E) + μ(F). Moreover, the notion of stability is based on slopes: For 1 ≤ l ≤ rk(E), denote by μ_{max}^{(l)} the supremum (in fact it is the maximum) of the slopes μ(E′) of subbundles E′ ⊆ E of rank l. E is said to be stable if μ_{max}^{(l)} < μ(E) holds for all l < rk(E), and semistable if μ_{max}^{(l)} ≤ μ(E) for all l. To each projective variety over K endowed with a metrized line bundle, one can associate a zeta function as in [5] or [1]. We recall its definition in the special case of Grassmannians associated to Arakelov bundles:

Definition 3.2 If E is an Arakelov bundle over X and l ≤ rk(E) is a positive integer, then we define

ζ_{E}^{(l)}(s) := ∑_{{E′⊆E \atop rk(E′)=l}} exp(s · deg(E′)).

The growth of these zeta functions is related to the stability of E. More precisely, we have the following asymptotic bound:

Lemma 3.3 There is a constant C = C(E) such that

ζ_{E}^{(l)}(s) ≤ C · exp(s · lμ_{max}^{(l)}(E))

for all sufficiently large real numbers s.

Proof: Fix E and l. Denote by N(T) the number of subbundles E′ ⊆ E of rank l and degree at least −T. There are C_{1}, C_{2} ∈ ℝ such that

N(T) ≤ exp(C_{1}T + C_{2})

holds for all T ∈ ℝ. (Embedding the Grassmannian into a projective space, this follows easily from [9]. See [6], lemma 3.4.8 for more details.) If we order the summands of ζ_{E}^{(l)} according to their magnitude, we thus get

ζ_{E}^{(l)}(s) ≤ ∑_{ν=0}^{∞} N (−lμ_{max}^{(l)}(E) + ν + 1) · exp (s · (lμ_{max}^{(l)}(E) − ν))

≤ exp(s · lμ_{max}^{(l)}(E)) · ∑_{ν=0}^{∞} \frac{C_{3}}{exp((s − C_{1})ν)}.

But the last sum is a convergent geometric series for all s > C_{1} and decreases as s grows, so it is bounded for s ≥ C_{1} + 1. □

4 The main theorem

The global sections of an Arakelov bundle E over X = Spec(O_{K}) ∪ X_{∞} are by definition the elements of the finite set

Γ(E) := E_{K} ∩ E_{O_{𝔸}} ⊆ E_{𝔸}.

Note that in the special case K = ℚ, an Arakelov bundle without nonzero global sections is nothing but a (lattice) sphere packing: Γ(E) = 0 means that the (closed) balls of radius 1/2 centered at the points of the lattice E_{ℤ} are disjoint. Here larger degree corresponds to denser packings.

Theorem 4.1 Let E be an Arakelov bundle over the arithmetic curve X. If an integer n > rk(E) and an Arakelov line bundle L satisfy

1 > ∑_{l=1}^{rk(E)} 𝔡^{−nl/2} · λ^{nl} (\frac{K^{∗}O_{𝔸}^{nl}}{K^{∗}} ) · ζ_{E}^{(l)}(n) exp(l deg(L)),

then there is an Arakelov bundle F of rank n and determinant L such that

Γ(E ⊗ F) = 0.

Proof: Note that any global section of E ⊗ F is already a global section of E′ ⊗ F for a unique minimal subbundle E′ ⊆ E, namely the subbundle whose generic fibre is the image of the induced map (F_{K})^{dual} → E_{K}. We are going to average the number of these sections (up to K^{∗}) for a fixed subbundle E′ of rank l. Fix one particular Arakelov bundle F of rank n and determinant L. Choose linear isomorphisms ϕ_{E′} : K^{l} → E′_{K} and ϕ_{F} : K^{n} → F_{K} and let

ϕ : Mat_{n×l}(K) ⟶^{∼} (E′ ⊗ F)_{K}

be their tensor product. Our notation will not distinguish these maps from their canonical extensions to completions or adeles. For each g ∈ Sl_{n}(𝔸), we denote by gF the Arakelov bundle corresponding to the K-lattice ϕ_{F}(gK^{n}) ⊆ F_{𝔸}. More precisely, gF is the unique Arakelov bundle satisfying (gF)_{𝔸} = F_{𝔸}, (gF)_{O_{𝔸}} = F_{O_{𝔸}} and (gF)_{K} = ϕ_{F}(gK^{n}). This gives the usual identification between Sl_{n}(𝔸)/Sl_{n}(K) and the space of Arakelov bundles of rank n and fixed determinant together with local trivialisations. Observe that the generic fibre of E′ ⊗ gF is ϕ(gMat_{n×l}(K)). A generic section is not in E′′ ⊗ gF for any E′′ ⊊ E′ if and only if the corresponding matrix has rank l. So according to the mean value formula of section 2, the average number of global sections

∫_{Sl_{n}(𝔸)/Sl_{n}(K)} card (\frac{K^{∗}Γ(E′ ⊗ gF)}{K^{∗}} \ ⋃_{E′′⊊E′} \frac{K^{∗}Γ(E′′ ⊗ gF)}{K^{∗}} ) dτ(g)

Documenta Mathematica 8 (2003) 115–123

is equal to the integral

𝔡^{−nl/2} ∫_{Mat_{n×l}(𝔸)} (f_{K} ∘ div_{E′⊗F} ∘ ϕ) dλ^{n×l}.  (2)

Here the function f_{K} : ℝ_{≥0}^{X} → ℝ_{≥0} is defined by

f_{K}(D) := \{ \begin{matrix}1/card\{a ∈ K^{∗} : div(a) · D ≤ 1\} & if D ≤ 1 \\ 0 & otherwise \end{matrix}

with the convention 1/∞ = 0. In order to compute (2), we start with the local transformation formula

λ_{v}^{n×l} ({M ∈ Mat_{n×l}(K_{v}) : c_{1} ≤ ∥ϕ(M)∥_{E′⊗F,v} ≤ c_{2}}) =
λ_{v}^{nl} ({M ∈  K_{v}^{nl}  : c_{1} ≤  ∥M∥  ≤ c_{2}}) · ∥ det(ϕ)∥_{det(E′⊗F),v}^{−1}

for all c_{1}, c_{2} ∈ ℝ_{≥0}. Regarding this as a relation between measures on ℝ_{≥0} and taking the product over all places v ∈ X, we get the equation

(div_{E′⊗F} ∘ ϕ)_{∗}λ^{n×l} = exp deg(E′ ⊗ F) · (div_{O^{nl}})_{∗}λ^{nl}  (3)

of measures on ℝ_{≥0}^{X}. Hence the integrals of f_{K} with respect to these measures also coincide:

∫_{Mat_{n×l}(𝔸)} (f_{K} ∘ div_{E′⊗F} ∘ ϕ) dλ^{n×l} = exp(n deg(E′) + l deg(F)) · λ^{nl} (\frac{K^{∗}O_{𝔸}^{nl}}{K^{∗}} ) .

We substitute this for the integral in (2). A summation over all nonzero subbundles E′ ⊆ E yields

∫_{Sl_{n}(𝔸)/Sl_{n}(K)} card (\frac{K^{∗}Γ(E ⊗ gF) \ 0}{K^{∗}}) dτ(g) =

= ∑_{l=1}^{rk(E)} 𝔡^{−nl/2} · ζ_{E}^{(l)}(n) exp(l deg(F)) · λ^{nl} (\frac{K^{∗}O_{𝔸}^{nl}}{K^{∗}} ) .

But the right hand side was assumed to be less than one, so there there has to be a g ∈ Sl_{n}(𝔸) with Γ(E ⊗ gF) = 0. □

In order to apply this theorem, one needs to compute λ^{N}(K^{∗}O_{𝔸}^{N}/K^{∗}) for N ≥ 2. We start with the special case K = ℚ. Here each adele a ∈ O_{𝔸}^{N} outside a set of measure zero has a rational multiple in O_{𝔸}^{N} with valuation one at all finite places, and this multiple is unique up to sign. Hence we conclude

λ^{N} (\frac{ℚ^{∗}O_{𝔸}^{N}}{ℚ^{∗}} ) = \frac{V_{N}}{2} · ∏_{p prime} λ_{p}^{N}(ℤ_{p}^{N} \ pℤ_{p}^{N}) = \frac{V_{N}}{2ζ(N)}.

In particular, the special case K = ℚ and E = O of the theorem above is precisely the Minkowski-Hlawka existence theorem for sphere packings [8], §15. For a general number field K, we note that the roots of unity preserve O_{𝔸}^{N}. Then we apply Stirling’s formula to the factorials occurring via the unit ball volumes and get

λ^{N} (\frac{K^{∗}O_{𝔸}^{N}}{K^{∗}} ) ≤ \frac{λ^{N}(O_{𝔸}^{N})}{w(K)} ≤ (\frac{2πe}{N})^{dN/2} · (\frac{1}{πN} )^{(r_{1}+r_{2})/2} · \frac{1}{2^{r_{2}/2}w(K)}.

Using such a bound and the asymptotic statement 3.3 about ζ_{E}^{(l)}, one can deduce the following corollary of theorem 4.1.

Corollary 4.2 Let the Arakelov bundle E over X be given. If n is a sufficiently large integer and μ is a real number satisfying

μ_{max}^{(l)}(E) + μ ≤ \frac{d}{2}(log n + log l − log π − 1 − log 2) + \frac{log 𝔡}{2}

for all 1 ≤ l ≤ rk(E), then there is an Arakelov bundle F of rank n and slope larger than μ such that Γ(E ⊗ F) = 0.

If E is semistable, this gives the theorem 0.1 stated in the introduction. Here is some evidence that these bounds are not too far from being optimal:

Proposition 4.3 Assume given ϵ > 0 and a nonzero Arakelov bundle E. Let n > n(ϵ) be a sufficiently large integer, and let μ be a real number such that

μ_{max}^{(l)}(E) + μ ≥ \frac{d}{2}(log n + log l − log π − 1 + log 2 + ϵ) + \frac{log 𝔡}{2}

holds for at least one integer 1 ≤ l ≤ rk(E). Then there is no Arakelov bundle F of rank n and slope μ with Γ(E ⊗ F) = 0.

Proof: Fix such an l and a subbundle E′ ⊆ E of rank l and slope μ_{max}^{(l)}(E). For each F of rank n and slope μ, we consider the Arakelov bundle F′ := E′ ⊗ F of rank nl. By Stirling’s formula, the hypotheses on n and μ imply

exp deg(F′) · λ^{nl}(O_{𝔸}^{nl}) > 2^{nld} · 𝔡^{nl/2}.

Now choose a K-linear isomorphism ϕ : K^{nl} ⟶^{∼} F′_{K} and extend it to adeles. Applying the global transformation formula (3), we get

λ^{nl}(ϕ^{−1}F′_{O_{𝔸}}) > 2^{nld} · λ^{nl}(𝔸^{nl}/K^{nl}).

According to Minkowski’s theorem on lattice points in convex sets (in an adelic version like [11], theorem 3), ϕ^{−1}(F′_{O_{𝔸}}) ∩ K^{nl} ≠ {0} follows. This means that F′ — and hence E ⊗ F — must have a nonzero global section. □

Observe that the lower bound 4.2 and the upper bound 4.3 differ only by the constant d log 2. So up to this constant, the maximal slope of such tensor products without global sections is determined by the stability of E, more precisely by the μ_{max}^{(l)}(E). Taking E = O, we get lower and upper bounds for the maximal slope of Arakelov bundles without global sections, as mentioned in the introduction. In the special case E = O and K = ℚ of lattice sphere packings, [2] states that no essential improvement of corollary 4.2 is known whereas several people have improved the other bound 4.3 by constants.

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Norbert Hoffmann

Mathematisches Institut

Bunsenstraße 3–5 D-37073 Göttingen Germany hoffmann@uni-math.gwdg.de 124

Documenta Mathematica 8 (2003)