COHOMOLOGICAL DIVISIBILITY AND POINT COUNT DIVISIBILITY

HÉLÈNE ESNAULT AND NICHOLAS M. KATZ

Abstract. Let X ⊂ ℙ^{n} be a closed scheme defined by r homogeneous equations of degrees d_{1} ≥ d_{2} ≥ … ≥ d_{r} over the finite field 𝔽_{q}, with complement U := ℙ^{n} \ X. Let κ be the maximum of 0 and the integral part of the rational number \frac{n−d_{2}−...−d_{r}}{d_{1}} . We show that the eigenvalues of the geometric Frobenius acting on the ℓ-adic cohomology H_{c}^{i}(U ×_{𝔽_{q}} \overline{𝔽_{q}}, ℚ_{ℓ}) with compact supports are divisible by q^{κ} as algebraic integers.

1. Introduction

Nearly seventy years have passed since the theorem of Chevalley-Warning [7], [26]: over a finite field k = 𝔽_{q} of characteristic p, if f ∈ k[X_{1}, …, X_{n}] is a polynomial in n ≥ 1 variables of degree d < n, then the number of solutions in 𝔸^{n}(k) of f = 0 is divisible by p. For d ≥ 1, Ax [2] improved this to divisibility by q^{μ}, where μ is the least non-negative integer which is ≥ \frac{n−d}{d}. Thus

(1.1) μ = max(0, Ceiling(\frac{n − d}{d})).

In this form, the result remains true, if trivially so, without the assumption that n > d. Ax also noted that, by an elementary inclusionexclusion argument, his result implied that if one had r ≥ 1 polynomials f_{i} ∈ k[X_{1}, …, X_{n}], with f_{i} of degree d_{i} ≥ 1, then the number of solutions in 𝔸^{n}(k) of the system of equations f_{1} = … = f_{r} = 0 is divisible by q^{λ}, where λ is the least non-negative integer which is ≥ \frac{n−∑_{i} d_{i}}{∑ d_{i}}, i.e.,

(1.2) λ = max(0, Ceiling(\frac{n − ∑_{i} d_{i}}{∑ d_{i}})).

Date: November 10, 2003.

Katz [20] improved this to divisibility by q^{μ}, where μ is the least nonnegative integer which is ≥ \frac{n−∑_{i} d_{i}}{max_{i}d_{i}}, i.e.,

(1.3) μ := μ(n; d_{1}, …, d_{r}) := max(0, Ceiling(\frac{n − ∑_{i} d_{i}}{max_{i}d_{i}})).

An elementary proof of this result was given by Wan [23]. Adolphson and Sperber [1] gave a divisibility estimate for additive character sums which, they showed, included this result. Wan [24] gave a divisibility estimate for multiplicative character sums which, he showed, included this result as well.

del Angel noted in [8] that if one reverses the inclusion-exclusion argument which Ax used to pass from a result for one polynomial to a result for several polynomials, then the estimate (1.3) for several polynomials gives the following result for one polynomial which is reducible: if one has r ≥ 1 polynomials f_{i} ∈ k[X_{1}, …, X_{n}], with f_{i} of degree d_{i} ≥ 1, then the number of solutions in 𝔸^{n}(k) of the single equation f_{1} ⋯ f_{r} = 0 is divisible by q^{μ}, for μ = q^{μ(n;d_{1},...,d_{r})} as above. If one replaces f_{1} ⋯ f_{r} by any product f_{1}^{a_{1}} ⋯ f_{r}^{a_{r}} with all a_{i} ≥ 1, one has the same solutions in 𝔸^{n}(k), so the same divisibility. Notice that if one omits some of the factors in f_{1}^{a_{1}} ⋯ f_{r}^{a_{r}}, i.e., if one allows some a_{i} to be zero, then the number of solutions of f_{1}^{a_{1}} ⋯ f_{r}^{a_{r}} = 0 in 𝔸^{n}(k) remains divisible by the same q^{μ} [simply because if one removes some of the f_{i}’s from consideration, the numerator n − ∑_{i} d_{i} of μ increases, and its denominator max_{i}d_{i} decreases].

With this result in hand, del Angel did inclusion-exclusion a la Ax and got the following result. Suppose one has a list of r ≥ 1 polynomials f_{i} ∈ k[X_{1}, …, X_{n}], with f_{i} of degree d_{i} ≥ 1. Take any R ≥ 1, and any list of R ≥ 1 polynomials g_{j} ∈ k[X_{1}, …, X_{n}], such that g_{j} is a product g_{j} = ∏_{i} f_{i}^{a_{i,j}}, with all a_{i,j} ≥ 0. Then the equations g_{1} = … = g_{R} = 0 define a closed subscheme X ⊂ 𝔸^{n}, whose number of k-points is divisible by q^{μ(n;d_{1},...,d_{r})}.

There is an obvious projective version of these affine results. If the f_{i} are all homogeneous, then all the g_{j} are homogeneous, and the same equations g_{1} = … = g_{R} = 0 also define a closed subscheme X^{proj} ⊂ ℙ^{n−1} whose affine cone is the X above. The number of 𝔽_{q}-rational points of U := ℙ^{n−1} \ X^{proj} is easily seen to be divisible by q^{μ(n;d_{1},...,d_{r})}.

The common feature of these results is that one has a separated kscheme X/k of finite type, and an integer κ ≥ 1, such that for each finite extension 𝔽_{q^{ν}} of 𝔽_{q}, |X(𝔽_{q^{ν}})| is divisible by q^{ν·κ}. Some years ago, Deligne posed the problem of characterizing those situations (X/k, κ) in which the divisibilities of numbers of points result from corresponding divisibilities of all the eigenvalues of Frobenius on all the compact ℓ-adic cohomology groups, ℓ any prime other than p, of the variety X/k in question. As explained by Ax [2], such divisibilities of numbers of points imply (and indeed are equivalent to) the corresponding divisibility of all reciprocal zeros and poles of the zeta function. But for any given ℓ ≠ p, the ℓ-adic cohomological expression of the zeta function may well have cancellation: the problem is that there might be pairs of Frobenius eigenvalues on different ℓ-adic groups which cancel, but which both have insufficient divisibility. An example of such a X is the disjoint union of a point of degree one over 𝔽_{q} with 𝔾_{m} × ℙ^{n−1}. Then is has exactly q^{n} rational points while Frobenius acts trivially on the zeroth and the first cohomology with compact supports.

In this note, we show that if (X/k, κ) arises as an instance of the point count divisibility theorems ([20]) explained above, then in fact all its Frobenius eigenvalues have the corresponding divisibilty.

We should emphasize that in proving the asserted divisibilities of Frobenius eigenvalues, we are not giving a new proof of the above-cited divisibilities of point counts. Rather, those point count divisibilities are an essential starting point of our proof, see (3.6). (For another approach to these questions, see [16], which treats the case of projective complete intersections. For a discussion of a motivic approach, see the survey [5].) In section 5.3, we explain how our methods can be used to sharpen known results on the analogous question of Hodge type of complex projective varieties defined by equations of low degree, cf. [12], [13], [14], [8], [15].

Acknowledgements: We warmly thank Spencer Bloch for his interest in and for his incisive comments on the work leading up to this article.

2. Statement of results: the affine case

We fix a finite field k = 𝔽_{q} of characteristic p and a prime number ℓ ≠ p. For a separated scheme X/k of finite type, we abbreviate

H^{i}(X) := H^{i}(X ×_{k} \overline{k}, ℚ_{ℓ}), H_{c}^{i}(X) := H_{c}^{i}(X ×_{k} \overline{k}, ℚ_{ℓ}). (2.1)

We denote by F = F_{k} the geometric Frobenius endomorphism of these cohomology groups. By the Grothendieck-Lefschetz trace formula [19]

(2.2) |X(k)| = ∑_{i}(−1)^{i}Trace(F; H_{c}^{i}(X)), and

|X(k_{ν})| = ∑_{i}(−1)^{i}Trace(F^{ν}; H_{c}^{i}(X))

for k_{ν} = 𝔽_{q^{ν}}. Recall [10], Corollaire 5.5.3, that all the eigenvalues of F acting on H_{c}^{i}(X) are algebraic integers, i.e. elements of \overline{ℚ_{ℓ}} which are integral over ℤ.

Theorem 2.1. Suppose one has a list of r ≥ 1 polynomials f_{i} ∈ k[X_{1}, …, X_{n}], with f_{i} of degree d_{i} ≥ 1. Take any integer R ≥ 1, and any list of R ≥ 1 polynomials g_{j} ∈ k[X_{1}, …, X_{n}], such that each g_{j} is a product g_{j} = ∏_{i} f_{i}^{a_{i,j}}, with all a_{i,j} ≥ 0. Denote by X/k the closed subscheme of 𝔸^{n}(k) defined by g_{1} = … = g_{R} = 0. Then every eigenvalue of F on every compact cohomology group H_{c}^{i}(X) is divisible as an algebraic integer by q^{μ(n;d_{1},...,d_{r})}.

From the excision sequence

… → H_{c}^{i}(𝔸^{n} \ X) → H_{c}^{i}(𝔸^{n}) → H_{c}^{i}(X) → …, (2.3)

and the well-known cohomological structure of 𝔸^{n}, namely

H_{c}^{i}(𝔸^{n}) = 0 for i ≠ 2n, H_{c}^{2n}(𝔸^{n}) = ℚ_{ℓ}(−n), (2.4)

where ℚ_{ℓ}(−n) is the 1 dimensional vector space over ℚ_{ℓ} on which F acts as q^{n}, one sees that Theorem 2.1 is equivalent to

Theorem 2.2. Under the hypotheses of Theorem 2.1, every eigenvalue of F on every compact cohomology group H_{c}^{i}(𝔸^{n} \ X) is divisible as an algebraic integer by q^{μ(n;d_{1},...,d_{r})}.

Theorem 2.2 itself results from the following slightly sharper result, suggested by [10], Corollaire 5.3.3, (ii), and by [25], [15]. For each integer j ≥ 0, let us define

μ_{j}(n; d_{1}, …, d_{r}) := j + μ(n − j; d_{1}, …, d_{r}). (2.5)

Thus

μ_{0}(n; d_{1}, …, d_{r}) = μ(n; d_{1}, …, d_{r}), (2.6)

and we have the obvious inequalities

μ_{j+1}(n; d_{1}, …, d_{r}) ≥ μ_{j}(n; d_{1}, …, d_{r}). (2.7)

Theorem 2.3. Under the hypotheses of Theorem 2.1, we have the following results.

1) Every eigenvalue of F on every group H_{c}^{i}(𝔸^{n} \ X) is divisible as an algebraic integer by q^{μ(n;d_{1},...,d_{r})}.

2) For each j ≥ 1, every eigenvalue of F on H_{c}^{n+j}(𝔸^{n} \ X) is divisible as an algebraic integer by q^{μ_{j}(n;d_{1},...,d_{r})}.

3. The proof of Theorem 2.3

We first reduce to the case R = 1. For j, with 1 ≤ j ≤ R, we denote by G_{j} ⊂ 𝔸^{n} the hypersurface in 𝔸^{n} defined by g_{j} = 0. Then

X = ∩_{j=1}^{R}G_{j}, hence (3.1)

𝔸^{n} \ X = ∪_{j=1}^{R}U_{j}, with U_{j} = 𝔸^{n} \ G_{j}.

One considers the Mayer-Vietoris spectral sequence for compact cohomology [22], [9] 6.2.10.3,

E_{1}^{−a,b} = ⊕_{1≤j_{1}<j_{2}<...<j_{a+1}≤R}H_{c}^{b}(∩_{i=1}^{a+1}U_{j_{i}}) ⇒ H_{c}^{b−a}(𝔸^{n} \ X), (3.2)

which is the cohomological incarnation of inclusion-exclusion. Thus every eigenvalue of F on a given group H_{c}^{b}(𝔸^{n} \ X) is an eigenvalue of F on some group H_{c}^{b+a}(∩_{i=1}^{a+1}U_{j_{i}}) for some a ≥ 0. So it suffices to show the following two statements:

1) For every b ≥ 0 and every a ≥ 0, every eigenvalue of F on every group H_{c}^{b}(∩_{i=1}^{a+1}U_{j_{i}}) is divisible as an algebraic integer by q^{μ(n;d_{1},...,d_{r})}

2) For every j ≥ 1 and every a ≥ 0, every eigenvalue of F on every group H_{c}^{n+j}(∩_{i=1}^{a+1}U_{j_{i}}) is divisible as an algebraic integer by q^{μ_{j}(n;d_{1},...,d_{r})}.

But an (a + 1)-fold intersection ∩_{i=1}^{a+1}U_{j_{i}} is the complement of the hypersurface of equation ∏_{i=1}^{a+1} g_{j_{i}} = 0, which falls under the case R = 1 of Theorem 2.3. So it suffices to prove Theorem 2.3 for R = 1. We use the weak Lefschetz theorem and induction on n, the number of variables. For n = 1, we have μ(1; d_{1}, …, d_{r}) = 0 (because ∑_{i} d_{i} ≥ 1), so assertion 1) of the theorem is an (easy) instance of Deligne’s integrality theorem [10], Corollaire 5.5.3. Assertion 2) of the theorem holds for n = 1 because the only possibly nonzero group H_{c}^{1+j}(𝔸^{1} \ X) with j ≥ 1 is H_{c}^{2}(𝔸^{1} \ X), which is ℚ_{ℓ}(−1). So suppose the theorem universally true for n − 1, in the case R = 1.

Whatever the nonempty hypersurface G ⊂ 𝔸^{n}, 𝔸^{n} \ G is smooth and geometrically connected of dimension n, so Poincaré duality tells us that the cup product

H_{c}^{i}(𝔸^{n} \ G) × H^{2n−i}(𝔸^{n} \ G) → H_{c}^{2n}(𝔸^{n} \ G) ≅ ℚ_{ℓ}(−n) (3.3)

is a perfect pairing of F -modules. Since 𝔸^{n} \ G is affine of dimension n, the affine Lefschetz theorem tells us that

H^{i}(𝔸^{n} \ G) = 0 for i > n. (3.4)

So by duality we have

H_{c}^{i}(𝔸^{n} \ G) = 0 for i < n. (3.5)

We now take G defined by ∏_{i} f_{i}^{a_{i}} = 0.

We first prove assertion 1) of the theorem. As noted by Wan and exploited in [6], Introduction and in [16], given the divisibility of

|(𝔸^{n} \ G)(k_{ν})| = ∑_{i≥n}(−1)^{i}Trace(F^{ν}; H_{c}^{i}(𝔸^{n} \ G)) (3.6)

by q^{ν·μ} for every ν ≥ 1, it suffices to prove that for all but a single value of i we have the asserted divisibility of Frobenius eigenvalues on H_{c}^{i}(𝔸^{n} \ G). We choose i = n. So we need only show the divisibility by q^{μ(n;d_{1},...,d_{r})} of Frobenius eigenvalues on H_{c}^{j}(𝔸^{n} \ G) for j > n.

This divisibility is invariant under finite extension of the ground field; after a degree d extension of the ground field, both eigenvalues and q are replaced by their d-th power. For a sufficiently general affine linear hyperplane A ⊂ 𝔸^{n} defined by an equation of the form

A : ∑_{i=1}^{n} α_{i}X_{i} + β_{i} = 0 (3.7)

with coefficients in a finite extension of k, (which, by extending k, we may assume to be k itself), the restriction maps on ordinary cohomology

H^{i}(𝔸^{n} \ G) → H^{i}(A \ A ∩ G) (3.8)

are injective for i < n. This is an instance of the weak Lefschetz theorem (see [21], 3.4.1, applied to V = 𝔸^{n} \ G, with π the inclusion and f the function 0.) The Poincaré dual of this injection is a surjection

H_{c}^{i−2}(A \ A ∩ G)(−1) ↠ H_{c}^{i}(𝔸^{n} \ G) (3.9)

for i > n. For A general, each f_{i}|_{A} will be a polynomial of the same degree d_{i}. So by induction on n, we know that every eigenvalue of F on every group H_{c}^{i−2}(A \ A ∩ G) is divisible by q^{μ(n−1;d_{1},...,d_{r})}. So every eigenvalue of F on every group H_{c}^{i−2}(A \ A ∩ G)(−1) is divisible by q^{μ(n−1;d_{1},...,d_{r})+1}. But we have the inequality

μ(n − 1; d_{1}, …, d_{r}) + 1 = μ_{1}(n; d_{1}, …, d_{r}) ≥ μ(n; d_{1}, …, d_{r}). (3.10)

This concludes the proof of assertion 1) of Theorem 2.3.

This same induction proves assertion 2) of Theorem 2.3. Indeed, for any integer j ≥ 1, the induction shows that for i = n + j, every eigenvalue of F on H_{c}^{n+j}(𝔸^{n} \ G) is also an eigenvalue of F on

H^{(n−1)+(j−1)}(A \ G ∩ A)(−1). (3.11)

By induction, every eigenvalue of F on (3.11) is divisible by q^{1+μ_{j−1}(n;d_{1},...,d_{r})}. And by definition we have

1 + μ_{j−1}(n; d_{1}, …, d_{r}) = μ_{j}(n; d_{1}, …, d_{r}). (3.12)

This concludes the proof of Theorem 2.3.

4. The projective case

Here is the projective version. Suppose now that the f_{i} are all homogeneous forms. Then each g_{j} is again homogeneous, and the closed X/k in 𝔸^{n} defined by g_{1} = … = g_{R} = 0 is the affine cone over a closed X^{proj} in ℙ^{n−1}, defined by the same equations.

Theorem 4.1. Under the hypotheses of Theorem 2.1, suppose in addition all the f_{i} are homogeneous. Then we have the following results.

1) Every eigenvalue of F on every group H_{c}^{i}(ℙ^{n−1} \ X^{proj}) is divis-

ible as an algebraic integer by q^{μ(n;d_{1},...,d_{r})}.

2) For each j ≥ 1, every eigenvalue of F on H_{c}^{n+j}(ℙ^{n−1} \ X^{proj}) is

divisible as an algebraic integer by q^{μ_{j}(n;d_{1},...,d_{r})}.

Proof. To prove the theorem, repeat the proof of Theorem 2.3 essentially verbatim. First reduce to the case R = 1 by the same spectral sequence argument. Then do induction on n. Again, the case n = 1 holds trivially. The open variety ℙ^{n−1}\X^{proj} is affine, smooth, and geometrically connected of dimension n − 1. The appropriate reference for the weak Lefschetz theorem in this context is [21] Appendix (d’après Deligne), A5, applied with its X := ℙ^{n−1} \ X^{proj}, and f the inclusion into ℙ^{n−1}. □

5. Concluding remarks

5.1. Concordance. In other work on this subject, e.g. in [13], [14], [15], one sees a different but equivalent expression of the divisibility, in terms of the floor, or integral part [x] of a real number x. One renumbers so that d_{1} ≥ d_{2} ≥ … ≥ d_{r}, and then one defines

κ(n; d_{1}, …, d_{r}) := max(0, [\frac{n − d_{2} − … − d_{r}}{d_{1}} ]). (5.1)

This κ is related to μ by the identity

κ(n; d_{1}, …, d_{r}) = μ(n + 1; d_{1}, …, d_{r}), (5.2)

whose verification we leave to the reader.

5.2. Axiomatization. The arguments we give here could easily be axiomatized, and then applied to any pair of cohomology theories (H^{⋆}, H_{c}^{⋆}), defined on smooth separated \overline{k}-schemes of finite type, with values in finite-dimensional vector spaces over a field K of characteristic zero, with the following properties:

i) zeta function formula for H_{c}^{⋆}, ii) Poincaré duality between H^{⋆} and H_{c}^{⋆}, iii) affine Lefschetz and weak Lefschetz theorems for H^{⋆}, iv) Mayer-Vietoris spectral sequence of an open covering for H_{c}^{⋆} .

An initial run through our arguments with μ = 0 shows that all Frobenius eigenvalues on H_{c}^{⋆} in any such theory are algebraic integers. Berthelot’s rigid cohomology and rigid cohomology with compact supports are conjecturally such a pair of theories. At present, (i), (ii) and (iv) are proven in [17], [4] and [3] respectively. Only (iii) remains open. Once (iii) has been proven, then Theorems 2.3 and 4.1 will hold for the eigenvalues of Frobenius on compact rigid cohomology, with divisibility by q^{μ} as algebraic integers and hence with slopes of Frobenius ≥ μ.

5.3. Application to Hodge type. Deligne has also formulated a Hodge theoretic version of the problem considered here. Given a nonempty separated ℂ-scheme X/ℂ of finite type, recall that its compact support Hodge type α(X/ℂ) is the largest integer α such that the Hodge filtration Fil, for the mixed Hodge structure on H_{c}^{⋆}(X), satisfies

Fil^{α}H_{c}^{⋆}(X) = H_{c}^{⋆}(X). (5.3)

We can also ”spread out” X/ℂ to a separated scheme X /A of finite type, with A a subring of ℂ which is finitely generated as a ℤ-algebra. For each such spreading out X /A, we define β(X /A) to be the largest integer β with the following property: for every finite field 𝔽_{q}, and for every ring homomorphism ϕ : A → 𝔽_{q}, the number of points on the scheme X_{ϕ}/𝔽_{q} obtained from X /A by the extension of scalars ϕ is divisible by q^{β}. We then define β(X/ℂ) to be the maximum of β(X /A) for all spreadings out of X/ℂ. One sees easily that β(X/ℂ) ≤ dim(X). The problem is to characterize those X/ℂ for which α(X/ℂ) = β(X/ℂ). [Presumably, when this equality holds, it holds because α(X/ℂ) and β(X/ℂ) are both equal to some intrinsic ”motivic” invariant γ(X/ℂ), cf. [18], sections 9-10 for Grothendieck’s discussion of this sort of question.] In general, not much is known. There are simple cases where this equality fails, e.g. the disjoint union of a rational point with the product 𝔾_{m} × ℙ^{n−1}. But in cases where the divisibility theorems of Ax et al give an estimate β(X/ℂ) ≥ κ, one knows that α(X/ℂ) ≥ κ, cf. [12], [14], [8], [15].

The methods we use here allow us to sharpen these last results. Given the known results on the Hodge type of hypersurfaces as input ([12], [13]), the arguments proving Theorems 2.3 and 4.1 also prove their analogues over ℂ, with eigenvalue divisibility replaced by Hodge type. We should emphasize that we are not giving a new proof of the known results on the Hodge type of hypersurfaces, but rather using them as a starting point.

Consider first the projective case. The first assertion of the Hodge analogue of Theorem 4.1 is the result [12], [13], [14], [8]. To prove the second assertion, reduce to the R = 1 case by the spectral sequence, and then use induction and the weak Lefschetz theorem exactly as in the finite field case. Consider now the affine case, i.e., the Hodge analogue of Theorem 2.3. The spectral sequence reduces us to the case R = 1. Once we know the first assertion in the case R = 1, we get the second assertion in that case by using induction and the affine weak Lefschetz theorem, exactly as in the finite field case. To prove the first assertion in the R = 1 case, we reduce it to the projective case. For this reduction, for each f_{i} ∈ ℂ[X_{1}, …, X_{n}], with f_{i} of degree d_{i} ≥ 1, denote by F_{i} ∈ ℂ[X_{0}, …, X_{n}] the homogeneous form of the same degree d_{i} such that

f_{i}(X_{1}, …, X_{n}) = F_{i}(1, X_{1}, …, X_{n}), (5.4)

and denote by F_{i,0} ∈ ℂ[X_{1}, …, X_{n}] the leading form of f_{i}, i.e.,

F_{i,0}(X_{1}, …, X_{n}) = F_{i}(0, X_{1}, …, X_{n}). (5.5)

Given G the hypersurface in 𝔸^{n} defined by ∏_{i} f_{i}^{a_{i}} = 0, denote by G^{proj} the hypersurface in ℙ^{n} defined by ∏_{i} F_{i}^{a_{i}} = 0, and denote by G_{0}^{proj} the hypersurface in ℙ^{n−1} defined by ∏_{i} F_{i,0}^{a_{i}} = 0. Then we have an excision sequence

… → H_{c}^{i−1}(ℙ^{n−1} \ G_{0}^{proj}) → H_{c}^{i}(𝔸^{n} \ G) → H_{c}^{i}(ℙ^{n} \ G^{proj}) → …. (5.6)

But H_{c}^{⋆}(ℙ^{n−1} \ G_{0}^{proj}) has Hodge type at least μ(n; d_{1}, …, d_{r}), and H_{c}^{⋆}(ℙ^{n} \ G^{proj}) has Hodge type at least

μ(n + 1; d_{1}, …, d_{r}) ≥ μ(n; d_{1}, …, d_{r}). (5.7)

Thus H_{c}^{⋆}(𝔸^{n} \ G) has Hodge type at least μ(n; d_{1}, …, d_{r}), as required.

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Universität Essen, FB6, Mathematik, 45117 Essen, Germany

E-mail address: esnault@uni-essen.de

Princeton University, Mathematics, Fine Hall, NJ 08544-1000, USA

E-mail address: nmk@math.princeton.edu