Proceedings of Symposia in Pure Mathematics

Integral Homology of P GL_{2} over Elliptic Curves

Kevin P. Knudson

The Friedlander–Milnor Conjecture [1] asserts that if G is a reductive algebraic group over an algebraically closed field k, then the comparison map

H_{ét}^{•}(BG_{k}, ℤ/p) ⟶ H^{•}(BG, ℤ/p)

is an isomorphism for all primes p not equal to the characteristic of k. Gabber’s rigidity theorem [2] implies that this map is indeed an isomorphism for the stable general linear group GL (this is due to Suslin [6] for k = ℂ and to Jardine [3] for arbitrary k). Similarly, a proof of an unstable version of rigidity would lead to a proof of the unstable Friedlander–Milnor Conjecture.

In this note we consider unstable rigidity for the group P GL_{2} over an elliptic curve E. We assume that E is defined by the equation F (x, y) = 0, where

F (x, y) = y^{2} + a_{1}xy + a_{3}y − x^{3} − a_{2}x^{2} − a_{4}x − a_{6},

and the a_{i} lie in an infinite field k. Denote by \overline{E} the projective curve E ∪{∞}. Denote by A the coordinate ring of the affine curve E. If l ∈ k and F (l, y) = 0 has no rational solutions, denote by k(ω) the quadratic extension of k inside the algebraic closure \overline{k} for which F (l, ω) = 0. Our main result is the following.

Theorem. For all i ≥ 1,

H_{i}(P GL_{2}(A), ℤ) = ⨁_{\begin{matrix}p∈\overline{E} \\ 2p=0 \end{matrix}} H_{i}(P GL_{2}(k), ℤ) ⊕ ⨁_{\begin{matrix}p∈\overline{E},2p≠0 \\ p∼−p \end{matrix}} H_{i}(k^{×}, ℤ)

⊕ ⨁_{l∈k,F (l,y)=0 has no rat. sol.} H_{i}(k(ω)^{×}/k^{×}, ℤ).

Here, p ∼ −p means that we identify the factors corresponding to the points p and −p.

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

1991 Mathematics Subject Classification. 20G10.

Supported by an NSF Postdoctoral Fellowship, grant no. DMS–9627503.

Figure 1. F (l, y) = 0 has no rational solutions

Figure 2. F (l, y) = 0 has a unique rational solution

This is proved by examining the action of P GL_{2}(A) on the Bruhat–Tits tree X associated to a two-dimensional vector space over the function field of E (see, e.g., Serre’s book [5]). The quotient graph P GL_{2}(A)\X is a tree (this is due to S. Takahashi [8]) and the various simplex stabilizers are easily described.

As a corollary, we have the following rigidity result.

Corollary. Suppose the field k is algebraically closed and let x, y be distinct points on \overline{E}. Then the specialization homomorphisms

s_{x}, s_{y} : H_{•}(P GL_{2}(A), ℤ) ⟶ H_{•}(P GL_{2}(k), ℤ)

coincide.

1. The Quotient and the Stabilizers

In [8], Takahashi described a fundamental domain for the P GL_{2}(A)- action on X; denote this subtree by D. There is a distinguished vertex o ∈ D. For each l in k ∪ {∞} there is a vertex v(l) adjacent to o. The rest of D may be described as follows. Let D(l) denote the subtree of D−{o} which contains v(l). The tree D is the union of o and the various D(l) (which are disjoint). The trees D(l) are as follows.

(1) Suppose F (x, y) = 0 has no rational solution with x = l. Then D(l) consists only of v(l) (see Figure 1).

(2) Suppose l = ∞ or F (x, y) = 0 has a unique rational solution with x = l. Let p be the point at infinity of E or the rational point corresponding to the solution. Note that p is a point of order 2. Then D(l) consists of an infinite path c(p, 1), c(p, 2), … and an extra vertex e(p) (see Figure 2).

(3) Suppose F (x, y) = 0 has two different solutions such that x = l. Let p, q be the corresponding points on E. Then D(l) consists of two infinite paths c(p, 1), c(p, 2), … and c(q, 1), c(q, 2), … (see Figure 3).

The infinite path c(p, 1), c(p, 2), … is called a cusp. Note that there is a one-to-one correspondence between cusps and the rational points of \overline{E}.

Since X is contractible, we have a spectral sequence with E^{1}–term

E_{p,q}^{1} = ⨁_{σ^{(p)}⊂D} H_{q}(Γ_{σ}, ℤ) ⟹ H_{p+q}(P GL_{2}(A), ℤ)

HOMOLOGY OF P GL_{2} 3

Figure 3. F (l, y) = 0 has two distinct solutions

where Γ_{σ} is the stabilizer of the p–simplex σ in P GL_{2}(A).

The various stabilizers of the GL_{2}(A) action were described in [8]. Denote these by Γ̃_{σ}.

Proposition 1.1 (cf. [8],Theorem 5). The stabilizers Γ̃_{σ} are (up to isomorphism)

Γ̃_{o} ≅ k^{×}

Γ̃_{v(l)} ≅ \begin{cases}k(ω)^{×} & in case (1) \\ k^{×} × k & in case (2) \\ k^{×} × k^{×} & in case (3) \end{cases}

Γ̃_{e(p)} ≅ GL_{2}(k)
Γ̃_{c(p,n)} ≅ (k^{n} ×_{θ} k^{×}) × k^{×}

where k^{n} ×_{θ} k^{×} is the semidirect product of k^{n} and k^{×} and θ is the automorphism of k^{n} given by coordinate-wise multiplication by elements of k^{×}. Furthermore, the stabilizer of an edge is the intersection of the stabilizers of its vertices.

The groups Γ̃_{c(p,n)} are of the form

\{( \begin{matrix}p & q \\ 0 & s \end{matrix} ) : p, s ∈ k^{×}, q ∈ k^{n}\}.

Denote the diagonal subgroups of these Γ̃ by L̃. By Theorem 1.11 of [4], these stabilizers satisfy

H_{•}(Γ̃, ℤ) ≅ H_{•}(L̃, ℤ),

the isomorphism being induced by the inclusion L̃ → Γ̃.

Corollary 1.2. The stabilizers Γ_{σ} satisfy

H_{•}(Γ_{o}, ℤ) ≅ H_{•}({1}, ℤ)

H_{•}(Γ_{v(l)}, ℤ) ≅ \begin{cases}H_{•}(k(ω)^{×}/k^{×}, ℤ) & in case (1) \\ H_{•}(k, ℤ) & in case (2) \\ H_{•}(k^{×}, ℤ) & in case (3) \end{cases}

H_{•}(Γ_{c(p,n)}, ℤ) ≅ H_{•}(k^{×}, ℤ)
H_{•}(Γ_{e(p)}, ℤ) ≅ H_{•}(P GL_{2}(k), ℤ).

2. The Main Theorem

Note that our spectral sequence consists of two columns and that each row E_{•,q}^{1} is the chain complex C_{•}(D, H_{q}), where H_{q} is the coefficient system σ ↦ H_{q}(Γ_{σ}). Fix a positive integer q ≥ 1. Note that H_{q}(Γ_{o}, ℤ) = 0 and that H_{q}(Γ_{e}, ℤ) = 0 for any edge incident with o. It follows that the chain complex C_{•}(D, H_{q}) is a direct sum of chain complexes

C_{•}(D, H_{q}) = ⨁_{l∈k∪{∞}} C_{•}(D(l), H_{q}).

Proposition 2.1. Suppose F (x, y) = 0 has no rational solutions with x = l. Then

H_{i}(D(l), H_{q}) = \begin{cases}H_{q}(k(ω)^{×}/k^{×}, ℤ) & i = 0 \\ 0 & i > 0. \end{cases}

Proof. This is clear since C_{•}(D(l), H_{q}) consists only of the single group H_{q}(k(ω)^{×}/k^{×}, ℤ) sitting in degree zero.

Proposition 2.2. Suppose l = ∞ or F (x, y) = 0 has a unique rational solution with x = l. Then

H_{i}(D(l), H_{q}) = \begin{cases}H_{q}(P GL_{2}(k), ℤ) & i = 0 \\ 0 & i > 0. \end{cases}

Proof. The stabilizer of v(l) and of the edge joining v(l) to c(p, 1) is isomorphic to k. The map Γ_{⟨v(l),c(p,1)⟩} → Γ_{v(l)} is an isomorphism on homology and the map Γ_{⟨v(l),c(p,1)⟩} → Γ_{c(p,1)} induces the zero map on homology. It follows that H_{•}(D(l), H_{q}) ≅ H_{•}(D(l)′, H_{q}) where D(l)′ is the tree obtained by deleting v(l) and the edge joining it to c(p, 1). One checks easily that the map

C_{1}(D(l), H_{q}) ⟶ C_{0}(D(l), H_{q})

is injective with cokernel H_{q}(P GL_{2}(k), ℤ) (or equivalently, check that the relative homology groups H_{•}(D(l), e(p); H_{q}) vanish).

Proposition 2.3. Suppose F (x, y) = 0 has two distinct solutions with x = l. Then

H_{i}(D(l), H_{q}) = \begin{cases}H_{q}(k^{×}, ℤ) & i = 0 \\ 0 & i > 0. \end{cases}

Proof. In this case, D(l) is a tree and the stabilizer of each vertex and each edge is k^{×}. The maps Γ_{e} → Γ_{v} induce isomorphisms on homology for each edge and vertex. It follows that C_{•}(D(l), H_{q}) is a chain complex with constant coefficients. Since D(l) is contractible, the result follows.

Theorem 2.4. For all i ≥ 1,

H_{i}(P GL_{2}(A), ℤ) ≅ ⨁_{\begin{matrix}p∈\overline{E} \\ 2p=0 \end{matrix}} H_{i}(P GL_{2}(k), ℤ) ⊕ ⨁_{\begin{matrix}p∈\overline{E},2p≠0 \\ p∼q \end{matrix}} H_{i}(k^{×}, ℤ)

⊕ ⨁_{l∈k,F (l,y)=0 has no solutions} H_{i}(k(ω)^{×}/k^{×}, ℤ).

Proof. Note that since D is contractible, E_{0,0}^{2} = ℤ and E_{1,0}^{2} = 0. The preceding propositions show that

H_{0}(D, H_{q}) = ⨁_{l∈k∪{∞}} H_{0}(D(l), H_{q})

and H_{1}(D, H_{q}) = 0. It remains to identify the various direct summands with points of \overline{E}.

Those l for which F (l, y) = 0 has a unique solution (or l = ∞) correspond to points of order 2 in \overline{E}. Those l for which F (l, y) = 0 has two distinct solutions correspond to pairs of points on \overline{E}. For such a pair p, q, the groups H_{i}(k^{×}, ℤ) arising from the cusps associated to p, q are identified together since they are both adjacent to v(l). The claimed direct sum decomposition follows.

3. Rigidity

Suppose now that the field k is algebraically closed. Note that the projective curve \overline{E} is isomorphic to the group Pic^{0}\overline{E} of degree zero line bundles on \overline{E}. Moreover, after a suitable linear change of coordinates, the points p, q corresponding to the two solutions of F (l, y) = 0 may be assumed to satisfy q = −p.

Corollary 3.1. If k is algebraically closed, then for all i ≥ 1,

H_{i}(P GL_{2}(A), ℤ) ≅ ⨁_{\begin{matrix}L∈Pic^{0}\overline{E} \\ 2L=0 \end{matrix}} H_{i}(P GL_{2}(k), ℤ) ⊕ ⨁_{\begin{matrix}L∈Pic^{0}\overline{E},2L≠0 \\ L∼−L \end{matrix}} H_{i}(k^{×}, ℤ).

Proof. This is obvious once one notes that the factors H_{i}(k(ω)^{×}/k^{×}, ℤ) do not enter the picture when k is algebraically closed.

As a consequence, we have the following rigidity result.

Corollary 3.2. Suppose k is algebraically closed and let x, y be distinct points on \overline{E}. Then the corresponding specialization homomorphisms

s_{x}, s_{y} : H_{i}(P GL_{2}(A), ℤ) ⟶ H_{i}(P GL_{2}(k), ℤ)

coincide for all i ≥ 0.

Proof. This follows from the direct sum decomposition of Corollary 3.1. Since the groups which appear as summands do not involve rational functions on \overline{E}, the homomorphisms s_{x} and s_{y} must agree on each summand.

Acknowledgments

I thank Andrei Suslin for pointing out that rigidity follows from the main theorem. I also thank Rick Jardine for many interesting discussions about the Friedlander–Milnor Conjecture.

References

[1] E. Friedlander, G. Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helv. 49 (1984), 347–361.

[2] O. Gabber, K-theory of Henselian local rings and Henselian pairs, Contemp. Math. 126 (1992), 59–70.

[3] J. Jardine, Simplicial objects in a Grothendieck topos, Contemp. Math. 55 (1986), 193–239.

[4] Yu. Nesterenko, A. Suslin, Homology of the full linear group over a local ring, and Milnor’s K–theory, Math. USSR Izvestiya 34 (1990), 121–145.

[5] J.-P. Serre, Trees, Springer–Verlag, Berlin/ Heidelberg, New York, 1980.

[6] A. Suslin, On the K-theory of local fields, J. Pure Appl. Algebra 34 (1984), 301–318.

[7] A. Suslin, Algebraic K-theory of fields, Proc. ICM, Berkeley, 1986.

[8] S. Takahashi, The fundamental domain of the tree of GL(2) over the function field of an elliptic curve, Duke Math. J. 73 (1993), 85–97.

Department of Mathematics, Northwestern University, Evanston, IL 60208

   E-mail address: knudson@math.nwu.edu