Integral Homology of PGL2 over Elliptic Curves

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(BGk,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 PGL2 over an elliptic curve E. We assume that E is defined by the equation F (x, y) = 0, where F (x, y) = y + a1xy + a3y − x 3 − a2x 2 − a4x− a6, and the ai lie in an infinite field k. Denote by 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 k for which F (l, ω) = 0. Our main result is the following.