By James Mellaart

Thick layers of unburnt plaster

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There is a second rather general approach due to Schmidt [65]. In order to describe this method, we require some notation. When F(x) E Q[Xl," . , x B ] is a form of degree d > 1, write h(F) for the least number h such that F may be written in the form F = AlBl + A2B2 + ... + AhBh, with Ai, Bi forms in Q[x] of positive degree (1 ~ i ~ h). There is an analogous concept for systems of forms which we avoid describing in the interest of saving space. 4 (Schmidt). Let d be an integer exceeding 1, and write X(d) = d'l2 4d d!.

More precisely, a symbol on F is a bimultiplicative map to an abelian group A (here written multiplicatively) such that c(x, 1 - x) = 1, x 1= 0,1. By Matsumoto, there is a (universal) symbol (x,y) {x,y} ~ such that composition gives a bijection { hOmOmOrPhism} K 2 (F) ---+ A. t---+ { symbol on F } with values in A . In other words, any symbol c on F with values in A is ofthe form c(x, y) for a unique homomorphism ¢: K 2 (F) ---+ A. = ¢( {x, y} ) Examples. 1) If F = Qp then the Hilbert symbol, with values in ±1, is defined as (a,b)p ={ I if x 2 -1 otherwise - ay2 - bz2 = 0 has a solution in Qp 2) Given a discrete valuation v on F with maximal ideal M and residue field k, Tate defined the tame symbol at v V(lI) (x y) == (_I)v(z)v(lI)_x_ , v yv(z) Note that in particular (x, y)v = 1 if v(x) We will let the corresponding map be 3) The analogue of 1) for F mod M.

Let P E Q[x±1, y±1] be a tempered Laurent polynomial, irreducible over Q and not vanishing on the torus T2. As in nO 8, let C /Q be the normalization of the projective closure of the curve in 'lI' determined by P = 0 and regard x, Y as functions in Q( C). Theorem. With the above notation m(P) = 2~ f({x,Y})(['Yl) for some non-trivial cycle 'Y E Hl (C, Z) fixed by complex conjugation. The homology class ['Y] is related to the class of the torus T in H 2(1P'2 \ C,Z) by the tube homomorphism (see [Grl).

### Catal Huyuk: A Neolithic Town in Anatolia by James Mellaart

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