A Gauss-Bonnet theorem for motivic cohomology
Let C be a smooth complete irreducible curve over \({\mathbb{F}}_ q\) with function field L. Let E be a smooth complete irreducible curve over L. Then there exists a regular curve \({\mathcal E}\) proper flat and of finite type over C with generic fibre L-isomorphic to E. \({\mathcal E}\) is a smooth projective surface over \({\mathbb{F}}_ q\). Assume the \({\mathbb{F}}_ q\)- rational projection p: \({\mathcal E}\to C\) has an \({\mathbb{F}}_ q\)-rational section \(\sigma: C\to {\mathcal E}\). The fibration p: \({\mathcal E}\to C\) with section \(\sigma\) gives rise to a constant M called the covolume. It is constructed by means of the Tamagawa measures (for the places v of L) of suitable quotients of the \(SL_ 2(L_ v)\). M can be calculated: \[ M=(\log q)\zeta (C,2)^{-1}\cdot \prod_{v}K_ v \cdot Res_{s=2}\zeta ({\mathcal E},s), \] where the product runs over the \(v\in C\) such that \(p^{-1}(v)\) is singular, and \(K_ v=\zeta_{k({\mathcal E}_{v,\sigma})}(2)\zeta ({\mathcal E}_ v,2)^{-1}\). Here k(\({\mathcal E}_{v,\sigma})\) is the residue field of the local ring of the component of the fibre \({\mathcal E}_ v\) of p: \({\mathcal E}\to C\) over v containing \(\sigma\) v. On the other hand, the arithmetic Euler-Poincaré characteristic \(\chi\) (\({\mathcal E},{\mathbb{Z}})\) of \({\mathcal E}\) is defined by \[ \chi ({\mathcal E},{\mathbb{Z}})=\frac{[H^ 0({\mathcal E},{\mathbb{Z}})_{tor}][H^ 2({\mathcal E},{\mathbb{Z}})_{\cot or}][H^ 4({\mathcal E},{\mathbb{Z}})]}{[H^ 1({\mathcal E},{\mathbb{Z}})][H^ 3({\mathcal E},{\mathbb{Z}})][H^ 5({\mathcal E},{\mathbb{Z}})]}, \] where \(H^ i({\mathcal E},{\mathbb{Z}})\) is étale cohomology and [G] denotes the order of the (finite) group G. \(\chi\) (\({\mathcal E},{\mathbb{Z}})\) is an intrinsic invariant of the surface \({\mathcal E}.\) The Gauß-Bonnet theorem takes the form: \(\chi\) (\({\mathcal E},{\mathbb{Z}})=Mq^{-\chi}\zeta (C,2)\cdot \prod_{v}K_ v^{-1} \), where \(\chi\) is the self-intersection number of the diagonal on \({\mathcal E}\times {\mathcal E}\). The interest with respect to motivic cohomology comes from the fact that several terms in the Gauß-Bonnet formula have interpretations in K-theory. - The proof of the theorem relies on the calculated value of M and, via the functional equation, on a result, due to J. S. Milne, which relates the Euler-Poincaré characteristic to the residue of the zeta-function of the surface at \(s=0\). In case E is an elliptic curve over L and p: \({\mathcal E}\to C\) is the complete Néron model of E, such that the fibres of p are semistable, the formula is worked out explicitly.
arithmetic Euler-Poincaré characteristic
scientific article; zbMATH DE number 4200370