Motivic orthogonal two-dimensional representations of \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\)
The computation of the \(\zeta\)-function of a variety \(V\) can always be carried through, if one motivically can split the cohomology into one-dimensional pieces. The method is essentially to write down a finite number of possibilities for the \(\zeta\)-function and test them by reduction \(\text{mod } p\) until all but one is excluded. In the article under review, the author considers motives of rank two over \(\mathbb{Q}\) with a non-degenerate symmetric pairing, and shows how in this case one may reduce the computation to the first simple case. He applies this to the computation of the \(\zeta\)-function of the diagonal quartic \(\sum X_i= \sum X_i^{- 1}= 0\) K-3 surface, earlier computed by \textit{C. Peters}, \textit{J. Top} and \textit{M. van der Vlugt} [J. Reine Angew. Math. 432, 151-176 (1992; Zbl 0749.14037)], getting a simpler algorithm only needing to reduce for \(p= 2\). The proofs make essential use of \(p\)-adic Hodge theory as explained by \textit{G. Faltings} [J. Am. Math. Soc. 1, No. 1, 255-299 (1988; Zbl 0764.14012)].
motivic orthogonal two-dimensional representations
motives of rank two with a non-degenerate symmetric pairing
scientific article; zbMATH DE number 836159