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Motivic orthogonal two-dimensional representations of \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\). (English) Zbl 0847.11035
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 C. Peters, J. Top and 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 G. Faltings [J. Am. Math. Soc. 1, No. 1, 255-299 (1988; Zbl 0764.14012)].

11G18 Arithmetic aspects of modular and Shimura varieties
11F70 Representation-theoretic methods; automorphic representations over local and global fields
14G20 Local ground fields in algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
Full Text: DOI
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