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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)].

MSC:
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
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[1] [BPV] W. Barth, C. Peters and A. Van de Ven,Compact Complex surfaces, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984. · Zbl 0718.14023
[2] [Fa] G. Faltings,p-adic Hodge theory, Journal of the American Mathematical Society1 (1988), 255–299. · Zbl 0764.14012
[3] [L] R. Livné,Cubic exponential sums and Galois representations, inCurrent Trends in Arithmetic Algebraic Geometry, Contemporary Mathematics, Volume 67, AMS, 1987.
[4] [PTV] C. Peters, J. Top and M. van der Vlugt,The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes, Journal für die reine und angewandte Mathematik432 (1992), 151–176. · Zbl 0749.14037
[5] [Dur] J.-P. Serre,Modular forms of weight one and Galois representations, inAlgebraic Number Fields (A. Frohlich, ed.), Academic Press, New York, 1977, pp. 193–297.
[6] [ALR] J.-P. Serre,Abelian -Adic Representations and Elliptic Curves, Benjamin, New York, Amsterdam, 1968.
[7] [S-I] T. Shioda and H. Inose,On singular K3 surfaces, inComplex and Algebraic Geometry, Cambridge University Press, 1977, pp. 117–136.
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