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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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##### References:
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