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Motives and mirror symmetry for Calabi-Yau orbifolds. (English) Zbl 1167.14023

Yui, Noriko (ed.) et al., Modular forms and string duality. Proceedings of a workshop, Banff, Canada, June 3–8, 2006. Providence, RI: American Mathematical Society (AMS); Toronto: The Fields Institute for Research in Mathematical Sciences (ISBN 978-0-8218-4484-7/hbk). Fields Institute Communications 54, 3-46 (2008).
The authors use motivic methods to analyse mirror symmetry for certain Calabi-Yau threefolds \(V\) defined by weighted Fermat equations in \(\mathbb{P}^4\). The main result is a one-to-one correspondence between motivic factors and so-called monomials for mirror pairs of such Calabi-Yau threefolds. This is established by counting the number of rational points for the reduction of \(Y\) at a prime number.
For the entire collection see [Zbl 1147.11005].

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F80 Galois representations
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