MOTIVIC L-FUNCTION IDENTITIES FROM CFT AND ARITHMETIC MIRROR SYMMETRY

Abstract

Exactly solvable mirror pairs of Calabi–Yau threefolds of hypersurface type exist in the class of Gepner models that include nondiagonal affine invariants. Motivated by the string theoretic automorphy established previously for models in this class, it is natural to ask whether the arithmetic structure of mirror pairs varieties reflects the fact that as conformal field theories, they are isomorphic. Mirror symmetry in particular predicts that the L-functions of the Ω-motives of such pairs are identical. In this paper this prediction is confirmed by showing that the Ω-motives of exactly solvable mirror pairs are isomorphic. This follows as a corollary of the proof of a more general result establishing an isomorphism between nondiagonally and diagonally induced motives in this class of varieties. The motivic approach formulated here circumvents the difficulty that no mirror construction of the Hasse–Weil zeta function is known.