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New examples of modular rigid Calabi-Yau threefolds. (English) Zbl 1062.14050
A smooth projective threefold \(X\) is called Calabi-Yau if it satisfies the two conditions: (i) the canonical bundle is trivial, and (ii) \(H^1(X,{\mathcal O}_X)=H^2(X,{\mathcal O}_X)=0.\) Furthermore, a Calabi-Yau threefold \(X\) is called rigid if \(H^1(X,{\mathcal T}_X)=0\). Thus, any rigid Calabi-Yau threefold has the third Betti number \(B_3(X)=2\). Assume that \(X\) is defined over \({\mathbb Q}\). Then the natural action of the absolute Galois group \(\text{Gal}(\bar{\mathbb Q}/{\mathbb Q})\) on the etale \(\ell\)-adic cohomology group \(H^3_{\text{ét}}(X,{\mathbb Q}_{\ell})\) induces an \(\ell\)-adic Galois representation \(\rho_X\) of dimension \(2\). Define the \(L\)-series of \(X\) as that of \(H^3_{\text{ét}}(X,{\mathbb Q}_{\ell})\), i.e., as an Euler product \[ L(X,s)=L(H^3_{\text{ét}}(X,{\mathbb Q}_{\ell}),s)=L^*(s)\prod_p (\text{det}({\mathbf 1}-\rho_X(\text{Frob}_p)p^{-s})^{-1} \] where the product runs over the good primes, and \(L^*(s)\) corresponds to the bad primes. Since \(\rho_X(\text{Frob}_p)\) is represented by a \(2\times 2\) matrix in \(GL_2({\mathbb Q}_{\ell})\), the characteristic polynomial has the form: \[ \text{det}({\mathbf 1}-\rho_X(\text{Frob}_p)T)=1-tr\rho_X (\text{Frob}_p)T+ \text{det}\rho_X(\text{Frob}_p)T^2 \in 1+{\mathbb{Z}}[T]. \] The modularity conjecture for rigid Calabi-Yau threefolds over \({\mathbb Q}\) claims that any rigid Calabi-Yau threefolds over \({\mathbb Q}\) is modular, that is, its \(L\)-series coincides up to finitely many Euler factors with the Mellin transform \(L(f,s)\) of a cusp form \(f\) of weight \(4\) on some \(\Gamma_0(N)\), where \(N\) is only divisible by primes of bad reduction. (This precise form of the conjecture was formulated in the paper of M.-H. Saito and N. Yui [J. Math. Kyoto Univ. 41, 403–419 (2001; Zbl 1077.14546)]). The main result of this paper is to give examples of rigid Calabi-Yau threefolds over \({\mathbb Q}\) for which the modularity conjecture holds true. The construction of the rigid Calabi-Yau threefolds discussed here is a generalization of that of C. Schoen [Math. Z. 197, 177–199 (1988; Zbl 0631.14032)]. Let \(S_1(6)\) denote the elliptic modular surface of level \(6\), i.e., the universal elliptic curve over the modular curve \(X_1(6)\). The new examples are twisted self-fiber products of \(S_1(6)\).
Theorem. Let \(\pi\) be a non-trivial automorphism of \({\mathbb{P}}^1\), interchanging \(0,\, 1\) and \(\infty\). Then a small resolution of the twisted self-fiber product \((S_1(6),pr)\times_{{\mathbb{P}}^1} (S_1(6),\pi\circ pr)\) is a modular rigid Calabi-Yau threefold, associated to a newform \(f\) of weight \(4\) and level \(10,17,21\) or \(73\).
The proof follows the standard approach based on Livné’s theorem [R. Livné, in: Current trends in arithmetical algebraic geometry. Proc. Summer Res. Conf. Arcata, Contemp. Math. 67, 247–261 (1987; Zbl 0621.14019)], that is, compare the traces \(\text{tr}\rho_X(\text{Frob}_p)\) with the coefficients of \(f\) for sufficiently many good primes \(p\).

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
11G18 Arithmetic aspects of modular and Shimura varieties
11F11 Holomorphic modular forms of integral weight
11F23 Relations with algebraic geometry and topology
14G35 Modular and Shimura varieties
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