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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 $$p$$.

##### MSC:
 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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