Arithmetic of certain Calabi-Yau varieties and mirror symmetry. (Q2768084)

This set of lecture notes serves as an overview of the arithmetic study of Calabi-Yau varieties. The article is divided into three parts; the first part discusses the modularity question for arithmetic Calabi-Yaus. The second part focuses on orbifold Calabi-Yaus and the third part focuses on \(K3\) surfaces and the mirror moonshine phenomenon. NEWLINENEWLINEFor the purposes of this article, a Calabi-Yau variety \(X\) is a smooth projective variety of some dimension \(d\) defined over a field \(K\) such that the cohomology groups \(H^i(X, \mathcal{O}_X)\) vanish for \(0 < i < d\) and the canonical sheaf \(K_X\) is trivial. Calabi-Yau varieties of dimensions 1 and 2 are elliptic curves and \(K3\) surfaces respectively. Since the focus is on the arithmetic of these varieties, we disregard the metric properties of Calabi-Yaus over \(\mathbb{C}\). NEWLINENEWLINEDue to the work of \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 443--551 (1995; Zbl 0823.11029)], \textit{R. Taylor} and \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 553--572 (1995; Zbl 0823.11030)], and \textit{C. Breuil, B. Conrad, F. Diamond} and \textit{R. Taylor} [J. Am. Math. Soc. 14, No. 4, 843--939 (2001; Zbl 0982.11033)], it is known that every elliptic curve \(E\) defined over \(\mathbb{Q}\) is modular. Among other things, this means that the \(L\)-series of the two-dimensional Galois representation \(H^1_{\text{ét}}(X, \mathbb{Q}_l)\) is the Mellin transform of a modular form of weight 2 and level \(N\) equal to the conductor of \(E\). It is conjectured that Calabi-Yau threefolds \(X\) which are rigid (\(b^3 = 2\)) are also modular; that is, the \(L\)-series of \(H^3_{\text{ét}}(X, \mathbb{Q}_l)\) (up to some factors at the primes of bad reduction of \(X\)) is the Mellin transform of a modular form of weight 4 and level \(N\) divisible only by the bad primes of \(X\). If \(X\) is not rigid, we may ask if \(H^3(X, \mathbb{Q}_l)\) splits into motivic factors \(\mathcal{I} \oplus \mathcal{M}\) where the Hodge structure of \(M\) is of type \((3,0) + (0,3)\) and \(h^{3,0}(\mathcal{I}) = 0\). We may then ask if \(M\) corresponds to a modular form as before. This conjecture is essentially a special case of the Fontaine-Mazur conjecture [\textit{J.-M. Fontaine} and \textit{B. Mazur}, Ser. Number Theory 1, 41--78 (1995; Zbl 0839.14011)], which states that all irreducible odd 2-dimensional Galois representations ``coming from geometry'' are modular. Note that we skip the \(K3\) \((d = 2)\) case for now; this is because the presence of algebraic cycles in \(H^2_{\text{ét}}(X, \mathbb{Q}_l)\) requires separate treatment. NEWLINENEWLINEThe article presents examples of modular rigid Calabi-Yaus due to \textit{C. Schoen} [J. Reine. Angew. Math. 364, 85--111 (1986; Zbl 0568.14022)], Hirzebruch, \textit{B. van Geemen} and \textit{N. O. Nygaard} [J. Number Theory 53, No. 1, 45--87 (1995: Zbl 0838.11047)], \textit{H. Verrill} [J. Number. Theory 81, No. 2, 310--334 (2000; Zbl 0971.14034)], and \textit{M.-H. Saito} and \textit{N. Yui} [J. Math. Kyoto Univ. 41, No. 2, 403--419 (2001; Zbl 1077.14546)]. Three techniques for proving the modularity of these Calabi-Yaus are given. First, one may exhibit a birational equivalence between the variety \(X\) and another variety \(Y\) already known to be modular; for example, \(Y\) may be a fiber product of elliptic surfaces. Second, one may use the Faltings-Serre-Livné criterion [\textit{R. Livné}, Contemp. Math. 67, 247--261 (1987; Zbl 0621.14019)] for establishing the equivalence of two 2-dimensional Galois representations; in practice, this technique reduces the task of proving modularity to counting the number of \(\mathbb{F}_p\)-rational points of \(X\) for a specific finite set of primes \(p\) and checking them against the corresponding coefficients of the modular form. Third, one may use Wiles's technique of establishing the modularity of the residual Galois representation of \(X\) at some good prime \(\ell\); in fact, this technique was used by \textit{L. Dieulefait} and \textit{J. Manoharmayum} [Fields Inst. Commun. 38, 159--166 (2003; Zbl 1096.14015)] to show that all rigid Calabi-Yau threefolds satisfying some mild conditions are modular. NEWLINENEWLINEIn the remainder of the first part of the article, the author presents two additional conjectures. The first is that a rigid Calabi-Yau \(X\) defined over \(\mathbb{Q}\) is modular if and only if its intermediate Jacobian (an elliptic curve) is also define over \(\mathbb{Q}\) (and thus modular); the second is that if \(X\) is a Calabi-Yau over \(K\) (not necessarily rigid), that the order of vanishing of the \(L\)-series of \(X\) at \(s = 2\) is equal to the rank of the kernel of the cycle class map \(\text{CH}^2(X_K)_{\text{hom}} = \ker [\text{CH}^2(X_K) \rightarrow H^4(X_{\overline{K}}, \mathbb{Z}_l(2))]\) (the assertion that \(\text{CH}^2(X_K)_{\text{hom}}\) is finitely generated is part of the conjecture). The second conjecture is due to Beilinson-Bloch [\textit{A. A. Beilinson}, J. Sov. Math. 30, 2036--2070 (1985; Zbl 0588.14013); \textit{S. Bloch}, Duke Math. J. 52, 379--397 (1985; Zbl 0628.14006)] and may be thought of as a generalization of the conjecture of \textit{B. J. Birch} and \textit{H. P. F. Swinnerton-Dyer} [J. Reine. Angew. Math. 212, 7--25 (1963; Zbl 0118.27601); J. Reine. Angew. Math. 218, 79--108 (1965; Zbl 0147.02506)]. NEWLINENEWLINEThe second part of the article presents some of the author's results on the arithmetic of Fermat-type Calabi-Yau varieties and their mirrors, obtained from the Greene-Plesser orbifolding construction. Their \(L\)-series are computed in terms of Hecke \(L\)-series with Grössencharacter and Dedekind zeta-functions, and these varieties are used to give some evidence in support of the Beilinson-Bloch conjecture. Some questions regarding the \(L\)-series of a Fermat-type Calabi-Yau and its mirror are posed at the end. NEWLINENEWLINEThe third part of the article focuses on the arithmetic of two-dimensional Calabi-Yaus, i.e. \(K3\) surfaces. Here the author quickly reviews the lattice structures on \(K3\) surfaces and Dolgachev's description of the moduli space of marked, lattice-polarized \(K3\) surfaces [\textit{I. V. Dolgachev}, J. Math. Sci. New York 81, No. 3, 2599--2630 (1996; Zbl 0890.14024)]. The author then presents the mirror moonshine conjecture of \textit{B. H. Lian} and \textit{S.-T. Yau} [Commun. Math. Phys. 176, No. 1, 163--191 (1996; Zbl 0867.14017)] and some concrete evidence in support of it. Further questions and open problems are collected at the end.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00034].