The modularity of certain non-rigid Calabi-Yau threefolds
The authors consider Calabi-Yau threefolds which possess a non-isotrivial semistable \(K3\)-fibration with 6 singular fibers, the minimal number of singular fibers of such a fibration [\textit{X. Sun, S.-L. Tan} and \textit{K. Zuo}, Math. Res. Lett. 10, No. 2--3, 323--342 (2003; Zbl 1100.14004)] by the Arakelov-Yau bound. In this case the base curve must be a modular curve. The authors start with the list of 9 torsion-free genus 0 congruence subgroups of \(\text{PSL}(2,\mathbb{Z})\) of index 24 and lift them to \(\text{SL}(2,\mathbb{Z})\). In four of these examples, the lifted group \(\tilde{\Gamma}\) contains \(\Gamma(2)\). In these four cases, the authors construct Calabi-Yau threefolds using a Kummer construction: start with the universal elliptic curve \(Y\) over \(\tilde{\Gamma}\) (itself an extremal \(K3\) surface), take its product with an elliptic curve \(E\), and then quotient out by the involution \((-1)\). Blowing up the double point locus in the quotient variety \(Y \times E / (-1)\) yields a smooth Calabi-Yau variety \(X\). The authors explicitly compute the \(L\)-series of \(H^3(X)\). In each case, \(L(H^3(X), s)\) has the form \(L(g_3 \otimes g_2^E, s) L(g_2^E, s-1)^{n'_{-}} \Pi_{\delta'} L(E \otimes \chi_{\delta'}, s-1)^{n''_{-}}\), where \(g_3\) is the weight 3 modular form associated to \(Y\), \(g_2^E\) is the weight 2 modular form associated to \(E\), \(\chi_{\delta'}\) is the quadratic character cut out by \(\mathbb{Q}(\sqrt{\delta'})\), and \(n'_{-}\) and \(n''_{-}\) represent the number of algebraic cycles on \(Y\) anti-invariant under \((-1)\) and defined over \(\mathbb{Q}\) and \(\mathbb{Q}(\sqrt{\delta'})\) respectively. As a follow-up to this article, \textit{K. Hulek} and \textit{H. Verrill} construct non-rigid Calabi-Yau threefolds associated to the other five congruence subggroups above and show that their \(L\)-functions are also associated to modular forms [J. Math. Kyoto Univ. 45, No. 4, 667--681 (2005; Zbl 1106.14023)].
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