Rigid realizations of modular forms in Calabi-Yau threefolds
A classical construction due to Eichler and Shimura associates to a Hecke eigenform of weight 2 with eigenvalues in \(\mathbb Z\) an elliptic curve over \(\mathbb Q\) (the converse being, of course, the celebrated result of Wiles, Taylor and others). For higher weight \(k\), there is Deligne's geometric realization via fibre products of universal elliptic curves [\textit{P. Deligne}, in: Sémin. Bourbaki 1968/69, No. 355, 139--172 (1971; Zbl 0206.49901)], but no uniform approach seems to be known (ideally involving varieties of dimension \(k-1\) with geometric genus \(p_g=1\)). In this direction, there is the following question formulated independently by Mazur and van Straten: Is every weight \(k>2\) Hecke eigenform with eigenvalues in \(\mathbb Z\) realized by a \((k-1)\)-dimensional Calabi-Yau variety over \(\mathbb Q\)? The weight \(3\) case is known due to work of \textit{N. D. Elkies} and the reviewer [Adv. Math. 240, 106--131 (2013; Zbl 1314.14069)] (greatly helped by the fact that the odd weight forces the Hecke eigenform to have complex multiplication by an old result of \textit{K. A. Ribet} [Lect. Notes Math. 601, 17--52 (1977; Zbl 0363.10015)], but already in weight \(4\) the problem seems to be wide open (let alone the question whether the number of Hecke eigenforms in question is finite or infinite up to twists). A great many examples were found as double octics by \textit{C. Meyer} [Modular Calabi-Yau threefolds. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1096.14032)]. Going one step further even, one might ask for the Calabi-Yau threefold \(X\) to be rigid; that is, it admits no infinitesimal deformations. Then it follows that \(b_3(X)=2\), so \(X\) (if defined over \(\mathbb Q\) or number fields) has the very rare feature that it automatically comes with a system of \(\ell\)-adic 2-dimensional Galois representations attached to the third \(\ell\)-adic étale cohomology. In fact, if \(X\) is defined over \(\mathbb Q\), then it can be seen as a consequence of Serre's conjecture [\textit{C. Khare} and \textit{J.-P. Wintenberger}, Invent. Math. 178, No. 3, 485--504 (2009; Zbl 1304.11041)] that \(X\) is modular, i.e. the above system of Galois representations agrees with that associated to a certain weight \(4\) form. Again, very few examples are known, and the present paper provides a welcome addition by working out some 7 new examples of rigid Calabi-Yau threefolds over \(\mathbb Q\). They arise as quotients of non-rigid Calabi-Yau threefolds by a symplectic involution (where the original threefolds occur in one-dimensional families of double octics and were already shown to be modular by Meyer). Besides rigidity, the main point of the argument is the existence of a crepant resotion, and this is discussed in greater generality by analysing the fixed loci of involutions on double octics.
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