Rigid realizations of modular forms in Calabi-Yau threefolds
type: Item
instance of: scholarly article
title: Rigid realizations of modular forms in Calabi-Yau threefolds
zbMATH Open document ID: 1401.14175
author: Dominik Burek
published in: Journal of Pure and Applied Algebra
publication date: 2018-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1705.04059
review text: 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…
reviewed by: Matthias Schütt
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 14J15
zbMATH DE Number: 6950413
zbMATH Keywords: Calabi-Yau threefold
zbMATH Keywords: rigid
zbMATH Keywords: modular form
zbMATH Keywords: double octic
MaRDI profile type: Publication
OpenAlex ID: W2963886047
arXiv ID: 1705.04059
cites work: A new cohomology theory of orbifold
cites work: A Calabi–Yau threefold with 192 automorphisms
cites work: Modularity of some non-rigid double octic Calabi-Yau threefolds
cites work: Non-liftable Calabi-Yau spaces
cites work: Q4233743
cites work: Q4462779
cites work: Rigid Calabi-Yau threefolds over \(\mathbb Q\) are modular
cites work: On modularity of rigid and nonrigid Calabi-Yau varieties associated to the Root Lattice A4
cites work: Q3438253
cites work: Serre's modularity conjecture. I
cites work: Serre's modularity conjecture. II
cites work: Modular Calabi-Yau threefolds
cites work: Q4462789
Wikidata QID: Q129970904
DOI: 10.1016/J.JPAA.2018.04.005
Recommended article: Rigid Calabi-Yau threefolds over \(\mathbb Q\) are modular
Recommended article: Modularity of a certain Calabi-Yau threefold
Recommended article: Modularity of Maschke's octic and Calabi-Yau threefold
Recommended article: Periods of singular double octic Calabi–Yau threefolds and modular forms
Recommended article: Generalised Kummer constructions and Weil restrictions
Recommended article: On modularity of rigid and nonrigid Calabi-Yau varieties associated to the Root Lattice A4
Recommended article: Quadratic Twists of Rigid Calabi–Yau Threefolds Over ℚ
Recommended article: Arithmetic of certain Calabi-Yau varieties and mirror symmetry.
Recommended article: MODULAR CALABI–YAU THREEFOLDS OF LEVEL EIGHT
Recommended article: New examples of modular rigid Calabi-Yau threefolds
description: scientific article