In a recent paper by F. Gouvea and N. Yui, a detailed account is given of a patching argument due to Serre that proves that the modularity of all rigid Calabi–Yau threefolds defined over \( \mathbb{Q} \) follows from Serre’s modularity conjecture (now a theorem). In this note, we give an alternative proof of this implication. The main difference with Serre’s argument is that instead of using as a main input residual modularity in infinitely many characteristics, we just require residual modularity in a suitable characteristic. This is combined with the effective Chebotarev theorem.
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References
F. Diamond, M. Flah, and L. Guo, “The Tamagawa number conjeture of adjoint motives of modular forms,” Ann. Si. École Norm. Sup., 37, 663–727 (2004).
L. V. Dieulefait, “Remarks on Serre’s modularity conjeture,” preprint (2006).
L. V. Dieulefait and J. Manoharmayum, “Modularity of rigid Calabi-Yau threefolds over \( \mathbb{Q} \),” in: Calabi-Yau Varieties and Mirror Symmetry, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, Rhode Island (2003), pp. 159–166.
F. Gouvea and N. Yui, “Rigid Calabi-Yau threefolds over \( \mathbb{Q} \) are modular: a footnote to Serre,” preprint (2009).
K. Hulek, R. Kloosterman, and M. Schuett, “Modularity of Calabi-Yau varieties,” in: F. Catanese, H. Esnault, A. Hukleberry, K. Hulek, and T. Peternell (eds.), Global Aspects of Complex Geometry, Springer-Verlag, Berlin (2006), pp. 271-309.
C. Khare and J.-P. Wintenberger, “Serre’s modularity conjeture. I,” Invent. Math., 178, 485–504 (2009).
C. Khare and J.-P. Wintenberger, “Serre’s modularity conjeture. II,” Invent. Math., 178, 505–586 (2009).
M. Kisin, “Modularity of 2-adic Barsotti-Tate representations,” Invent. Math., 178, 587–634 (2009).
K. Ribet, “On l-adic representations attahed to modular forms. II,” Glasgow Math. J., 27, 185–194 (1985).
J.-P. Serre, “Quelques appliations du théorème de densité de Chebotarev,” in: Oeuvres, Vol. III, Springer-Verlag, Berlin (1986).
J.-P. Serre, “Sur les représentations modulaires de degré 2 de Gal\(\bar{\mathbb{Q}}/\mathbb{Q} \),” Duke Math. J., 54, 179–230 (1987).
R. Taylor, “On the meromorphic continuation of degree two L-funtions”, Doc. Math., Extra Vol.: John Coates’ Sixtieth Birthday, 729-779 (2006).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 377, 2010, pp. 44–49.
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Dieulefait, L. On the modularity of rigid Calabi−Yau threefolds: epilogue. J Math Sci 171, 725–727 (2010). https://doi.org/10.1007/s10958-010-0175-8
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DOI: https://doi.org/10.1007/s10958-010-0175-8