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New realizations of modular forms in Calabi-Yau threefolds arising from \(\phi^{4}\) theory. (English) Zbl 1420.11077
Summary: F. Brown and O. Schnetz [Commun. Number Theory Phys. 7, No. 2, 293–325 (2013; Zbl 1290.81083)] found that the number of points over \(\mathbb{F}_p\) of a graph hypersurface is often related to the coefficients of a modular form. We set some of the reduction techniques used to discover such relations in a general geometric context. We also prove the relation for two examples of modular forms of weight 3 and two of weight 4, refine the statement and suggest a method of proving it for three more of weight 4, and use one of the proved examples to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight 4 (one provably and one conjecturally).

MSC:
11F23 Relations with algebraic geometry and topology
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
Software:
GENREG; Magma
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[1] Belkale, P.; Brosnan, P., Matroids, motives, and a conjecture of Kontsevich, Duke Math. J., 116, 1, 147-188, (2003) · Zbl 1076.14026
[2] Bloch, S.; Esnault, H.; Kreimer, D., On motives associated to graph polynomials, Comm. Math. Phys., 267, 1, 181-225, (2006) · Zbl 1109.81059
[3] Bosma, W.; Cannon, J.; Playoust, C., The magma algebra system. I. the user language, J. Symbolic Comput., 24, 235-265, (1997) · Zbl 0898.68039
[4] Brown, F.; Schnetz, O., A K3 in \(\phi^4\), Duke Math. J., 161, 10, 1817-1862, (2012) · Zbl 1253.14024
[5] Brown, F.; Schnetz, O., Modular forms in quantum field theory, Commun. Number Theory Phys., 7, 293-325, (2013) · Zbl 1290.81083
[6] Brown, F.; Schnetz, O.; Yeats, K., Properties of \(c_2\) invariants of Feynman graphs, Adv. Theor. Math. Phys., 18, 2, 323-362, (2014) · Zbl 1309.81174
[7] Cynk, S.; Schütt, M., Generalised Kummer constructions and Weil restrictions, J. Number Theory, 129, 1965-1975, (2009) · Zbl 1168.14015
[8] Cynk, S.; van Straten, D., Infinitesimal deformations of double covers of smooth algebraic varieties, Math. Nachr., 219, 716-726, (2006) · Zbl 1101.14006
[9] Cynk, S.; Szemberg, T., Double covers of \(\mathbb{P}^3\) and Calabi-Yau varieties, Banach Center Publ., 44, 93-101, (1998) · Zbl 0915.14025
[10] Deligne, P., Formes modulaires et répresentations -adiques, Séminaire Bourbaki, 355, 139-172, (1968-69)
[11] Dieulefait, L., On the modularity of rigid Calabi-Yau threefolds: epilogue, J. Math. Sci., 171, 6, 725-727, (2010) · Zbl 1290.14029
[12] Dieulefait, L.; Manoharmayum, J., Modularity of rigid Calabi-Yau threefolds over \(\mathbb{Q}\), (Yui, N.; Lewis, J. D., Calabi-Yau Varieties and Mirror Symmetry, (2003), American Mathematical Society), 159-166 · Zbl 1096.14015
[13] Doran, C.; Harder, A.; Novoseltsev, A.; Thompson, A., Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces · Zbl 1339.14024
[14] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math., vol. 150, (1995), Springer-Verlag · Zbl 0819.13001
[15] Elkies, N.; Schütt, M., Modular forms and K3 surfaces, Adv. Math., 240, 106-131, (2013) · Zbl 1314.14069
[16] Gouvêa, F.; Yui, N., Rigid Calabi-Yau threefolds over \(\mathbb{Q}\) are modular, Exp. Math., 29, 1, 142-149, (2011) · Zbl 1230.14056
[17] Itzykson, C.; Zuber, J. B., Quantum field theory, (2006), Dover Publications
[18] Iyama, O.; Wemyss, M., Singular derived categories of \(\mathbb{Q}\)-factorial terminalizations and maximal modification algebras, Adv. Math., 261, 85-121, (2014) · Zbl 1326.14033
[19] Jones, J. W.; Roberts, D. P., A database of number fields, LMS J. Comput. Math., 17, 1, 595-618, (2014) · Zbl 1360.11121
[20] Marcolli, M., Feynman integrals and motives · Zbl 1192.81241
[21] Meringer, M., Fast generation of regular graphs and construction of cages, J. Graph Theory, 30, 137-146, (1999) · Zbl 0918.05062
[22] Meyer, C., A dictionary of modular threefolds, (2005), Johannes Gutenberg-Universität in Mainz, available for download at
[23] Reid, M., Canonical 3-folds, (Beauville, A., Journées de géométrie algébrique d’Angers, (1980), Sijthoff and Nordhooff), 273-310
[24] Schütt, M., On the modularity of three Calabi-Yau threefolds with bad reduction at 11 · Zbl 1115.14032
[25] Schütt, M., CM newforms with rational coefficients, Ramanujan J. Math., 19, 2, 187-205, (2009) · Zbl 1226.11057
[26] Serre, J.-P., A course in arithmetic, Grad. Texts in Math., vol. 7, (1973), Springer-Verlag · Zbl 0256.12001
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