The ring of modular forms for the even unimodular lattice of signature (2,10)

Hashimoto, Kenji; Ueda, Kazushi

doi:10.1090/proc/15667

The ring of modular forms for the even unimodular lattice of signature (2,10)
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by Kenji Hashimoto and Kazushi Ueda PDF

Proc. Amer. Math. Soc. 150 (2022), 547-558 Request permission

Abstract:

We show that the ring of modular forms with characters for the even unimodular lattice of signature (2,10) is generated by forms of weights 4, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42, and 252 with one relation of weight 504. The proof is based on the comparison of the orbifold quotient of the symmetric domain with the root stack of the coarse moduli space.

References

Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. MR 2450211, DOI 10.1353/ajm.0.0017
Dan Burns Jr. and Michael Rapoport, On the Torelli problem for kählerian $K-3$ surfaces, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 2, 235–273. MR 447635
Egbert Brieskorn, The unfolding of exceptional singularities, Nova Acta Leopoldina (N.F.) 52 (1981), no. 240, 65–93. Leopoldina Symposium: Singularities (Thüringen, 1978). MR 642697
Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. MR 2306040, DOI 10.1353/ajm.2007.0007
C. Dieckmann, A. Krieg, and M. Woitalla, The graded ring of modular forms on the Cayley half-space of degree two, Ramanujan J. 48 (2019), no. 2, 385–398. MR 3911795, DOI 10.1007/s11139-017-9970-x
I. V. Dolgachev, Mirror symmetry for lattice polarized $K3$ surfaces, J. Math. Sci. 81 (1996), no. 3, 2599–2630. Algebraic geometry, 4. MR 1420220, DOI 10.1007/BF02362332
V. Gritsenko, K. Hulek, and G. K. Sankaran, Abelianisation of orthogonal groups and the fundamental group of modular varieties, J. Algebra 322 (2009), no. 2, 463–478. MR 2529099, DOI 10.1016/j.jalgebra.2009.01.037
E. Looijenga, The smoothing components of a triangle singularity. II, Math. Ann. 269 (1984), no. 3, 357–387. MR 761312, DOI 10.1007/BF01450700
Rick Miranda, The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research], ETS Editrice, Pisa, 1989. MR 1078016
V. V. Nikulin, Finite groups of automorphisms of Kählerian $K3$ surfaces, Trudy Moskov. Mat. Obshch. 38 (1979), 75–137 (Russian). MR 544937
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type $\textrm {K}3$, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572 (Russian). MR 0284440
Hironori Shiga, One attempt to the $K3$ modular function. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 157–182. MR 616904
Matthias Schütt and Tetsuji Shioda, Elliptic surfaces, Algebraic geometry in East Asia—Seoul 2008, Adv. Stud. Pure Math., vol. 60, Math. Soc. Japan, Tokyo, 2010, pp. 51–160. MR 2732092, DOI 10.2969/aspm/06010051
Andrei N. Todorov, Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of $K3$ surfaces, Invent. Math. 61 (1980), no. 3, 251–265. MR 592693, DOI 10.1007/BF01390067