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We determine explicit generators for the ring of modular forms associated with the moduli spaces of K3 surfaces with automorphism group $(\mathbb {Z}/2\mathbb {Z})^2$ and of Picard rank 13 and higher. The K3 surfaces in question carry a canonical Jacobian elliptic fibration and the modular form generators appear as coefficients in the Weierstrass-type equations describing these fibrations.

Summary

We treat some more advanced topics: monstrous (and other) moonshine, Monster and E_8, Niemeier lattices, the triangle property, generalized line graphs, quiver representations, cluster algebras, von Neumann algebras, catastrophes, Calabi–Yau, elliptic fibrations.

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi–Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim (Y)\leq 5$ and $Y$ is not of product type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of Kawamata log terminal pairs $(X, \Delta )$ with $K_X+\Delta$ numerically trivial and not of product type, in dimension at most four.

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K3 SURFACES AND ORTHOGONAL MODULAR FORMS

5 - Advanced Miscellany

Summary

Birational boundedness of low-dimensional elliptic Calabi–Yau varieties with a section

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3 results

K3 SURFACES AND ORTHOGONAL MODULAR FORMS

5 - Advanced Miscellany

Summary

Birational boundedness of low-dimensional elliptic Calabi–Yau varieties with a section