In this talk we will see that there are only finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we will show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we will see that it must be a twist of one of four curves.

This is joint work with Samuele Anni and Eran Assaf.[-]

A pair consisting of a K3 surface and a non-symplectic automorphism of order three is called an Eisenstein K3 surface. We introduce an invariant of Eisenstein K3 surfaces, which we obtain using the equivariant analytic torsion of an Eisenstein K3 surface and the analytic torsion of its fixed locus. Then this invariant gives rise to a function on the moduli space of Eisenstein K3 surfaces, which consists of 24 connected components and each of which is a complex ball quotient depending on the topological type of the automorphism of order three. Our main result is that, for each topological type, the invariant is expressed as the product of the Petersson norms of two kinds of automorphic forms, one is an automorphic form on the complex ball and the other is a Siegel modular form. In many cases, the automorphic form on the complex ball obtained in this way is a so-called reflective modular form. In some cases, this automorphic form is obtained as the restriction of an explicit Borcherds product to the complex ball. This is a joint work with Shu Kawaguchi.[-]