Number Theory Seminar

We study the ring of the scalar-valued automorphic forms on the Deligne–Mostow Shimura variety associated to a family of cyclic covers of the projective line. This Shimura surface has complex multiplication by $\mathbb{Q}(\zeta_{5})$ with signature (1,1,2,0). Moreover, it is isomorphic to the closure of the Torelli image of the Hurwitz space of totally ramified $\mathbb{Z}/5\mathbb{Z}$ covers of $\mathbb{P}^1$. We compute the pullback of the Hodge line bundle in terms of the boundary divisors of the Hurwitz space and use this to describe the graded ring structure of scalar-valued automorphic forms at specific levels.