We study the ring of the algebraic automorphic forms of parallel weights on the Deligne–Mostow Shimura variety associated to a family of cyclic covers of the projective line. This Shimura surface defined over the cyclotomic field Q(ζ₅) with signature (3,0) and (2,1). Moreover, it is isomorphic to the closure of the Torelli image of the Hurwitz space of ramified Z/5Z covers of P¹. We compute the pullback of the Hodge line bundle in terms of the boundary divisors of the Hurwitz space, and identify a specific level supported at ramified primes, for which a finite cover of the Hurwitz space embeds into. We use these results to obtain the graded ring structure of algebraic automorphic forms at specific levels. As an application, we express the well-known Klein j-function on the sub-Shimura curve in terms of the generators of this graded ring. We also explore the general framework to compute such algebraic automorphic forms on other Deligne-Mostow Shimura varieties.
Yuxin Lin (Tue,) studied this question.