Theta characteristics and noncongruence modular forms

The Hodge bundle over a modular curve is a square-root of the canonical bundle twisted by the cuspidal divisor, or a theta characteristic, due to the Kodaira--Spencer isomorphism. We prove that, in most cases, a section of a theta characteristic (or any odd power of it) different from is a noncongruence modular form. On the other hand, we show how gives rise to a ``twisted'' analogue of the diagonal period map to a Siegel threefold, whose difference attributes to the stackiness of the moduli of abelian surfaces A₂. Some questions on the Brill--Noether theory of the modular curves are answered.