This question is related to my other question here: Examples of subspaces singled out by modular forms.

Here I am wondering if there is a philosophical explanation about why the supersingular locus on the modular curve mod p is the zero locus of a modular form.

Even better, are there sufficient conditions ensuring that a subset of the modular curve is the zero locus of a modular form?

Thank you for your help.

This question is related to my other question here: Examples of subspaces singled out by modular forms.

Here I am wondering if there is a philosophical explanation about why the supersingular locus on the modular curve mod p is the zero locus of a modular form.

Even better, are there sufficient conditions ensuring that a subset of the modular curve is the zero locus of a modular form?

Thank you for your help.