Analyze arithmetic of polynomials of j-invariants of norm modular form zeros

Study, for each congruence subgroup Γ and even weight k, the arithmetic of the polynomial whose roots are the j-invariants of zeros of the norm modular form \mathcal N_{Γ,k}, including determining its discriminant factorization, its Galois group, and congruence properties modulo primes related to the weight, and compare with the function-field analogues.

Background

The paper defines the norm modular form \mathcal N_{Γ,k} whose zeros collect all zeros of the Eisenstein series used, providing a natural divisor on X(1). The corresponding j-polynomial encapsulates algebraic data of all zero orbits.

The authors point to rich arithmetic questions: discriminant factorizations, Galois groups, and congruences, with a precedent in the function-field case of Drinfeld modular forms.