The Eisenstein ideal at prime-square level has constant rank

Area of Science:

• Number Theory
• Algebraic Number Theory

Background:

• Cuspidal forms (f) are central objects in number theory.
• Previous work established the existence of a cuspform f with specific properties modulo primes above p.
• The properties relate to the ℓ-th Fourier coefficients of f.

Purpose of the Study:

• To establish the uniqueness of the cuspform f up to Galois conjugacy.
• To determine the exact field extension generated by the coefficients of f.
• To generalize these findings to cases where higher powers of p divide the level of the cuspform.

Main Methods:

• Galois theory
• Theory of modular forms
• Analysis of Fourier coefficients

Main Results:

• The cuspform f is proven to be unique up to Galois conjugacy.
• The field extension generated by the coefficients of f is precisely Q(ζp).
• Similar uniqueness and field extension results are established for higher powers of p.

Conclusions:

• The study provides a definitive characterization of a specific cuspform and its associated number field.
• These results have implications for understanding the arithmetic properties of modular forms.