The index of a certain quotient of the Hecke algebra in its normalization

Let $Γ$ be a congruence subgroup of $SL_2(Z)$, and let $f$ be a normalized eigenform of weight $k$ on $Γ$. Let $K$ denote the number field generated over $Q$ by the Fourier coefficients of $f$. Let $R$ denote the the order in $K$ generated by the Fourier coefficients of $f$, which is contained in the ring of integers $O$ of $K$. We relate the primes that divide the index of $R$ in $O$ to primes $p$ such that $f$ is congruent to a conjugate of $f$ modulo a prime ideal of residue characteristic $p$. The index mentioned above is the same as the index of the quotient of the Hecke algebra by the annihilator ideal of $f$ in its normalization.

💡 Research Summary

The paper investigates the relationship between the index of the order generated by the Fourier coefficients of a normalized eigenform and the occurrence of congruences between that eigenform and its Galois conjugates. Let Γ be a congruence subgroup of SL₂(ℤ) and let f be a normalized eigenform of weight k on Γ. Denote by K the number field generated by the Fourier coefficients aₙ(f) and by 𝒪 the full ring of integers of K. The coefficients generate an order R⊂𝒪; in general R may be a proper suborder, and the index