Degree formulas for GL₂ Hecke operators

Shimura Theorem 3.24, identities 6 and 7: degree formulas for the GL₂ Hecke algebra.

Main results

• deg_T_diag_ppow — deg(T(pⁱ, p^{i+k})) = p^{k−1}(p+1) for k > 0
• deg_T_diag_scalar — deg(T(c,c)) = 1
• deg_T_sum_prime_pow — deg(T(pᵏ)) = 1 + p + ⋯ + pᵏ
• deg_T_sum — deg(T(m)) = σ₁(m)

References

• Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Theorem 3.24

Identity 6: Degree formulas (wrapping existing results)

theorem HeckeRing.GL2.deg_T_diag_ppow (p : ℕ) (hp : Nat.Prime p) (i k : ℕ) (hk : 0 < k) : HeckeCosetDeg (GLn.GLPair 2) (GLn.TDiag ![p ^ i, p ^ (i + k)]) = ↑(p ^ (k - 1) * (p + 1))

Theorem 3.24(6): deg(T(pⁱ, p^{i+k})) = p^{k-1}(p+1) for k > 0.

theorem HeckeRing.GL2.deg_T_diag_scalar (c : ℕ) (hc : 0 < c) : HeckeCosetDeg (GLn.GLPair 2) (GLn.TDiag fun (x : Fin 2) => c) = 1

Scalar case: deg(T(c, c)) = 1.

Identity 7: Degree of T(m)

theorem HeckeRing.GL2.deg_T_sum_prime_pow (p : ℕ) (hp : Nat.Prime p) (k : ℕ) : (deg (GLn.GLPair 2)) (TSum ⟨p ^ k, ⋯⟩) = ∑ j ∈ Finset.range (k + 1), ↑p ^ j

deg(T(pᵏ)) = 1 + p + ⋯ + pᵏ. Proof by strong induction: for k >= 2, split the expansion at i=0 to get deg = p^{k-1}(p+1) + deg_tail, where the tail's degree equals deg(TSum(p^{k-2})) by a shift argument (the degree of TAd(p^{i+1}, p^{k-i-1}) equals that of TAd(p^i, p^{k-2-i}) since both have the same diagonal ratio).

theorem HeckeRing.GL2.deg_T_sum (m : ℕ+) : (deg (GLn.GLPair 2)) (TSum m) = ↑((ArithmeticFunction.sigma 1) ↑m)

Theorem 3.24(7): deg(T(m)) = σ₁(m). By prime factorization + coprime multiplicativity + prime-power case.