Stability of the new subspace under the bad-prime operators

For a prime p dividing the level N, the operator U_p = T_p maps the new subspace S_k(Γ₁(N))ⁿᵉʷ into itself. With the good primes, the new subspace is stable under T_p at every prime. Together with its stability under the diamond operators, and that of the old subspace under every T_p and every diamond operator, this is Diamond–Shurman's Proposition 5.6.2: both subspaces are stable under the whole Hecke algebra.

At a good prime the stability of the new subspace comes from that of the old subspace, since the Petersson adjoint of T_p is ⟨p⟩⁻¹ T_p. At p ∣ N that argument is not available: the adjoint of U_p is not a multiple of U_p, and the stability of the old subspace only gives the stability of the new subspace under the adjoint. Instead, the newforms span the new subspace (HeckeRing.GL2.Newform.span_range_toCuspForm_eq_cuspFormsNew), and each of them is a U_p-eigenvector (HeckeRing.GL2.Newform.heckeUCuspNat_eq_qExpansion_coeff_smul, proved without any bad-prime stability), so U_p carries a spanning family of the new subspace into it.

On a character space S_k(N, χ), where the Hecke ring of Γ₀(N) acts, the new part is then stable under every prime generator, hence under every T_n (HeckeRing.GL2.heckeRingHomCuspCharSpace_heckeTCompositeGamma0_mem_of_forall_prime_dvd).

Main results

• TauCeti.heckeTCuspNat_mem_cuspFormsNew: T_p maps the new subspace into itself, for every prime p, with TauCeti.cuspFormsNew_map_heckeTCuspNat_le its Submodule.map form.
• TauCeti.heckeUCuspNat_mem_cuspFormsNew: its specialization to U_p, for a prime p ∣ N, with TauCeti.cuspFormsNew_map_heckeUCuspNat_le its Submodule.map form.
• TauCeti.coe_heckeRingHomCuspCharSpace_heckeTCompositeGamma0_mem_cuspFormsNew: the new part of S_k(N, χ) is stable under every T_n, with TauCeti.cuspFormsNew_comap_map_heckeRingHomCuspCharSpace_heckeTCompositeGamma0_le its Submodule.map form.

References

• F. Diamond and J. Shurman, A first course in modular forms, Proposition 5.6.2.
• T. Miyake, Modular forms, Theorem 4.6.13.

theorem TauCeti.heckeTCuspNat_mem_cuspFormsNew {k : ℤ} {N p : ℕ} [NeZero N] (hp : Nat.Prime p) {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ cuspFormsNew N k) : (HeckeRing.GL2.heckeTCuspNat k p) f ∈ cuspFormsNew N k

The new subspace is stable under every T_p: for every prime p, whether or not it divides the level, T_p maps S_k(Γ₁(N))ⁿᵉʷ into itself. This is the new-space half of Diamond–Shurman's Proposition 5.6.2 for the operators T_p. At a good prime it is heckeTCuspNat_mem_cuspFormsNew_of_coprime; at p ∣ N the newforms span the new subspace and T_p = U_p scales each of them.

theorem TauCeti.cuspFormsNew_map_heckeTCuspNat_le {N p : ℕ} [NeZero N] (hp : Nat.Prime p) (k : ℤ) : Submodule.map (HeckeRing.GL2.heckeTCuspNat k p) (cuspFormsNew N k) ≤ cuspFormsNew N k

The new subspace is stable under every T_p, in the Submodule.map form.

theorem TauCeti.heckeUCuspNat_mem_cuspFormsNew {k : ℤ} {N p : ℕ} [NeZero N] (hp : Nat.Prime p) (hpN : p ∣ N) {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ cuspFormsNew N k) : (HeckeRing.GL2.heckeUCuspNat k p hp hpN) f ∈ cuspFormsNew N k

The new subspace is stable under the bad-prime operator U_p: for a prime p dividing the level, U_p = T_p maps S_k(Γ₁(N))ⁿᵉʷ into itself. This is the specialization of heckeTCuspNat_mem_cuspFormsNew to the primes dividing the level.

theorem TauCeti.cuspFormsNew_map_heckeUCuspNat_le {N p : ℕ} [NeZero N] (hp : Nat.Prime p) (hpN : p ∣ N) (k : ℤ) : Submodule.map (HeckeRing.GL2.heckeUCuspNat k p hp hpN) (cuspFormsNew N k) ≤ cuspFormsNew N k

The new subspace is stable under U_p for p ∣ N, in the Submodule.map form.

theorem TauCeti.coe_heckeRingHomCuspCharSpace_heckeTCompositeGamma0_mem_cuspFormsNew {k : ℤ} {N : ℕ} [NeZero N] {χ : (ZMod N)ˣ →* ℂˣ} (n : ℕ) {F : ↥(cuspFormCharSpace k χ)} (hF : ↑F ∈ cuspFormsNew N k) : ↑(((HeckeRing.GL2.heckeRingHomCuspCharSpace k χ) (HeckeRing.GL2.heckeTCompositeGamma0 N n)) F) ∈ cuspFormsNew N k

The new part of S_k(N, χ) is stable under every Hecke operator T_n, the composite element heckeTCompositeGamma0 N n of the Γ₀(N) Hecke ring acting on S_k(N, χ), whether or not n shares a factor with the level. This is the new-space half of Diamond–Shurman's Proposition 5.6.2 for the operators T_n; the old-space half is coe_heckeRingHomCuspCharSpace_heckeTCompositeGamma0_mem_cuspFormsOld.

theorem TauCeti.cuspFormsNew_comap_map_heckeRingHomCuspCharSpace_heckeTCompositeGamma0_le {N : ℕ} [NeZero N] (χ : (ZMod N)ˣ →* ℂˣ) (n : ℕ) (k : ℤ) : Submodule.map ((HeckeRing.GL2.heckeRingHomCuspCharSpace k χ) (HeckeRing.GL2.heckeTCompositeGamma0 N n)) (Submodule.comap (cuspFormCharSpace k χ).subtype (cuspFormsNew N k)) ≤ Submodule.comap (cuspFormCharSpace k χ).subtype (cuspFormsNew N k)

The new part of S_k(N, χ) is stable under every T_n, in Submodule.map form.