Vanishing of the bad-prime eigenvalue when p² ∣ N and χ is defined modulo N / p

For a prime p with p² ∣ N and a nebentypus χ defined modulo N / p, the operator U_p on S_k(N, χ) lowers the level: U_p f is a cusp form of level Γ₁(N / p). Since p ∣ N / p, the p upper-triangular matrices [1, b; 0, p] defining U_p are exactly Miyake's descent family at p² ∣ N, whose slash sum is Γ₀(N / p)-equivariant with the lowered nebentypus (TauCeti.descendCuspForm). So U_p f is old.

For a newform f this forces a_p(f) = 0: U_p f = a_p(f) • f is both old and new, hence zero, while f ≠ 0. This is the vanishing case of the classification of the bad-prime eigenvalues of a newform (Atkin–Lehner for trivial nebentypus, Li in general; Miyake, Theorem 4.6.17): in terms of c = v_p(cond χ), the hypotheses say v_p(N) ≥ 2 and c < v_p(N). In the remaining cases the classification gives a_p ≠ 0; those cases are not treated here. Since the newforms of nebentypus χ span the new part of S_k(N, χ), U_p is zero on that whole new part.

Main results

• HeckeRing.GL2.heckeUCuspNat_eq_ofLe_descendCuspForm: for p² ∣ N and χ pulled back from χ₀ modulo N / p, U_p f is the descent of f, a cusp form in S_k(Γ₁(N / p), χ₀), read at level N.
• HeckeRing.GL2.heckeUCuspNat_mem_cuspFormsOld_of_sq_dvd: in that situation U_p f is old.
• HeckeRing.GL2.Newform.qExpansion_coeff_prime_eq_zero_of_sq_dvd: a newform whose nebentypus factors through N / p, with p² ∣ N, has a_p = 0.
• HeckeRing.GL2.heckeUCuspNat_eq_zero_of_mem_cuspFormsNew_of_sq_dvd: U_p vanishes on the new part of S_k(N, χ) when p² ∣ N and χ factors through N / p.

References

• T. Miyake, Modular forms, Lemma 4.6.14 and Theorem 4.6.17.
• A. O. L. Atkin and J. Lehner, Hecke operators on Γ₀(m), Math. Ann. 185 (1970), 134–160.
• W.-C. W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315.

U_p lowers the level

theorem HeckeRing.GL2.heckeUCuspNat_eq_ofLe_descendCuspForm {N p : ℕ} [NeZero N] (k : ℤ) (hp : Nat.Prime p) (hpsq : p ^ 2 ∣ N) {χ : (ZMod N)ˣ →* ℂˣ} {χ₀ : (ZMod (N / p))ˣ →* ℂˣ} (hcomp : χ = χ₀.comp (ZMod.unitsMap ⋯)) {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ cuspFormCharSpace k χ) : (heckeUCuspNat k p hp ⋯) f = CuspForm.ofLe ⋯ (TauCeti.descendCuspForm k hp ⋯ hcomp hf)

U_p lowers the level when p² ∣ N and the nebentypus is defined modulo N / p. For f ∈ S_k(Γ₁(N), χ) with χ the pull-back of χ₀ modulo N / p, the form U_p f is the descent descendCuspForm of f, a cusp form in S_k(Γ₁(N / p), χ₀) (descendCuspForm_mem_cuspFormCharSpace), read at level N.

theorem HeckeRing.GL2.heckeUCuspNat_mem_cuspFormsOld_of_sq_dvd {N p : ℕ} [NeZero N] (k : ℤ) (hp : Nat.Prime p) (hpsq : p ^ 2 ∣ N) {χ : (ZMod N)ˣ →* ℂˣ} {χ₀ : (ZMod (N / p))ˣ →* ℂˣ} (hcomp : χ = χ₀.comp (ZMod.unitsMap ⋯)) {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ cuspFormCharSpace k χ) : (heckeUCuspNat k p hp ⋯) f ∈ TauCeti.cuspFormsOld N k

U_p maps into the old subspace when p² ∣ N and the nebentypus is defined modulo N / p: U_p f comes from the proper divisor level N / p (heckeUCuspNat_eq_ofLe_descendCuspForm).

The vanishing of a_p

@[simp]

theorem HeckeRing.GL2.Newform.qExpansion_coeff_prime_eq_zero_of_sq_dvd {N p : ℕ} [NeZero N] {k : ℤ} (f : Newform N k) (hp : Nat.Prime p) (hpsq : p ^ 2 ∣ N) (hχ : f.dirichletLift.FactorsThrough (N / p)) : (PowerSeries.coeff p) (UpperHalfPlane.qExpansion 1 ⇑f.toCuspForm) = 0

A newform has a_p = 0 when p² ∣ N and its nebentypus is defined modulo N / p (Atkin–Lehner for trivial nebentypus; Li; Miyake, Theorem 4.6.17). With Newform.heckeUCuspNat_eq_qExpansion_coeff_smul this says U_p f = 0.

theorem HeckeRing.GL2.heckeUCuspNat_eq_zero_of_mem_cuspFormsNew_of_sq_dvd {N p : ℕ} [NeZero N] (k : ℤ) (hp : Nat.Prime p) (hpsq : p ^ 2 ∣ N) {χ : (ZMod N)ˣ →* ℂˣ} (hχ : DirichletCharacter.FactorsThrough (MulChar.ofUnitHom χ) (N / p)) {f : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 N)) k} (hf : f ∈ TauCeti.cuspFormsNew N k ⊓ cuspFormCharSpace k χ) : (heckeUCuspNat k p hp ⋯) f = 0

U_p vanishes on the new part of S_k(N, χ) when p² ∣ N and χ is defined modulo N / p. In particular the new part of S_k(N, χ) is U_p-stable in this case.