Forms supported on multiples of an index prime to the level

A nonzero cusp form F ∈ S_k(Γ₁(M), χ) can have its q-expansion supported on the multiples of q when q divides the level — the level-raises V_q from level M / q are exactly such forms. When q is prime to M this cannot happen: such an F is 0 (TauCeti.eq_zero_of_qExpansionSupportedOnDvd_of_coprime).

The argument first descends F at the same level: read at level q M, where q divides the level, F is V_q G on coefficients for some G ∈ S_k(Γ₁(M), χ) (TauCeti.exists_mem_cuspFormCharSpace_qExpansion_coeff_eq_ite_dvd, which holds for every q ≠ 0). For a prime q ∤ M the Hecke operator T_q is U_q + χ(q) q^{k-1} V_q on coefficients, so V_q F = (χ(q) q^{k-1})⁻¹ (T_q F - G) is again a form of level M, and iterating gives V_{q^j} G ∈ S_k(Γ₁(M), χ) for every j. Their q-expansions vanish below q^j, so the Sturm bound forces G = 0, and then F = 0. A composite q ≠ 1 prime to M reduces to any of its prime factors.

The consequence used in newform theory is that a prime away from the level can be dropped from a coprimality condition on the coefficients: if a_n(h) = 0 at every n coprime to L M, then a_n(h) = 0 at every n coprime to M (TauCeti.qExpansion_coeff_eq_zero_of_forall_coprime_mul). This is what lets strong multiplicity one ignore finitely many primes outside the level.

Main results

• TauCeti.eq_zero_of_qExpansionSupportedOnDvd_of_coprime: for q ≠ 1 prime to M, a form of S_k(Γ₁(M), χ) supported on the multiples of q is 0.
• TauCeti.qExpansion_coeff_eq_zero_of_forall_coprime_mul: vanishing of the coefficients at the indices coprime to L M implies vanishing at those coprime to M.

References

• T. Miyake, Modular forms, Lemma 4.6.8 and Theorem 4.6.12, where the auxiliary level L of strong multiplicity one is reduced to the level of the form.
• F. Diamond and J. Shurman, A first course in modular forms, Proposition 5.3.1 (the action of T_p on q-expansions) and Section 5.7.

theorem TauCeti.eq_zero_of_qExpansionSupportedOnDvd_of_coprime {M : ℕ} [NeZero M] {k : ℤ} {q : ℕ} (hq : q ≠ 1) (hqM : q.Coprime M) {χ : (ZMod M)ˣ →* ℂˣ} {F : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 M)) k} (hFχ : F ∈ cuspFormCharSpace k χ) (hF : QExpansionSupportedOnDvd q F) : F = 0

Away from the level, no nonzero form is supported on the multiples of an index. For q ≠ 1 prime to M, a form F ∈ S_k(Γ₁(M), χ) whose q-expansion is supported on the multiples of q is 0. Compare the level-raises V_q from level M / q, which are nonzero forms of this kind when q ∣ M.

theorem TauCeti.qExpansion_coeff_eq_zero_of_forall_coprime_mul {M : ℕ} [NeZero M] {k : ℤ} {L : ℕ} (hL : L ≠ 0) {χ : (ZMod M)ˣ →* ℂˣ} {h : CuspForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) (CongruenceSubgroup.Gamma1 M)) k} (hχ : h ∈ cuspFormCharSpace k χ) (hvan : ∀ (n : ℕ), n.Coprime (L * M) → (PowerSeries.coeff n) (UpperHalfPlane.qExpansion 1 ⇑h) = 0) (n : ℕ) (hn : n.Coprime M) : (PowerSeries.coeff n) (UpperHalfPlane.qExpansion 1 ⇑h) = 0

Vanishing of the coefficients prime to L M implies vanishing of those prime to M. If h ∈ S_k(Γ₁(M), χ) has a_n(h) = 0 at every n coprime to L M, for some L ≠ 0, then a_n(h) = 0 at every n coprime to M. The primes of L dividing M change nothing, and each prime of L away from M is dropped by TauCeti.eq_zero_of_qExpansionSupportedOnDvd_of_coprime.