Linear independence of the Eisenstein series with character

In weight k ≥ 3, the normalized raised Eisenstein series E_k^(psi, phi, t), indexed by primitive, parity-compatible characters psi modulo u and phi modulo v and a raising parameter t with tuv ∣ N, are linearly independent in M_k(Γ₁(N)) (Diamond–Shurman, Theorem 4.5.2). Consequently the Eisenstein subspace of M_k(N, chi) spanned by the series of nebentypus chi has dimension the number of their indices.

The proof reads off Fourier coefficients, and its core is a statement about sequences, shared with weight two. The nth coefficient of E_k^(psi, phi, t), n > 0, is the twisted divisor sum sigma_(k-1)^(psi,phi)(n / t) when t ∣ n and 0 otherwise. Given a vanishing finite combination, let t₀ be the least raising parameter carrying a nonzero coefficient. At the indices t₀ m with m prime to N only the series with t = t₀ contribute, since every t divides N; so the twisted divisor sums of their character pairs satisfy a linear relation on the integers prime to N. Raised to level N, distinct primitive characters stay distinct, because the conductor of a primitive character is its modulus; hence that relation is trivial by the independence of twisted divisor sums (DirichletCharacter.linearIndependent_twistedDivisorSum).

At exponent 0, the exponent of weight one, the twisted divisor sums are symmetric in the two characters, so the sequences of an index (psi, phi, t) and of its exchange (phi, psi, t) coincide. The same argument, with the independence of these sums over distinct unordered pairs of characters (DirichletCharacter.linearIndependent_twistedDivisorSum_zero), shows that this is the only coincidence.

Main results

• TauCeti.EisensteinSeries.CharIndex.linearIndependent_twistedDivisorSum: the sequences n ↦ sigma_e^(psi,phi)(n / t), supported on the multiples of t, of distinct indices are linearly independent.
• TauCeti.EisensteinSeries.CharIndex.linearIndepOn_twistedDivisorSum_zero: at exponent 0, the sequences of a set of indices containing no index together with a different exchange are linearly independent.
• TauCeti.EisensteinSeries.CharIndex.linearIndependent_form: the series E_k^(psi, phi, t) are linearly independent.
• TauCeti.EisensteinSeries.CharIndex.linearIndependent_inCharSpace: so are those of a fixed nebentypus chi, inside M_k(N, chi).
• TauCeti.finrank_eisensteinSubspace: the Eisenstein subspace of M_k(N, chi) has dimension the number of indices with nebentypus chi.

References

• F. Diamond and J. Shurman, A first course in modular forms, Section 4.5, Theorem 4.5.2.

theorem TauCeti.EisensteinSeries.CharIndex.linearIndependent_twistedDivisorSum {N k : ℕ} [NeZero N] {e : ℕ} (he : e ≠ 0) : LinearIndependent ℂ fun (a : CharIndex N k) (n : { n : ℕ // n ≠ 0 }) => if a.t ∣ ↑n then (DirichletCharacter.twistedDivisorSum e a.psi a.phi) (↑n / a.t) else 0

The positive Fourier coefficients of the raised Eisenstein series of the indices, as sequences: the sequences n ↦ sigma_e^(psi,phi)(n / t) on the multiples n of t, zero at the other positive integers, attached to distinct indices (psi, phi, t) are linearly independent for every exponent e ≠ 0.

theorem TauCeti.EisensteinSeries.CharIndex.linearIndepOn_twistedDivisorSum_zero {N k : ℕ} [NeZero N] {S : Set (CharIndex N k)} (hS : ∀ a ∈ S, a.swap ∈ S → a.swap = a) : LinearIndepOn ℂ (fun (a : CharIndex N k) (n : { n : ℕ // n ≠ 0 }) => if a.t ∣ ↑n then (DirichletCharacter.twistedDivisorSum 0 a.psi a.phi) (↑n / a.t) else 0) S

At exponent zero, where the twisted divisor sums are symmetric in the two characters, the sequences n ↦ sigma_0^(psi,phi)(n / t) on the multiples n of t, zero at the other positive integers, are linearly independent on any set S of indices containing no index together with a different exchange (phi, psi, t).

theorem TauCeti.EisensteinSeries.CharIndex.linearIndependent_form {N k : ℕ} [NeZero N] (hk : 3 ≤ ↑k) : LinearIndependent ℂ fun (a : CharIndex N k) => a.form hk

Linear independence of the Eisenstein series with character (Diamond–Shurman, Theorem 4.5.2): in weight k ≥ 3, the normalized raised series E_k^(psi, phi, t) attached to distinct indices are linearly independent.

theorem TauCeti.EisensteinSeries.CharIndex.linearIndependent_inCharSpace {N k : ℕ} [NeZero N] (chi : (ZMod N)ˣ →* ℂˣ) (hk : 3 ≤ ↑k) : LinearIndependent ℂ fun (a : { a : CharIndex N k // a.nebentypus = chi }) => (↑a).inCharSpace hk ⋯

In weight k ≥ 3, the Eisenstein series E_k^(psi, phi, t) of nebentypus chi are linearly independent in the character space M_k(N, chi).

theorem TauCeti.finrank_eisensteinSubspace {N k : ℕ} [NeZero N] (chi : (ZMod N)ˣ →* ℂˣ) (hk : 3 ≤ ↑k) : Module.finrank ℂ ↥(eisensteinSubspace chi hk) = Nat.card { a : EisensteinSeries.CharIndex N k // a.nebentypus = chi }

In weight k ≥ 3, the Eisenstein subspace of M_k(N, chi) has dimension the number of indices (psi, phi, t) with induced nebentypus chi.