Congruences between Eisenstein series and cusp forms
TheoremLu's congruence for the Eisenstein series \(G_k\)#●ready to prove
Let \(k \ge 12\) be even with \(k \ne 14\). For \(w, n \in \mathbb{N}\) let \(\sigma_{-1}(n)\) be \(1\) if \(n = 0\) and \(0\) otherwise; for \(w \ge 1\) let \(\sigma_{w-1}(0) = -B_w/(2w)\), where \(B_w\) is the \(w\)-th Bernoulli number, and \(\sigma_{w-1}(n) = \sum_{d \mid n} d^{w-1}\) for \(n \ge 1\). In \(\mathbb{Q}[[q]]\) let \(G_w = \sum_{n \ge 0} \sigma_{w-1}(n) q^n\), let \(E_w = G_w/\sigma_{w-1}(0)\) for even \(w\), and let \(\Delta = q \prod_{j \ge 1} (1 - q^j)^{24}\); for \(i, n \in \mathbb{N}\) write \(b_i(n)\) for the coefficient of \(q^n\) in \(\Delta^i E_{k-12i}\). Let \(N_w = 1\) for \(w \le 2\), let \(N_w\) be the absolute value of the numerator of \(\sigma_{w-1}(0)\) for \(w \ge 3\), and let \(N'_k\) be the largest divisor of \(N_k\) that is coprime to \(N_{k-12i}\) for every \(1 \le i \le \lfloor k/12 \rfloor\). For rationals \(x, y\) and a positive integer \(N\), \(x \equiv y \pmod{N}\) means that \((x - y)/N\) has denominator coprime to \(N\). Let \(m = \lfloor k/12 \rfloor - 1\) if \(k \equiv 2 \pmod{12}\) and \(m = \lfloor k/12 \rfloor\) otherwise, let \(C\) be the \(m \times m\) matrix \((b_i(j))_{1 \le i, j \le m}\), and let \((\gamma_1, \dots, \gamma_m) = (\sigma_{k-1}(1), \dots, \sigma_{k-1}(m))\, C^{-1}\). Then \(\sigma_{k-1}(n) \equiv \sum_{i=1}^m \gamma_i b_i(n) \pmod{N'_k}\) for every \(n \ge 0\), that is, \(G_k \equiv \sum_{i=1}^m \gamma_i \Delta^i E_{k-12i} \pmod{N'_k}\) coefficientwise.
public theorem_wanted eisenstein_congr_sum_delta_pow_mul_eisenstein (k : ℕ) (hk : 12 ≤ k)
(hke : Even k) (hk14 : k ≠ 14) :
let σ₀ : ℕ → ℕ → ℚ := fun w n => if w = 0 then (if n = 0 then 1 else 0)
else if n = 0 then -bernoulli w / (2 * w) else (ArithmeticFunction.sigma (w - 1) n : ℚ)
let Δ : ℚ⟦X⟧ := X * ∏' j : ℕ, (1 - X ^ (j + 1)) ^ 24
let b : ℕ → ℕ → ℚ := fun i n =>
coeff n (Δ ^ i * PowerSeries.mk (σ₀ (k - 12 * i))) / σ₀ (k - 12 * i) 0
let Nw : ℕ → ℕ := fun w => if w ≤ 2 then 1 else (σ₀ w 0).num.natAbs
let N' : ℕ := ((Nat.divisors (Nw k)).filter fun d =>
∀ i ∈ Finset.Icc 1 (k / 12), Nat.Coprime d (Nw (k - 12 * i))).sup id
let m : ℕ := if k % 12 = 2 then k / 12 - 1 else k / 12
let C : Matrix (Fin m) (Fin m) ℚ := fun i j => b (i.val + 1) (j.val + 1)
let γ : Fin m → ℚ := Matrix.vecMul (fun j => σ₀ k (j.val + 1)) C⁻¹
∀ n : ℕ, ((σ₀ k n - ∑ i : Fin m, γ i * b (i.val + 1) n) / N').den.Coprime N'Sources
- Qing Lu, Explicit Congruences for Modular Forms of Higher Weights, Journal of Integer Sequences 29 (2026), Article 26.5.1, Theorem 1 (
jis-2026-26.5.1-thm-1).
TheoremLu's congruence for the divisor sums \(\sigma_{k-1}(n)\)#●ready to prove
Let \(k \ge 12\) be even with \(k \ne 14\). For \(w, n \in \mathbb{N}\) let \(\sigma_{-1}(n)\) be \(1\) if \(n = 0\) and \(0\) otherwise; for \(w \ge 1\) let \(\sigma_{w-1}(0) = -B_w/(2w)\), where \(B_w\) is the \(w\)-th Bernoulli number, and \(\sigma_{w-1}(n) = \sum_{d \mid n} d^{w-1}\) for \(n \ge 1\). In \(\mathbb{Q}[[q]]\) let \(G_w = \sum_{n \ge 0} \sigma_{w-1}(n) q^n\) and \(\Delta = q \prod_{j \ge 1} (1 - q^j)^{24}\), and for \(i, n \in \mathbb{N}\) let \(c_i(n)\) be the coefficient of \(q^n\) in \(\Delta^i G_{k-12i}\), that is, the additive convolution \((\tau^{*i} * \sigma_{k-12i-1})(n)\), where \(\tau(n)\) is the coefficient of \(q^n\) in \(\Delta\). Let \(N_w = 1\) for \(w \le 2\), let \(N_w\) be the absolute value of the numerator of \(\sigma_{w-1}(0)\) for \(w \ge 3\), and let \(N'_k\) be the largest divisor of \(N_k\) that is coprime to \(N_{k-12i}\) for every \(1 \le i \le \lfloor k/12 \rfloor\). For rationals \(x, y\) and a positive integer \(N\), \(x \equiv y \pmod{N}\) means that \((x - y)/N\) has denominator coprime to \(N\). Let \(m = \lfloor k/12 \rfloor - 1\) if \(k \equiv 2 \pmod{12}\) and \(m = \lfloor k/12 \rfloor\) otherwise, let \(C\) be the \(m \times m\) matrix \(\bigl(c_i(j)/\sigma_{k-12i-1}(0)\bigr)_{1 \le i, j \le m}\), and let \((\gamma_1, \dots, \gamma_m) = (\sigma_{k-1}(1), \dots, \sigma_{k-1}(m))\, C^{-1}\). Then for every \(n \ge 1\), \(\sigma_{k-1}(n) \equiv \sum_{i=1}^m \bigl(\gamma_i/\sigma_{k-12i-1}(0)\bigr)\, c_i(n) \pmod{N'_k}\).
MetaMathlibExt.sigma_congr_sum_tau_conv_sigma · 15 lines
public theorem_wanted sigma_congr_sum_tau_conv_sigma (k : ℕ) (hk : 12 ≤ k) (hke : Even k)
(hk14 : k ≠ 14) :
let σ₀ : ℕ → ℕ → ℚ := fun w n => if w = 0 then (if n = 0 then 1 else 0)
else if n = 0 then -bernoulli w / (2 * w) else (ArithmeticFunction.sigma (w - 1) n : ℚ)
let Δ : ℚ⟦X⟧ := X * ∏' j : ℕ, (1 - X ^ (j + 1)) ^ 24
let conv : ℕ → ℕ → ℚ := fun i n => coeff n (Δ ^ i * PowerSeries.mk (σ₀ (k - 12 * i)))
let Nw : ℕ → ℕ := fun w => if w ≤ 2 then 1 else (σ₀ w 0).num.natAbs
let N' : ℕ := ((Nat.divisors (Nw k)).filter fun d =>
∀ i ∈ Finset.Icc 1 (k / 12), Nat.Coprime d (Nw (k - 12 * i))).sup id
let m : ℕ := if k % 12 = 2 then k / 12 - 1 else k / 12
let C : Matrix (Fin m) (Fin m) ℚ := fun i j =>
conv (i.val + 1) (j.val + 1) / σ₀ (k - 12 * (i.val + 1)) 0
let γ : Fin m → ℚ := Matrix.vecMul (fun j => σ₀ k (j.val + 1)) C⁻¹
∀ n : ℕ, 1 ≤ n → ((σ₀ k n - ∑ i : Fin m,
γ i / σ₀ (k - 12 * (i.val + 1)) 0 * conv (i.val + 1) n) / N').den.Coprime N'Sources
- Qing Lu, Explicit Congruences for Modular Forms of Higher Weights, Journal of Integer Sequences 29 (2026), Article 26.5.1, Corollary 3 (
jis-2026-26.5.1-cor-3).
TheoremLu's one-term congruence for \(G_k\)#●ready to prove
Let \(k \in \{12, 16, 18, 20, 22, 26\}\). For \(w, n \in \mathbb{N}\) let \(\sigma_{-1}(n)\) be \(1\) if \(n = 0\) and \(0\) otherwise; for \(w \ge 1\) let \(\sigma_{w-1}(0) = -B_w/(2w)\), where \(B_w\) is the \(w\)-th Bernoulli number, and \(\sigma_{w-1}(n) = \sum_{d \mid n} d^{w-1}\) for \(n \ge 1\). In \(\mathbb{Q}[[q]]\) let \(G_w = \sum_{n \ge 0} \sigma_{w-1}(n) q^n\), let \(E_w = G_w/\sigma_{w-1}(0)\) for even \(w\), and let \(\Delta = q \prod_{j \ge 1} (1 - q^j)^{24}\); for \(i, n \in \mathbb{N}\) write \(b_i(n)\) for the coefficient of \(q^n\) in \(\Delta^i E_{k-12i}\). Let \(N_w = 1\) for \(w \le 2\), let \(N_w\) be the absolute value of the numerator of \(\sigma_{w-1}(0)\) for \(w \ge 3\), and let \(N'_k\) be the largest divisor of \(N_k\) that is coprime to \(N_{k-12i}\) for every \(1 \le i \le \lfloor k/12 \rfloor\). For rationals \(x, y\) and a positive integer \(N\), \(x \equiv y \pmod{N}\) means that \((x - y)/N\) has denominator coprime to \(N\). Then \(\sigma_{k-1}(n) \equiv b_1(n) \pmod{N'_k}\) for every \(n \ge 0\), that is, \(G_k \equiv \Delta E_{k-12} \pmod{N'_k}\) coefficientwise; in particular the congruence holds for every \(n \ge 1\).
public theorem_wanted eisenstein_congr_delta_mul_eisenstein_dim_one (k : ℕ)
(hk : k ∈ ({12, 16, 18, 20, 22, 26} : Finset ℕ)) :
let σ₀ : ℕ → ℕ → ℚ := fun w n => if w = 0 then (if n = 0 then 1 else 0)
else if n = 0 then -bernoulli w / (2 * w) else (ArithmeticFunction.sigma (w - 1) n : ℚ)
let Δ : ℚ⟦X⟧ := X * ∏' j : ℕ, (1 - X ^ (j + 1)) ^ 24
let b : ℕ → ℕ → ℚ := fun i n =>
coeff n (Δ ^ i * PowerSeries.mk (σ₀ (k - 12 * i))) / σ₀ (k - 12 * i) 0
let Nw : ℕ → ℕ := fun w => if w ≤ 2 then 1 else (σ₀ w 0).num.natAbs
let N' : ℕ := ((Nat.divisors (Nw k)).filter fun d =>
∀ i ∈ Finset.Icc 1 (k / 12), Nat.Coprime d (Nw (k - 12 * i))).sup id
(∀ n : ℕ, ((σ₀ k n - b 1 n) / N').den.Coprime N') ∧
∀ n : ℕ, 1 ≤ n → ((σ₀ k n - b 1 n) / N').den.Coprime N'Sources
- Qing Lu, Explicit Congruences for Modular Forms of Higher Weights, Journal of Integer Sequences 29 (2026), Article 26.5.1, Corollary 5 (
jis-2026-26.5.1-cor-5).
TheoremLu's two-term congruence for \(G_k\)#●ready to prove
Let \(k \in \{24, 28, 30, 32, 34, 38\}\). For \(w, n \in \mathbb{N}\) let \(\sigma_{-1}(n)\) be \(1\) if \(n = 0\) and \(0\) otherwise; for \(w \ge 1\) let \(\sigma_{w-1}(0) = -B_w/(2w)\), where \(B_w\) is the \(w\)-th Bernoulli number, and \(\sigma_{w-1}(n) = \sum_{d \mid n} d^{w-1}\) for \(n \ge 1\). In \(\mathbb{Q}[[q]]\) let \(G_w = \sum_{n \ge 0} \sigma_{w-1}(n) q^n\), let \(E_w = G_w/\sigma_{w-1}(0)\) for even \(w\), and let \(\Delta = q \prod_{j \ge 1} (1 - q^j)^{24}\); for \(i, n \in \mathbb{N}\) write \(b_i(n)\) for the coefficient of \(q^n\) in \(\Delta^i E_{k-12i}\). Let \(N_w = 1\) for \(w \le 2\), let \(N_w\) be the absolute value of the numerator of \(\sigma_{w-1}(0)\) for \(w \ge 3\), and let \(N'_k\) be the largest divisor of \(N_k\) that is coprime to \(N_{k-12i}\) for every \(1 \le i \le \lfloor k/12 \rfloor\). For rationals \(x, y\) and a positive integer \(N\), \(x \equiv y \pmod{N}\) means that \((x - y)/N\) has denominator coprime to \(N\). Let \(\gamma_2 = 2^{k-1} + 25 - 1/\sigma_{k-13}(0)\). Then \(\sigma_{k-1}(n) \equiv b_1(n) + \gamma_2 b_2(n) \pmod{N'_k}\) for every \(n \ge 0\), that is, \(G_k \equiv \Delta E_{k-12} + \gamma_2 \Delta^2 E_{k-24} \pmod{N'_k}\) coefficientwise; in particular the congruence holds for every \(n \ge 1\).
public theorem_wanted eisenstein_congr_delta_mul_eisenstein_dim_two (k : ℕ)
(hk : k ∈ ({24, 28, 30, 32, 34, 38} : Finset ℕ)) :
let σ₀ : ℕ → ℕ → ℚ := fun w n => if w = 0 then (if n = 0 then 1 else 0)
else if n = 0 then -bernoulli w / (2 * w) else (ArithmeticFunction.sigma (w - 1) n : ℚ)
let Δ : ℚ⟦X⟧ := X * ∏' j : ℕ, (1 - X ^ (j + 1)) ^ 24
let b : ℕ → ℕ → ℚ := fun i n =>
coeff n (Δ ^ i * PowerSeries.mk (σ₀ (k - 12 * i))) / σ₀ (k - 12 * i) 0
let Nw : ℕ → ℕ := fun w => if w ≤ 2 then 1 else (σ₀ w 0).num.natAbs
let N' : ℕ := ((Nat.divisors (Nw k)).filter fun d =>
∀ i ∈ Finset.Icc 1 (k / 12), Nat.Coprime d (Nw (k - 12 * i))).sup id
let γ₂ : ℚ := 2 ^ (k - 1) + 25 - 1 / σ₀ (k - 12) 0
(∀ n : ℕ, ((σ₀ k n - (b 1 n + γ₂ * b 2 n)) / N').den.Coprime N') ∧
∀ n : ℕ, 1 ≤ n → ((σ₀ k n - (b 1 n + γ₂ * b 2 n)) / N').den.Coprime N'Sources
- Qing Lu, Explicit Congruences for Modular Forms of Higher Weights, Journal of Integer Sequences 29 (2026), Article 26.5.1, Corollary 6 (
jis-2026-26.5.1-cor-6).
TheoremLu's three-term congruence for \(G_k\)#●ready to prove
Let \(k \in \{36, 40, 42, 44, 46, 50\}\). For \(w, n \in \mathbb{N}\) let \(\sigma_{-1}(n)\) be \(1\) if \(n = 0\) and \(0\) otherwise; for \(w \ge 1\) let \(\sigma_{w-1}(0) = -B_w/(2w)\), where \(B_w\) is the \(w\)-th Bernoulli number, and \(\sigma_{w-1}(n) = \sum_{d \mid n} d^{w-1}\) for \(n \ge 1\). In \(\mathbb{Q}[[q]]\) let \(G_w = \sum_{n \ge 0} \sigma_{w-1}(n) q^n\), let \(E_w = G_w/\sigma_{w-1}(0)\) for even \(w\), and let \(\Delta = q \prod_{j \ge 1} (1 - q^j)^{24}\); for \(i, n \in \mathbb{N}\) write \(b_i(n)\) for the coefficient of \(q^n\) in \(\Delta^i E_{k-12i}\). Let \(N_w = 1\) for \(w \le 2\), let \(N_w\) be the absolute value of the numerator of \(\sigma_{w-1}(0)\) for \(w \ge 3\), and let \(N'_k\) be the largest divisor of \(N_k\) that is coprime to \(N_{k-12i}\) for every \(1 \le i \le \lfloor k/12 \rfloor\). For rationals \(x, y\) and a positive integer \(N\), \(x \equiv y \pmod{N}\) means that \((x - y)/N\) has denominator coprime to \(N\). Let \(\gamma_2 = 2^{k-1} + 25 - 1/\sigma_{k-13}(0)\) and \(\gamma_3 = 3^{k-1} + 48 \cdot 2^{k-1} + 949 - (2^{k-13} + 25)/\sigma_{k-13}(0) - (2^{k-1} + 25)/\sigma_{k-25}(0) + 1/\bigl(\sigma_{k-13}(0)\,\sigma_{k-25}(0)\bigr)\). Then \(\sigma_{k-1}(n) \equiv b_1(n) + \gamma_2 b_2(n) + \gamma_3 b_3(n) \pmod{N'_k}\) for every \(n \ge 0\), that is, \(G_k \equiv \Delta E_{k-12} + \gamma_2 \Delta^2 E_{k-24} + \gamma_3 \Delta^3 E_{k-36} \pmod{N'_k}\) coefficientwise; in particular the congruence holds for every \(n \ge 1\).
MetaMathlibExt.eisenstein_congr_delta_mul_eisenstein_dim_three · 15 lines
public theorem_wanted eisenstein_congr_delta_mul_eisenstein_dim_three (k : ℕ)
(hk : k ∈ ({36, 40, 42, 44, 46, 50} : Finset ℕ)) :
let σ₀ : ℕ → ℕ → ℚ := fun w n => if w = 0 then (if n = 0 then 1 else 0)
else if n = 0 then -bernoulli w / (2 * w) else (ArithmeticFunction.sigma (w - 1) n : ℚ)
let Δ : ℚ⟦X⟧ := X * ∏' j : ℕ, (1 - X ^ (j + 1)) ^ 24
let b : ℕ → ℕ → ℚ := fun i n =>
coeff n (Δ ^ i * PowerSeries.mk (σ₀ (k - 12 * i))) / σ₀ (k - 12 * i) 0
let Nw : ℕ → ℕ := fun w => if w ≤ 2 then 1 else (σ₀ w 0).num.natAbs
let N' : ℕ := ((Nat.divisors (Nw k)).filter fun d =>
∀ i ∈ Finset.Icc 1 (k / 12), Nat.Coprime d (Nw (k - 12 * i))).sup id
let γ₂ : ℚ := 2 ^ (k - 1) + 25 - 1 / σ₀ (k - 12) 0
let γ₃ : ℚ := 3 ^ (k - 1) + 2 ^ (k - 1) * 48 + 949 - (2 ^ (k - 13) + 25) / σ₀ (k - 12) 0 -
(2 ^ (k - 1) + 25) / σ₀ (k - 24) 0 + 1 / (σ₀ (k - 12) 0 * σ₀ (k - 24) 0)
(∀ n : ℕ, ((σ₀ k n - (b 1 n + γ₂ * b 2 n + γ₃ * b 3 n)) / N').den.Coprime N') ∧
∀ n : ℕ, 1 ≤ n → ((σ₀ k n - (b 1 n + γ₂ * b 2 n + γ₃ * b 3 n)) / N').den.Coprime N'Sources
- Qing Lu, Explicit Congruences for Modular Forms of Higher Weights, Journal of Integer Sequences 29 (2026), Article 26.5.1, Corollary 7 (
jis-2026-26.5.1-cor-7).
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