The coset-sum criterion for the period relation

Let ℛₙ = k[ℳₙ], with ℳₙ the projective determinant-n matrix module and Γ = PSL(2, ℤ) acting on it on both sides. For ξ ∈ ℛₙ and a right coset K = M Γ, write ⟨ξ, K⟩ = ∑_{M ∈ K} c_M. Popa and Zagier's §3 Theorem 2(b) says that a solution ξ of the exchange relations (B) satisfies the period relation

(A) (1 - S) ξ - Tₙ^∞ (1 - S) ∈ (1 - T) ℛₙ

if and only if ⟨ξ, K⟩ = -1 for every right coset K. This is how they deduce (A) for their explicit element from (B) and its coset identity.

Proof

Put X = (1 - S) ξ - Tₙ^∞ (1 - S), U = T S, and let S and U also act on the right.

• By the first exchange relation, ξ (1 + S) = (1 + U + U²) y for some y, and (1 - S) (1 + U + U²) = (1 - T) (-S (1 + U + U²)) in k[Γ], so X (1 + S) = (1 - T) a with a = -S (1 + U + U²) y. Since ξ (1 + S) has coset sums 2 ⟨ξ, K⟩, the coset sums of a are ⟨a, K⟩ = -2 ⟨ξ, S⁻¹ K⟩.
• By the second exchange relation (1 - S) ξ (1 + U + U²) = 0. On the right, (1 - S) (1 + U + U²) = -(1 - T) S (1 + U + U²), and Tₙ^∞ (1 - T) = (1 - T) ν, where ν is the sum of the classes of the matrices (a b; 0 d) with ad = n, a > 0 and 0 ≤ b < a. Hence X (1 + U + U²) = (1 - T) b with b = ν S (1 + U + U²). The matrices in ν represent the right cosets, each once, so ⟨b, K⟩ = 3.
• Left multiplication by 1 - T is injective on ℛₙ, as ⟨T⟩ acts on ℳₙ with infinite orbits. If X = (1 - T) m, then m (1 + S) = a and m (1 + U + U²) = b, so 2 ⟨m, K⟩ = -2 ⟨ξ, S⁻¹ K⟩ and 3 ⟨m, K⟩ = 3; hence ⟨ξ, K⟩ = -1.
• Conversely, if ⟨ξ, K⟩ = -1, then -a / 2 + b / 3 has vanishing coset sums, so it lies in ℛₙ (S - 1) + ℛₙ (U - 1), as S and U generate Γ. This produces m with m (1 + S) = -a and m (1 + U + U²) = -b. Then X + (1 - T) m is killed by 1 + S and by 1 + U + U² on the right, so it vanishes by the acyclicity of ℛₙ (Popa–Zagier, Lemma 2), and X = (1 - T) (-m).

The matrices of ν are the images of the upper-triangular representatives ℳₙ^∞ under the anti-involution (a b; c d) ↦ (d b; c a), which exchanges the left and right actions and fixes T. This gives both the identity Tₙ^∞ (1 - T) = (1 - T) ν and the coset sums of ν.

Main results

• ExchangeRelations.periodRelation_iff_mapDomain_orbitRel_mk_coeff_eq_neg_one: §3 Theorem 2(b), for n > 0 and 2, 3 invertible in k.

References

• A. Popa and D. Zagier, An elementary proof of the Eichler–Selberg trace formula, J. Reine Angew. Math. 762 (2020), 105–122, arXiv:1711.00327, §3, Lemma 2 and Theorem 2.

The anti-involution (a b; c d) ↦ (d b; c a) on ℳₙ

The right coset representatives ν

The two halves of the period defect

Popa–Zagier's Theorem 2(b)

theorem TauCeti.TraceFormulaMatrixModule.ExchangeRelations.periodRelation_iff_mapDomain_orbitRel_mk_coeff_eq_neg_one {k : Type u_1} {n : ℤ} [CommRing k] [Invertible 2] [Invertible 3] {ξ : MonoidAlgebra k (TraceFormulaMatrixModule n)} (hξ : ExchangeRelations k n ξ) (hn : 0 < n) : PeriodRelation k n ξ ↔ ∀ (x : TraceFormulaMatrixModule n), (Finsupp.mapDomain (Quotient.mk (MulAction.orbitRel (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℤ)ᵐᵒᵖ (TraceFormulaMatrixModule n))) ξ.coeff) ⟦x⟧ = -1

Popa–Zagier, §3 Theorem 2(b). For n > 0 and 2, 3 invertible in k, a solution ξ ∈ k[ℳₙ] of the exchange relations (B) satisfies the period relation (A), (1 - S) ξ - Tₙ^∞ (1 - S) ∈ (1 - T) k[ℳₙ], if and only if its coefficients sum to -1 over every right coset of PSL(2, ℤ) in ℳₙ: ⟨ξ, K⟩ = -1 for all K.