The exchange relations of Popa and Zagier #
Let ℛₙ = k[ℳₙ] be the free k-module on the projective determinant-n matrix module
ℳₙ = TauCeti.TraceFormulaMatrixModule n, with the left and right actions of Γ = PSL(2, ℤ).
Write S = [0, -1; 1, 0], T = [1, 1; 0, 1] and U = T S = [1, -1; 1, 0], so that
S² = U³ = 1 in Γ. An element ξ ∈ ℛₙ satisfies Popa and Zagier's exchange relations (B) if
ξ (1 + S) ∈ (1 + U + U²) ℛₙ and ξ (1 + U + U²) ∈ (1 + S) ℛₙ,
where ξ γ and γ ξ denote the right and left actions of γ ∈ k[Γ] on ℛₙ, induced by the
right and left actions of Γ on ℳₙ. Over ℚ, (B) says that ξ lies in the set ℬ of §3,
eq. (6).
For a right coset K = M Γ of a matrix M ∈ ℳₙ and ξ = ∑ c_M M, let ⟨ξ, K⟩ = ∑_{M ∈ K} c_M.
Left multiplication by g ∈ Γ permutes the right cosets. §3 Theorem 2(a) states that if ξ
satisfies (B), then ⟨ξ, g K⟩ = ⟨ξ, K⟩ for every g ∈ Γ and every right coset K; hence
⟨ξ, K⟩ is the same for all right cosets K in a double coset Γ M Γ.
Solutions of (B) that also satisfy the period relation (A) exist (§1, proved in §3). Let
𝓘 = (1 + S) ℛₙ + (1 + U + U²) ℛₙ, the right ideal of §3, eq. (5), and
𝒜 = {ξ | ξ (1 + S) ∈ 𝓘, ξ (1 + U + U²) ∈ 𝓘}. For n ≠ 0, every solution of (A) lies in 𝒜,
by the Choie–Zagier criterion (§3, Lemma 1); Lemma 3 then corrects it by an element of 𝓘 to a
solution of both (A) and (B).
Main definitions and results #
TauCeti.TraceFormulaMatrixModule.ExchangeRelations: the exchange relations (B).TauCeti.TraceFormulaMatrixModule.ExchangeRelations.mapDomain_orbitRel_mk_coeff_smul: §3 Theorem 2(a), the right coset sums of a solution of (B) are invariant under left multiplication.TauCeti.TraceFormulaMatrixModule.ExchangeRelations.mapDomain_orbitRel_mk_coeff_smul_op_smul: they are constant on every double coset.TauCeti.TraceFormulaMatrixModule.PeriodRelation.one_add_S_apply_mem_sup,TauCeti.TraceFormulaMatrixModule.PeriodRelation.one_add_U_add_U_sq_apply_mem_sup: forn ≠ 0, every solution of the period relation (A) lies in𝒜.TauCeti.TraceFormulaMatrixModule.exists_periodRelation_and_exchangeRelations: someξ ∈ ℛₙsatisfies both (A) and (B).
Implementation notes #
Right multiplication by g ∈ Γ on ℛₙ is Representation.ofMulAction k PSL(2, ℤ)ᵐᵒᵖ ℳₙ at
MulOpposite.op g; U is written (T : PSL(2, ℤ)) * S there. Left multiplication by the class of
g ∈ SL(2, ℤ) is Representation.ofMulAction k SL(2, ℤ) ℳₙ g, the simp-normal form given by
TauCeti.TraceFormulaMatrixModule.ofMulAction_coe.
The right coset M Γ is the orbit of M under PSL(2, ℤ)ᵐᵒᵖ, that is, its class in the
quotient by MulAction.orbitRel PSL(2, ℤ)ᵐᵒᵖ ℳₙ. The coset sums K ↦ ⟨ξ, K⟩ form the finitely
supported function ξ.coeff.mapDomain (Quotient.mk (MulAction.orbitRel PSL(2, ℤ)ᵐᵒᵖ ℳₙ)), the
coefficients of the orbit-sum map MonoidAlgebra.mapDomainLinearMap in simp-normal form.
Popa and Zagier work over ℚ. Theorem 2(a) only needs 2 and 3 to be cancellable in k,
which is assumed as IsSMulRegular k 2 and IsSMulRegular k 3; this holds, for instance, if k
is torsion-free or if 2 and 3 are invertible in k.
The ideal 𝓘 is the sum of the ranges of left multiplication by 1 + S and by 1 + U + U², with
U written T * S in SL(2, ℤ).
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: the relations (A) and (B), the
coset sums
⟨ξ, K⟩and the existence of solutions of (A) and (B) in §1; the ideal𝓘, eq. (5), the set𝒜, the setℬ, eq. (6), Lemmas 1–3 and Theorem 2(a) in §3.
The exchange relations (B) of Popa and Zagier for ξ ∈ k[ℳₙ]:
ξ (1 + S) ∈ (1 + U + U²) k[ℳₙ] and ξ (1 + U + U²) ∈ (1 + S) k[ℳₙ], where U = T S and the
products are right multiplication by 1 + S, 1 + U + U² and left multiplication by
1 + U + U², 1 + S.
- one_add_S : (1 + (Representation.ofMulAction k (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℤ)ᵐᵒᵖ (TraceFormulaMatrixModule n)) (MulOpposite.op ↑ModularGroup.S)) ξ ∈ (1 + (Representation.ofMulAction k (Matrix.SpecialLinearGroup (Fin 2) ℤ) (TraceFormulaMatrixModule n)) (ModularGroup.T * ModularGroup.S) + (Representation.ofMulAction k (Matrix.SpecialLinearGroup (Fin 2) ℤ) (TraceFormulaMatrixModule n)) (ModularGroup.T * ModularGroup.S) ^ 2).range
ξ (1 + S) ∈ (1 + U + U²) k[ℳₙ]. - one_add_U_add_U_sq : (1 + (Representation.ofMulAction k (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℤ)ᵐᵒᵖ (TraceFormulaMatrixModule n)) (MulOpposite.op (↑ModularGroup.T * ↑ModularGroup.S)) + (Representation.ofMulAction k (Matrix.ProjectiveSpecialLinearGroup (Fin 2) ℤ)ᵐᵒᵖ (TraceFormulaMatrixModule n)) (MulOpposite.op (↑ModularGroup.T * ↑ModularGroup.S)) ^ 2) ξ ∈ (1 + (Representation.ofMulAction k (Matrix.SpecialLinearGroup (Fin 2) ℤ) (TraceFormulaMatrixModule n)) ModularGroup.S).range
ξ (1 + U + U²) ∈ (1 + S) k[ℳₙ].
Instances For
Popa–Zagier, §3 Theorem 2(a). If ξ satisfies the exchange relations (B) and 2, 3 are
cancellable in k, then the right coset sums of ξ are invariant under left multiplication:
⟨ξ, g K⟩ = ⟨ξ, K⟩ for every g ∈ SL(2, ℤ) and every right coset K.
The right coset sums of a solution ξ of the exchange relations (B) are constant on every
double coset: ⟨ξ, g M h Γ⟩ = ⟨ξ, M Γ⟩ for g ∈ SL(2, ℤ) and h ∈ PSL(2, ℤ), if 2 and 3
are cancellable in k.
Solutions of the period relation and the exchange relations #
Left multiplication by g on k[ℳₙ] commutes with right multiplication by 1 + h.
Left multiplication by g on k[ℳₙ] commutes with right multiplication by 1 + h + h².
Solutions of (A) lie in 𝒜, the relation for S (Popa–Zagier, §3): for n ≠ 0 and 2,
3 invertible in k, if ξ satisfies the period relation (A), then
ξ (1 + S) ∈ (1 + S) k[ℳₙ] + (1 + U + U²) k[ℳₙ].
Solutions of (A) lie in 𝒜, the relation for U (Popa–Zagier, §3): for n ≠ 0 and 2,
3 invertible in k, if ξ satisfies the period relation (A), then
ξ (1 + U + U²) ∈ (1 + S) k[ℳₙ] + (1 + U + U²) k[ℳₙ].
Solutions of both (A) and (B) exist (Popa–Zagier, §1; proved in §3 after Lemma 3): if 2
and 3 are invertible in k, then for every n some ξ ∈ k[ℳₙ] satisfies both the period
relation (A) and the exchange relations (B).