Acyclicity of permutation modules of free PSL(2, ℤ)-sets #
In Γ = PSL(2, ℤ) the classes of S = (0 -1; 1 0) and U = T * S = (1 -1; 1 0) satisfy
S² = 1, U³ = 1 and U * S = T; these relations are in
TauCeti.NumberTheory.Modular.Relations, where U is written (T : PSL(2, ℤ)) * S. Let X be
a set on which Γ acts freely and k[X] its permutation module over a ring k. Then no nonzero
ξ ∈ k[X] is killed by both 1 + S and 1 + U + U²; Choie and Zagier call this acyclicity.
Popa and Zagier prove it for ℚ[ℳ], where ℳ is the set of integral matrices of positive
determinant modulo ±1; their descent applies to any free X and any ring k. Freeness of the
action is Mathlib's IsCancelSMul PSL(2, ℤ) X. The same holds for a free right action, that is,
a free action of PSL(2, ℤ)ᵐᵒᵖ, with S and U acting by right multiplication: this is the
left version for the free left action g • x = x · g⁻¹.
The proof is a descent. If ξ is killed by both operators then ξ = (T⁻¹ + T′⁻¹) ξ, where
T′ = U² * S is the class of (1 0; 1 1), so the coefficients of ξ satisfy
c(x) = c(T x) + c(T′ x). Every point of the support of ξ therefore has its T- or its
T′-translate in the support. Iterating, a point of the finite support returns to itself under
the class of a product g of copies of (1 1; 0 1) and (1 0; 1 1) in SL(2, ℤ). Such a
product has non-negative entries and is not ±1, so its class is not 1, contradicting
freeness.
Main results #
TauCeti.ModularGroup.disjoint_ker_one_add_S_ker_one_add_T_mul_S_add_sq: for a freePSL(2, ℤ)-setX, the kernels of1 + Sand1 + U + U²onk[X]are disjoint.TauCeti.ModularGroup.disjoint_ker_one_add_op_S_ker_one_add_op_T_mul_S_add_sq: the same for a free rightPSL(2, ℤ)-set.
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, Section 3, Lemma 2.
- Y. J. Choie and D. Zagier, Rational period functions for PSL(2, Z), Contemp. Math. 143 (1993), 89--108.
The descent #
TauCeti.ModularGroup.tPrime is Popa--Zagier's T′ = (1 0; 1 1), whose class in PSL(2, ℤ) is
U² * S. The descent is MulAction.eq_empty_of_forall_exists_smul_mem with
E = {T, T′} and P the set of classes of the matrices in SL(2, ℤ) with non-negative entries.
Acyclicity (Popa--Zagier, Lemma 2): if PSL(2, ℤ) acts freely on X, then on the
permutation module k[X] the kernels of 1 + S and 1 + U + U², with U = T * S, are
disjoint: their intersection is 0.
Freeness is needed: for a nontrivial ring k, on k[ℙ¹(ℚ)], where T fixes ∞, the nonzero
element [0] - [∞] lies in both kernels, since S swaps 0 and ∞ while U permutes 0,
∞, 1 cyclically.
Acyclicity for the right action (Popa--Zagier, Lemma 2): if PSL(2, ℤ) acts freely on
X from the right, that is, PSL(2, ℤ)ᵐᵒᵖ acts freely on X, then on the permutation module
k[X] the kernels of 1 + S and 1 + U + U², with S and U = T * S acting by right
multiplication, are disjoint: their intersection is 0.