Atkin–Lehner operators commute with Γ₀ double-coset slash operators #
For an exact divisor Q of N, the Atkin–Lehner operator W_Q on M_k(Γ₀(N)) commutes with
the Hecke operator [Γ₀(N) α Γ₀(N)] of every double coset whose determinant is coprime to Q,
and so does its normalization 𝒲_Q; likewise on S_k(Γ₀(N)). These results are strictly
coset-by-coset. Once a future identification theorem expresses the classical Tₙ and U_p
operators as the relevant sums of Γ₀ double-coset slash operators, they will imply the
corresponding commutation statements for those classical operators; that identification is not
proved here.
The proof is HeckeRing.GL2.heckeSlashSum_slash_of_mem_normalizer applied to the Atkin–Lehner
matrix read in GL(2, ℚ): that matrix normalizes Γ₀(N)
(TauCeti.IsAtkinLehnerMatrix.mem_normalizer_map_mapGL) and fixes each double coset of
determinant coprime to Q (TauCeti.IsAtkinLehnerMatrix.inv_mul_mul_mem_doubleCoset).
The statements are for every Atkin–Lehner matrix of a divisor Q ∣ N, not only the standard one
the operators Nat.IsExactDivisor.atkinLehnerOperator are built from.
Main results #
TauCeti.commute_atkinLehnerOperator_heckeSlashGamma0ModularFormEnd,TauCeti.commute_atkinLehnerOperatorCusp_heckeSlashGamma0CuspFormEnd: the raw operatorW_Qcommutes with each Hecke operator of determinant coprime toQ.- In the namespace
TauCeti.Nat.IsExactDivisor,commute_normalizedAtkinLehnerOperator_heckeSlashGamma0ModularFormEndandcommute_normalizedAtkinLehnerOperatorCusp_heckeSlashGamma0CuspFormEnd: so does the normalized operator𝒲_Q.
References #
- A. O. L. Atkin and J. Lehner, Hecke operators on
Γ₀(m), Math. Ann. 185 (1970), 134–160. - G. Shimura, Introduction to the arithmetic theory of automorphic functions, §3.4.
The Atkin–Lehner operator W_Q commutes with the Hecke operator of a double coset of
determinant coprime to Q, on M_k(Γ₀(N)).
The Atkin–Lehner operator W_Q commutes with the Hecke operator of a double coset of
determinant coprime to Q, on S_k(Γ₀(N)).
The normalized Atkin–Lehner operator 𝒲_Q commutes with the Hecke operator of a double
coset of determinant coprime to Q, on M_k(Γ₀(N)).
The normalized Atkin–Lehner operator 𝒲_Q commutes with the Hecke operator of a double
coset of determinant coprime to Q, on S_k(Γ₀(N)).