Strong multiplicity one without a nebentypus hypothesis #
The Fourier coefficients of a newform at the indices prime to its level determine its nebentypus:
at a good prime p the Hecke recurrence gives χ(p) = (a_p² - a_{p²}) / p^{k-1}
(TauCeti.char_prime_eq_coeff), and by Dirichlet's theorem on primes in arithmetic progressions a
character is determined by its values at all but finitely many primes
(DirichletCharacter.changeLevel_eq_changeLevel_of_eventually_apply_prime_eq). So two newforms
of levels N and M whose coefficients a_n agree at every n prime to N M outside a finite
set have nebentypes inducing the same character modulo every common multiple of N and M.
This removes the nebentypus hypothesis from strong multiplicity one stated on indices: such
newforms have the same level and are equal (Miyake, Theorem 4.6.19, for newforms), and at a fixed
level the coefficient and index forms of Miyake, Theorem 4.6.12, hold with no assumption on the
nebentypus. Agreement of the eigenvalues at the primes alone does not see a_{p²}, and the
prime form, HeckeRing.GL2.Newform.eq_of_forall_notMem_prime_eigenvalue_eq, keeps its nebentypus
hypothesis.
Main results #
All in the namespace HeckeRing.GL2.Newform:
eventually_dirichletLift_apply_prime_eq_of_forall_notMem_qExpansion_coeff_eq: newforms whose coefficients agree at the good indices outside a finite set have nebentypes agreeing at all but finitely many primes.χ_comp_unitsMap_eq_of_forall_notMem_qExpansion_coeff_eq: their nebentypes induce the same character modulo every common multiple of the levels.χ_eq_of_forall_notMem_qExpansion_coeff_eq: at a fixed level they have the same nebentypus.eq_of_forall_notMem_qExpansion_coeff_eqandeq_of_forall_notMem_eigenvalue_eq: strong multiplicity one at a fixed level, on the coefficients and on the eigenvalues at the good indices outside a finite set.level_eq_and_coe_eq_of_forall_notMem_qExpansion_coeff_eq: across levels, such newforms have the same level and are the same function.
References #
- T. Miyake, Modular forms, Theorem 4.6.12 and Theorem 4.6.19.
- F. Diamond and J. Shurman, A first course in modular forms, Proposition 5.8.5 for the Hecke recurrence.
The coefficients of a newform at its good indices determine its nebentypus at almost all
primes. If newforms f and g of levels N and M have the same Fourier coefficient at every
index n prime to N and M outside a finite set S, their zero-extended nebentypes agree at
all but finitely many primes: at a large prime p, both p and p² are good indices outside
S, and χ(p) = (a_p² - a_{p²}) / p^{k-1}.
The coefficients of a newform at its good indices determine its nebentypus. If newforms
f and g of levels N and M have the same Fourier coefficient at every index n prime to
N and M outside a finite set, their nebentypes induce the same character modulo every common
multiple L of N and M.
Newforms of the same level with the same coefficients at the good indices outside a finite set have the same nebentypus.
Strong multiplicity one, on Fourier coefficients (Miyake's own form of Theorem 4.6.12):
two newforms of level N and weight k whose q-expansion coefficients agree at every index
coprime to N outside a finite set are equal. No assumption is made on the nebentypus: the
coefficients determine it (Newform.χ_eq_of_forall_notMem_qExpansion_coeff_eq).
Strong multiplicity one (Miyake, Theorem 4.6.12, fixed level): two newforms of level N
and weight k whose eigenvalues agree at every index coprime to N outside a finite set are
equal. No assumption is made on the nebentypus.
Strong multiplicity one across levels, on Fourier coefficients (Miyake, Theorem 4.6.19,
for newforms). Let f, g be newforms of the same weight and of levels N, M whose
q-expansion coefficients agree at every index prime to N and M outside a finite set. Then
N = M and f and g are the same function on ℍ; their nebentypes then agree by
Newform.χ_comp_unitsMap_eq_of_forall_notMem_qExpansion_coeff_eq.