Sequences with a Hecke-type recurrence at the primes

A sequence a : ℕ → R satisfying, at every prime p coprime to an auxiliary L and every m coprime to L, a recurrence a_{pm} = c · a_m − d · a_{m/p} (the last term present only when p ∣ m) for some scalars c, d, vanishes at every n ≠ 0 coprime to L as soon as a₁ = 0. This is the combinatorial content of the vanishing of the Fourier coefficients of a Hecke eigenform with a₁ = 0 at the good indices, where c is the eigenvalue at p and d = χ(p) p^{k−1}.

Main results

• TauCeti.eq_zero_of_forall_prime_mul_eq_of_one_eq_zero_of_ne_zero_of_coprime: the vanishing at the indices coprime to L.
• TauCeti.prime_mul_eq_of_prime_pow_recurrence_of_coprime_mul_eq: conversely, a sequence that is multiplicative at coprime indices away from L and satisfies the recurrence along the powers of a single prime p satisfies it at every index coprime to L. Its hypotheses are the fixed-prime and L-restricted instances of conditions (2) and (3) of Diamond–Shurman's Proposition 5.8.5, and its conclusion is the fixed-prime instance of the recurrence the first lemma consumes — and, on a nebentypus space, the coefficient side of the eigen-relation at p.

References

• T. Miyake, Modular forms, §4.6 — the vanishing induction this lemma is the arithmetic core of.
• F. Diamond and J. Shurman, A first course in modular forms, §5.8 — in particular Proposition 5.8.5, whose conditions (2) and (3) are the hypotheses of the second lemma below.

theorem TauCeti.eq_zero_of_forall_prime_mul_eq_of_one_eq_zero_of_ne_zero_of_coprime {R : Type u_1} [NonUnitalNonAssocRing R] {a : ℕ → R} {L : ℕ} (ha : ∀ (p : ℕ), Nat.Prime p → p.Coprime L → ∃ (c : R) (d : R), ∀ (m : ℕ), m.Coprime L → a (p * m) = c * a m - if p ∣ m then d * a (m / p) else 0) (h1 : a 1 = 0) (n : ℕ) (hn0 : n ≠ 0) (hn : n.Coprime L) : a n = 0

A sequence with a Hecke-type prime recurrence and a₁ = 0 vanishes at the indices coprime to L. If at every prime p coprime to L there are scalars c, d with a_{pm} = c · a_m − d · a_{m/p} (the last term only when p ∣ m) for every m coprime to L, and a₁ = 0, then a_n = 0 for every n ≠ 0 coprime to L.

theorem TauCeti.prime_mul_eq_of_prime_pow_recurrence_of_coprime_mul_eq {R : Type u_2} [NonUnitalRing R] {a : ℕ → R} {L p : ℕ} {d : R} (hp : Nat.Prime p) (hpL : p.Coprime L) (hmul : ∀ (u v : ℕ), u.Coprime v → u.Coprime L → v.Coprime L → a (u * v) = a u * a v) (hrec : ∀ (r : ℕ), a (p ^ (r + 2)) = a p * a (p ^ (r + 1)) - d * a (p ^ r)) (m : ℕ) (hm0 : m ≠ 0) (hmL : m.Coprime L) : a (p * m) = a p * a m - if p ∣ m then d * a (m / p) else 0

The recurrence along the powers of p, plus multiplicativity, gives it at every index. Let a : ℕ → R be multiplicative at coprime indices away from L, and suppose that along the powers of a prime p ∤ L it satisfies a_{p^{r+2}} = a_p · a_{p^{r+1}} − d · a_{p^r}. Then it satisfies the full Hecke recurrence a_{pm} = a_p · a_m − d · a_{m/p} at every m ≠ 0 coprime to L, the last term present only when p ∣ m.

The hypotheses are the fixed-prime and L-restricted instances of conditions (3) and (2) of Diamond–Shurman's Proposition 5.8.5, whose own statements are global; the conclusion is the fixed-prime instance of the recurrence hypothesis of eq_zero_of_forall_prime_mul_eq_of_one_eq_zero_of_ne_zero_of_coprime above, which asks for it at every prime coprime to L — and, on a nebentypus space, the coefficient side of the Tₚ-eigen-relation.