Let p be a prime, and Mp the space of mod-p reductions of q-expansions of all modular forms of level one and weight divisible by p – 1 defined over Z. Then Mp is a filtered Fp-algebra under the weight filtration w, and w(fn) = n w(f) for all forms f in Mp.
Theorem: If f is a form in Mp and ℓ is a prime different from p then the sequence {Tℓ(fn)}n of elements in Mp satisfies a linear recursion of order ℓ + 1 over Mp. Moreover, the companion polynomial of this recursion
Proof: It's like the modular equation for j of level ℓ — so much so that Pℓ, j, defined analogously, is the modular equation mod p. See Proposition 6.2 in my thesis for details and a more general statement. The case p = 2 appears, in refined form, in Nicolas-Serre (see Théorème 3.1).
Since Mp is the algebra of regular functions on the complement of the supersingular locus in the j-line, we can always find a parameter y in Mp so that (ignoring the filtration) Mp is the polynomial algebra Fp[y] with finitely many elements inverted. As a consequence, the Hecke recursion polynomial Pℓ, y completely encodes the action of the Hecke operator Tℓ on all of Mp.
prime p | # ss j-inv | Mp | y | Filtration data | Recursion polynomials |
2 | 1 | F2[y] | y = Δ | w(y) = 12 |
Pℓ, y for ℓ < 200
Recovers computations of Nicolas-Serre-Deléglise. |
3 | 1 | F3[y] | y = Δ | w(y) = 12 | Pℓ, y for ℓ < 200 |
5 | 1 | F5[y] | y = Δ | w(y) = 12 | Pℓ, y for ℓ < 200 |
7 | 1 | F7[y] | y = Δ | w(y) = 12 | Pℓ, y for ℓ < 200 |
11 | 2 | F11[y, y–1] |
y = (E4)5 y–1 = (E6)5 |
w(y) = 20 w(y–1) = 30 |
Pℓ, y for ℓ < 100 |
13 | 1 | F13[y] | y = Δ | w(y) = 12 | Pℓ, y for ℓ < 200 |
17 | 2 | F17[y, y–1] |
y = (E4)4 |
w(y) = 16 w(y–1) = 48 |
Pℓ, y for ℓ < 100 |
19 | 2 | F19[y, y–1] |
y = (E6)3 |
w(y) = 18 w(y–1) = 36 |
Pℓ, y for ℓ < 100 |
23 | 3 | F23[y, y–1, (y + 16)–1] |
y = (E4)4E6 |
w(y) = 22 w(y–1) = 66 w((y + 16)–1) = 44 |
Pℓ, y for ℓ < 100 |
The Hecke recursion polynomials here were computed in SAGE by finding the kernel of the dimension-(ℓ + 2) Kronecker matrix over Fp(y) associated to each recurrence sequence.
Alternatively, we can deduce Pℓ, y from Pℓ, j, since in level one we always have Fp(y) = Fp(j). Andrew Sutherland has made the coefficients of the modular equations for j available on his webpage. This gives an independent check on the Hecke recursion polynomials above.