Let p be a prime, and M_{p} 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 M_{p} is a filtered F_{p}-algebra under the weight filtration w, and w(f^{n}) = n w(f) for all forms f in M_{p}.
Theorem: If f is a form in M_{p} and ℓ is a prime different from p then the sequence {T_{ℓ}(f^{n})}_{n} of elements in M_{p} satisfies a linear recursion of order ℓ + 1 over M_{p}. Moreover, the companion polynomial of this recursion
Proof: It's like the modular equation for j of level ℓ — so much so that P_{ℓ, 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 M_{p} 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 M_{p} so that (ignoring the filtration) M_{p} is the polynomial algebra F_{p}[y] with finitely many elements inverted. As a consequence, the Hecke recursion polynomial P_{ℓ, y} completely encodes the action of the Hecke operator T_{ℓ} on all of M_{p}.
prime p | # ss j-inv | M_{p} | y | Filtration data | Recursion polynomials |
2 | 1 | F_{2}[y] | y = Δ | w(y) = 12 |
P_{ℓ, y} for ℓ < 200
Recovers computations of Nicolas-Serre-Deléglise. |
3 | 1 | F_{3}[y] | y = Δ | w(y) = 12 | P_{ℓ, y} for ℓ < 200 |
5 | 1 | F_{5}[y] | y = Δ | w(y) = 12 | P_{ℓ, y} for ℓ < 200 |
7 | 1 | F_{7}[y] | y = Δ | w(y) = 12 | P_{ℓ, y} for ℓ < 200 |
11 | 2 | F_{11}[y, y^{–1}] |
y = (E_{4})^{5} y^{–1} = (E_{6})^{5} |
w(y) = 20 w(y^{–1}) = 30 |
P_{ℓ, y} for ℓ < 100 |
13 | 1 | F_{13}[y] | y = Δ | w(y) = 12 | P_{ℓ, y} for ℓ < 200 |
17 | 2 | F_{17}[y, y^{–1}] |
y = (E_{4})^{4} |
w(y) = 16 w(y^{–1}) = 48 |
P_{ℓ, y} for ℓ < 100 |
19 | 2 | F_{19}[y, y^{–1}] |
y = (E_{6})^{3} |
w(y) = 18 w(y^{–1}) = 36 |
P_{ℓ, y} for ℓ < 100 |
23 | 3 | F_{23}[y, y^{–1}, (y + 16)^{–1}] |
y = (E_{4})^{4}E_{6} |
w(y) = 22 w(y^{–1}) = 66 w((y + 16)^{–1}) = 44 |
P_{ℓ, 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 F_{p}(y) associated to each recurrence sequence.
Alternatively, we can deduce P_{ℓ, y} from P_{ℓ, j}, since in level one we always have F_{p}(y) = F_{p}(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.