Anna Medvedovsky: Hecke recursion polynomials

Hecke recursion polynomials

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

Pℓ, f = Xℓ + 1 + a1Xℓ + ⋯ + aℓX + aℓ + 1    in Mp[X]
satisfies w(ai) ≤ i w(f) for all i.

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).

The data

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.

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