Contemporary Mathematics

Volume 627, 2014

http://dx.doi.org/10.1090/conm/627/12528

Hecke grids and congruences for

weakly holomorphic modular forms

Scott Ahlgren and Nickolas Andersen

Abstract. Let U(p) denote the Atkin operator of prime index p. Honda and

Kaneko proved infinite families of congruences of the form f U(p) ≡ 0 (mod p)

for weakly holomorphic modular forms of low weight and level and primes p

in certain residue classes, and conjectured the existence of similar congruences

modulo higher powers of p. Partial results on some of these conjectures were

proved recently by Guerzhoy. We construct infinite families of weakly holomor-

phic modular forms on the Fricke groups Γ∗(N) for N = 1, 2, 3, 4 and describe

explicitly the action of the Hecke algebra on these forms. As a corollary, we

obtain strengthened versions of all of the congruences conjectured by Honda

and Kaneko.

1. Introduction

For a prime number p, let U(p) denote Atkin’s operator, which acts on power

series via

a(n)qn

U(p) :=

a(pn)qn.

In recent work, Honda and Kaneko [5] generalize a theorem of Garthwaite [2] in

order to establish infinite families of congruences of the form

f U(p) ≡ 0 (mod p)

for weakly holomorphic modular forms of low weight and level. For example, it is

shown that for any prime p ≡ 1 (mod 3) and any k ∈ {4, 6, 8, 10, 14} we have

(1.1)

Ek(6z)

η4(6z)

U(p) ≡ 0 (mod p).

For another example, if p ≡ 1 (mod 4), k ∈ {4, 6}, and f ∈ Mk(Γ0(2)) has p-

integral Fourier expansion, then it is shown that

(1.2)

f(4z)

η2(4z)η2(8z)

U(p) ≡ 0 (mod p).

Honda and Kaneko conjecture that these extend to congruences modulo higher pow-

ers of p. For example, they conjecture that for any p ≡ 1 (mod 3), the congruence

2010 Mathematics Subject Classification. Primary 11F33.

Key words and phrases. Atkin operator, Fricke groups, Hecke operators, modular forms.

The first author was supported by grant 208525 from the Simons Foundation.

c 2014 American Mathematical Society

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