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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