Congruences for coefficients of level 2 modular functions with poles at 0

Abstract

We give congruences modulo powers of p ∈ {3, 5, 7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and the first author and continuing work done by the authors and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at ∞.

Congruences for coefficients of modular functions in levels 3, 5, and 7 with poles at 0

“…Using Lemma 2.4, the smallest exponent of q appearing in U α p (φ (p) ) m is f α (p) (m). Lemma 2.5 provides a connection between γ p (m, α) and the integers f α (p) (m): γ p is counting the number of 0s and 1s to the left of the first nonzero digit in the base p expansion of m. The key difference between the following lemma and its corresponding lemma in [7] is that there are more digits in bases 3, 5, 7.…”

“…Implicitly, this proves the result by induction. This structure differs from [7] because it allows us to prove the polynomial step in a much cleaner way. Another approach to proving the base case can be found in [5,Lemma 4…”

“…to be the space of weakly holomorphic modular forms of weight k and level N that are holomorphic away from the cusp at ∞, and in the same notation as [7], we use M k (N ) to denote forms holomorphic away from the cusp at 0. For prime N , these are the only cusps.…”

“…Similarly, Jenkins, Andersen, and Thornton [3,8] proved congruences for the Fourier coefficients of elements of canonical bases for M 0 (p) with p = 2, 3, 5, 7. Jenkins, the author, and Moss [7] proved congruences for the Fourier coefficients of elements of a canonical basis for M 0 (2). It is natural to wonder whether a similar result holds for bases of M 0 (p) for the other genus zero primes, mirroring the results of M 0 (p).…”

“…The Jenkins, the author, and Moss in [7] proved congruences for the Fourier coefficients of φ (2) (z) and its powers. In this paper, we use similar techniques to obtain congruences for φ (p) (z) and its powers for p = 3, 5, 7.…”