Divisibility properties of coefficients of level $p$ modular functions for genus zero primes

Andersen, Nickolas; Jenkins, Paul A.

doi:10.1090/s0002-9939-2012-11434-0

Cited by 11 publications

(61 citation statements)

References 9 publications

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“…For higher levels, Lehner showed that similar congruences hold for the coefficients of modular functions in M ♯ 0 (p) with p ∈ {2, 3, 5, 7} if the functions have integral Fourier coefficients and the the order of the pole at infinity is bounded appropriately. Andersen and the first author [3] extended Lehner's theorem to include all elements of a canonical basis for M ♯ 0 (p), proving the following congruences, from which Lehner's results follow as a corollary.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 85%

Congruences for coefficients of modular functions

Jenkins

Thornton

2014

Ramanujan J

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We examine canonical bases for weakly holomorphic modular forms of weight 0 and level p = 2, 3, 5, 7, 13 with poles only at the cusp at ∞. We show that many of the Fourier coefficients for elements of these canonical bases are divisible by high powers of p, extending results of the first author and Andersen. Additionally, we prove similar congruences for elements of a canonical basis for the space of modular functions of level 4, and give congruences modulo arbitrary primes for coefficients of such modular functions in levels 1, 2, 3, 4, 5, 7, and 13.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 85%

Congruences for coefficients of modular functions

Jenkins

Thornton

2014

Ramanujan J

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“…There is some literature studying coefficient congruences of a related nature. The papers [1,26] discuss Hauptmoduln on Γ 0 (N ), and [40] discusses other coefficient congruences involving Hauptmoduln. However, there has not been a systematic study of p-adic annihilation for all of the monstrous moonshine Hauptmoduln.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

p-Adic Properties of Hauptmoduln with Applications to Moonshine

Chen

Marks

Tyler

2019

SIGMA

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The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the j-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coefficients of the j-function satisfy congruences modulo p n for p ∈ {2, 3, 5, 7, 11}, which led to the theory of p-adic modular forms. We combine these two aspects of the j-function to give a general theory of congruences modulo powers of primes satisfied by the Hauptmoduln appearing in monstrous moonshine. We prove that many of these Hauptmoduln satisfy such congruences, and we exhibit a relationship between these congruences and the group structure of the monster. We also find a distinguished class of subgroups of the monster with graded characters satisfying such congruences. and the right-hand sides are simple sums involving the dimensions of the irreducible representations of the monster group M:

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“…Let c m = 0, 2, 1 if m is congruent to 0, 1, 2 modulo 3 respectively. Then for the theorem to be true, we require (3.4) S m = 3 6m+5−4⌈m/3⌉+cm r for some r ∈ R (3) . We will prove this by induction.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…where Q 1 , Q 2 , Q 3 ∈ R (3) . Note that since g 1 and S 1 each are of the form 3 9 r for some r ∈ R (3) , we use the R (p) product lemma to quickly deduce that their product is of the form 3 18−4 r = 3 14 r for some r ∈ R (3) , from which we easily see S 2 = 3 14 (Q 2 (φ)). Similarly, 3 23 divides both g 1 S 2 and S 1 g 2 , which means that to keep g 1 S 2 and S 1 g 2 in R (p) , we lose 3 4 and get S 3 = 3 19 (Q 3 (φ)) using the same lemma.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%