Periodicities for Taylor coefficients of half-integral weight modular forms

For a prime p ≡ 3 (mod 4) and m ≥ 2, Romik raised a question about whether the Taylor coefficients around √ −1 of the classical Jacobi theta function θ 3 eventually vanish modulo p m . This question can be extended to a class of modular forms of half-integral weight on Γ 1 (4) and CM points; in this paper, we prove an affirmative answer to it for primes p ≥ 5. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on SL 2 (Z).

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“…around τ 0 = i (we refer to [2,8,10,15,16]). Romik defined (d(n)) ∞ n=0 to be the sequence such that…”

Section: Introduction and Resultsmentioning

confidence: 99%

$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\Gamma_1(4)$

Kim

Lee

2023

Taiwanese J. Math.

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“…This triply nested definition makes it rather unwieldy to work with the Taylor coefficients directly, though Romik conjectured several nice properties of the d(n) coefficients modulo any prime. This paper is dedicated towards refining the second half of Romik's conjecture, which was proven by the combined efforts of Scherer [8] and Guerzhoy-Mertens-Rolen [3]. Guerzhoy, Mertens, and Rolen in fact prove a stronger statement in the context of an arbitrary half integer weight modular form.…”

Section: Introductionmentioning

confidence: 97%

“…We essentially use the method of [8], who studied the p = 5 case; however, the proof for arbitrary p becomes significantly more technically complex. The modular form approach of Guerzhoy-Mertens-Rolen [3] is extremely beautiful, since it proves eventual periodicity for any weight 1/2 modular form. However, with our method we can show not just periodicity but also a finer algebraic relation between…”

Section: Introductionmentioning

confidence: 99%

Romik's conjecture for the Jacobi theta function

Wakhare

2020

Journal of Number Theory

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Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point i. He then conjectured that the Taylor coefficients d(n) either vanish or are periodic modulo any prime p; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, who considered arbitrary half integral weight modular forms. We refine previous work for p ≡ 1 (mod 4) by displaying a concise algebraic relation between d n + p−1 2 and d(n) related to the p-adic factorial, from which we can deduce periodicity with an effective period.

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“…The first problem is that Ω in Theorem 1.3 of [4] is nowhere defined. One may guess that Ω2 = ω, with the ω in the proof of Theorem 1.3 on pages 148/149 of [4]. We assume this guess in the following.…”

Section: Introductionmentioning

confidence: 99%

Congruence properties of coefficients of the eighth-order mock theta function $$V_0(q)$$

Hemanthkumar

2021

Ramanujan J

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In [Ramanujan J. 52 (2020), 275-290], Romik considered the Taylor expansion of Jacobi's theta function θ 3 (q) at q = e −π and encoded it in an integer sequence (d(n)) n≥0 for which he provided a recursive procedure to compute the terms of the sequence. He observed intriguing behaviour of d(n) modulo primes and prime powers. Here we prove (1) that d(n) eventually vanishes modulo any prime power p e with p ≡ 3 (mod 4), (2) that d(n) is eventually periodic modulo any prime power p e with p ≡ 1 (mod 4), and (3) that d(n) is purely periodic modulo any 2-power 2 e . Our results also provide more detailed information on period length, respectively from when on the sequence vanishes or becomes periodic. The corresponding bounds may not be optimal though, as computer data suggest. Our approach shows that the above congruence properties hold at a much finer, polynomial level. 2020 Mathematics Subject Classification. Primary 11F37; Secondary 11B83 14K25. Key words and phrases. Modular forms of half integer weight, Jacobi theta function, Taylor coefficients, congruences.

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Periodicities for Taylor coefficients of half-integral weight modular forms

Cited by 4 publications

References 25 publications

$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\Gamma_1(4)$

$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\Gamma_1(4)$

Romik's conjecture for the Jacobi theta function

Congruence properties of coefficients of the eighth-order mock theta function $$V_0(q)$$

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