Romik's conjecture for the Jacobi theta function

Wakhare, Tanay

doi:10.1016/j.jnt.2020.01.009

Cited by 3 publications

(6 citation statements)

References 9 publications

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“…We are now in a position to prove our first main result. The following theorem proves Conjecture 18(3) in [10], in the stronger form given in Theorem 1(1).…”

Section: The Sequence (V(n)) N≥0 Modulo Odd Prime Powersmentioning

confidence: 64%

“…Without any doubt, Equation (2.9) does provide a recursive way to compute the coefficients d(n). Indeed, Scherer [8] and Wakhare [10] used it for the proof of their results. However, in our opinion the suitability of (2.9) for the proof of congruence relations satisfied by the d(n)'s using inductive arguments is limited.…”

Section: Romik's Recursive Procedures For the Computation Of D(n)mentioning

confidence: 99%

“…He also obtained partial results on Items (2) and ( 3) by proving that d(n) ≡ (−1) n+1 (mod 5) for n ≥ 1, and that d(n) is odd for all n. Guerzhoy, Mertens and Rolen [4] claim to have proved Item (2) in full, as a special case of a more general result for a whole family of modular forms of half integer weight. 1 In [10], Wakhare revisited Item (2) for primes (i.e., for e = 1). He showed that d(n) is (eventually) periodic modulo any prime number p with 1 The authors of the present article admit that they are not able to comprehend what the result(s) in [4] say about Romik's sequence d(n) n≥0 .…”

Section: Introductionmentioning

confidence: 99%

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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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“…We are now in a position to prove our first main result. The following theorem proves Conjecture 18(3) in [10], in the stronger form given in Theorem 1(1).…”

Section: The Sequence (V(n)) N≥0 Modulo Odd Prime Powersmentioning

confidence: 64%

Section: Romik's Recursive Procedures For the Computation Of D(n)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Hemanthkumar

2021

Ramanujan J

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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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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 [8], [10], [2], [15], and [16]). Romik defined (d(n)) ∞ n=0 to be the sequence such that…”

mentioning

confidence: 99%

$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $Γ_1(4)$

Kim¹,

Lee²

2021

Preprint

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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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Romik's conjecture for the Jacobi theta function

Cited by 3 publications

References 9 publications

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

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

$p$-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 $Γ_1(4)$

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