p-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on Γ1(4)

被引:0
作者
Kim, Jigu [1 ]
Lee, Yoonjin [1 ,2 ]
机构
[1] Ewha Womans Univ, Inst Math Sci, Seoul, South Korea
[2] Korea Inst Adv Study, Seoul, South Korea
来源
TAIWANESE JOURNAL OF MATHEMATICS | 2023年 / 27卷 / 01期
基金
新加坡国家研究基金会;
关键词
modular forms; Taylor coefficients; congruences; CONGRUENCES;
D O I
10.11650/tjm/220802
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
For a prime p 3 (mod 4) and m >= 2, Romik raised a question about whether the Taylor coefficients around root-1 of the classical Jacobi theta function.3 eventually vanish modulo p(m). This question can be extended to a class of modular forms of half-integral weight on Gamma(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 SL2(Z).
引用
收藏
页码:23 / 38
页数:16
相关论文
共 18 条
[11]  
Serre Jean-Pierre, 1973, Lecture Notes in Math., V350, P191
[12]   MODULAR FORMS OF HALF INTEGRAL WEIGHT [J].
SHIMURA, G .
ANNALS OF MATHEMATICS, 1973, 97 (03) :440-481
[13]   Congruences for Γ1(4)-modular forms of half-integral weight [J].
Tupan, Alexandru .
RAMANUJAN JOURNAL, 2006, 11 (02) :165-173
[14]   Computing Power Series Expansions of Modular Forms [J].
Voight, John ;
Willis, John .
COMPUTATIONS WITH MODULAR FORMS, 2014, 6 :331-361
[15]   Taylor coefficients of the Jacobi θ3(q) function [J].
Wakhare, Tanay ;
Vignat, Christophe .
JOURNAL OF NUMBER THEORY, 2020, 216 :280-306
[16]   Romik's conjecture for the Jacobi theta function [J].
Wakhare, Tanay .
JOURNAL OF NUMBER THEORY, 2020, 215 :275-296
[17]   MODULAR-FORMS AND DIFFERENTIAL-OPERATORS [J].
ZAGIER, D .
PROCEEDINGS OF THE INDIAN ACADEMY OF SCIENCES-MATHEMATICAL SCIENCES, 1994, 104 (01) :57-75
[18]  
Zagier D., 2008, The 1-2-3 of Modular Forms, P1