{"title":"$\\Gamma_1(4)$上半积分权模形式泰勒系数的$p$-adic性质","authors":"Jigu Kim, Yoonjin Lee","doi":"10.11650/tjm/220802","DOIUrl":null,"url":null,"abstract":". 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 ).","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\\\\Gamma_1(4)$\",\"authors\":\"Jigu Kim, Yoonjin Lee\",\"doi\":\"10.11650/tjm/220802\",\"DOIUrl\":null,\"url\":null,\"abstract\":\". 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 ).\",\"PeriodicalId\":22176,\"journal\":{\"name\":\"Taiwanese Journal of Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2021-09-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Taiwanese Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.11650/tjm/220802\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Taiwanese Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.11650/tjm/220802","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\Gamma_1(4)$
. 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 ).
期刊介绍：
The Taiwanese Journal of Mathematics, published by the Mathematical Society of the Republic of China (Taiwan), is a continuation of the former Chinese Journal of Mathematics (1973-1996). It aims to publish original research papers and survey articles in all areas of mathematics. It will also occasionally publish proceedings of conferences co-organized by the Society. The purpose is to reflect the progress of the mathematical research in Taiwan and, by providing an international forum, to stimulate its further developments. The journal appears bimonthly each year beginning from 2008.
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