{"title":"On the modulo p zeros of modular forms congruent to theta series","authors":"Berend Ringeling","doi":"10.1016/j.jnt.2024.03.019","DOIUrl":null,"url":null,"abstract":"<div><p>For a prime <em>p</em> larger than 7, the Eisenstein series of weight <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span> has some remarkable congruence properties modulo <em>p</em>. Those imply, for example, that the <em>j</em>-invariants of its zeros (which are known to be real algebraic numbers in the interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1728</mn><mo>]</mo></math></span>), are at most quadratic over the field with <em>p</em> elements and are congruent modulo <em>p</em> to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span> for the full modular group as the modular forms for which the first <span><math><mi>dim</mi><mo></mo><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the <em>j</em>-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo <em>p</em> all in the ground field with <em>p</em> elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with <em>p</em> elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 386-407"},"PeriodicalIF":0.6000,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24000921/pdfft?md5=039b6bb5c8fac784d3fa69a2ccefbe61&pid=1-s2.0-S0022314X24000921-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24000921","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For a prime p larger than 7, the Eisenstein series of weight has some remarkable congruence properties modulo p. Those imply, for example, that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval ), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight for the full modular group as the modular forms for which the first Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
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