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<p>Recently, Allen, Grove, Long, and Tu proposed an explicit method to compute modularity for certain hypergeometric series, which gives a concrete link between certain hypergeometric objects and modular forms. The theory is exemplified by a collection of 199 weight 3 modular forms. Among other properties, their process shows that the <i>L</i>-value of such a modular form at 1 is an explicit multiple of a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40687_2026_639_IEq1_HTML.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="120" Type="Linedraw" Width="41" /> </InlineMediaObject> </InlineEquation> hypergeometric series. Using the framework of a finite Coxeter group governing the invariance group of normalized <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40687_2026_639_IEq2_HTML.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="120" Type="Linedraw" Width="41" /> </InlineMediaObject> </InlineEquation> series, this paper fully classifies and describes the possible Hecke eigenforms whose <i>L</i>-values can be obtained using this method. In addition, we determine when these modular forms differ by twist of a finite-order character using the perspective of hypergeometric functions. As one application, we reinterpret a classical identity of hypergeometric series as a formula involving <i>L</i>-values of two Hecke eigenforms that differ by a twist.</p>

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L-values of certain weight 3 modular forms and transformations of hypergeometric series

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月15日

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  • Esme Rosen

摘要

Recently, Allen, Grove, Long, and Tu proposed an explicit method to compute modularity for certain hypergeometric series, which gives a concrete link between certain hypergeometric objects and modular forms. The theory is exemplified by a collection of 199 weight 3 modular forms. Among other properties, their process shows that the L-value of such a modular form at 1 is an explicit multiple of a hypergeometric series. Using the framework of a finite Coxeter group governing the invariance group of normalized series, this paper fully classifies and describes the possible Hecke eigenforms whose L-values can be obtained using this method. In addition, we determine when these modular forms differ by twist of a finite-order character using the perspective of hypergeometric functions. As one application, we reinterpret a classical identity of hypergeometric series as a formula involving L-values of two Hecke eigenforms that differ by a twist.