For <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_659_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$N \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_659_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$S_{2}^{\text { new }}(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>S</mi> <mrow> <mn>2</mn> </mrow> <mi mathvariant="normal">new</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the newspace of cuspidal modular forms of weight 2 and level N. In 2004, Greg Martin conjectured that as a sequence in N, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_659_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$\dim S_2^{\text { new }}(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>dim</mo> <msubsup> <mi>S</mi> <mn>2</mn> <mi mathvariant="normal">new</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> takes on all possible natural numbers. In this paper, we investigate several generalizations and variations of this type of problem. In each case, we provide a complete characterization of when such a property holds.

Dimension sequences of modular forms

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2025年8月7日

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Erick Ross

摘要

For $N \ge 1$ $N \geq 1$ , let $S_{2}^{\text { new }}(N)$ $S_{2}^{new} (N)$ denote the newspace of cuspidal modular forms of weight 2 and level N. In 2004, Greg Martin conjectured that as a sequence in N, $\dim S_2^{\text { new }}(N)$ $dim S_{2}^{new} (N)$ takes on all possible natural numbers. In this paper, we investigate several generalizations and variations of this type of problem. In each case, we provide a complete characterization of when such a property holds.