Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {O}_L$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>L</mi> </msub> </math></EquationSource> </InlineEquation> be the ring of integers of a number field L, and let <Equation ID="Equ6"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_Equ6.gif" Format="GIF" Height="48" Rendition="HTML" Resolution="72" Type="Linedraw" Width="243" /> </MediaObject> <EquationSource Format="TEX">$ f(z)=\sum _{n=1}^{\infty } a_f(n)q^n\in S_k\cap \mathcal {O}_L[[q]] $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </munderover> <msub> <mi>a</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo>∈</mo> <msub> <mi>S</mi> <mi>k</mi> </msub> <mo>∩</mo> <msub> <mi mathvariant="script">O</mi> <mi>L</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mrow> <mo stretchy="false">[</mo> <mi>q</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </Equation>be a normalized Hecke eigenform for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_2(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We say f is non-ordinary at a prime p if there exists a prime ideal <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {p}\subset \mathcal {O}_L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">p</mi> <mo>⊂</mo> <msub> <mi mathvariant="script">O</mi> <mi>L</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> above p such that <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">$a_f(p)\equiv 0 \pmod {\mathfrak {p}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>f</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi mathvariant="fraktur">p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Previous work has shown that f is non-ordinary at p if <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">$(p-1)\mid (k-m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>-</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="178" /> </InlineMediaObject> <EquationSource Format="TEX">$m\in A=\{4, 6, 8, 10, 14\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <mi>A</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>14</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we focus on the case <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$m=12$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>12</mn> </mrow> </math></EquationSource> </InlineEquation>, which does not belong to the set A. For <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="124" /> </InlineMediaObject> <EquationSource Format="TEX">$(p-1)\mid (k-12)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∣</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>-</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we show that at least one of the following holds: 1. f is non-ordinary at p; 2. <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">$f\equiv \Delta \pmod {p}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>≡</mo> <mi mathvariant="normal">Δ</mi> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation> denotes the unique normalized cusp form of weight 12 for <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1188_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{SL}_2(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as usual.

Non-ordinary primes and $\Delta $ congruences for modular forms

期刊论文
发表日期：

2025年8月7日

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Wenjun Ma

摘要

Let $\mathcal {O}_L$ $O_{L}$ be the ring of integers of a number field L, and let $ f(z)=\sum _{n=1}^{\infty } a_f(n)q^n\in S_k\cap \mathcal {O}_L[[q]] $ $f (z) = \sum_{n = 1}^{\infty} a_{f} (n) q^{n} \in S_{k} \cap O_{L} [[q]]$ be a normalized Hecke eigenform for $\textrm{SL}_2(\mathbb {Z})$ ${SL}_{2} (Z)$ . We say f is non-ordinary at a prime p if there exists a prime ideal $\mathfrak {p}\subset \mathcal {O}_L$ $p \subset O_{L}$ above p such that $a_f(p)\equiv 0 \pmod {\mathfrak {p}}$ $a_{f} (p) \equiv 0 (mod p)$ . Previous work has shown that f is non-ordinary at p if $(p-1)\mid (k-m)$ $(p - 1) ∣ (k - m)$ for some $m\in A=\{4, 6, 8, 10, 14\}$ $m \in A = {4, 6, 8, 10, 14}$ . In this paper, we focus on the case $m=12$ $m = 12$ , which does not belong to the set A. For $(p-1)\mid (k-12)$ $(p - 1) ∣ (k - 12)$ , we show that at least one of the following holds: 1. f is non-ordinary at p; 2. $f\equiv \Delta \pmod {p}$ $f \equiv Δ (mod p)$ , where $\Delta $ $Δ$ denotes the unique normalized cusp form of weight 12 for $\textrm{SL}_2(\mathbb {Z})$ ${SL}_{2} (Z)$ as usual.

Non-ordinary primes and \(\Delta \) congruences for modular forms

摘要