AI 中文总结
本文利用$\textrm{PGL}_2(\boldsymbol{\textsf{F}}_p)$的模表示论提出新方法，计数模$\textrm{-}\boldsymbol{\textsf{l}}$同余，可在剩余不可约/可约情形下证明同余，且无需艾森斯坦级数常数项计算等工具。
AI 中文摘要
我们引入一种新方法，通过$\boldsymbol{\textrm{PGL}_2(\boldsymbol{\textsf{F}}_p)}$的模表示论研究特征形式之间的模$\boldsymbol{\textrm{-}\boldsymbol{\textsf{l}}}$同余。当$\boldsymbol{p \not\neq \boldsymbol{\textsf{1}} \boldsymbol{\textsf{mod}} \boldsymbol{\textsf{l}}}$时，我们利用该理论在$\boldsymbol{\textsf{mod-}\boldsymbol{\textsf{l}}}$伽罗瓦表示固定、$\boldsymbol{\textsf{level}}$为$\boldsymbol{\boldsymbol{\textsf{Γ}}_0(\boldsymbol{\textsf{p}}^2)}$的模形式空间上构造并描述额外结构。所得结构结果可视为经典升水平定理的精细化，因为它们不仅能证明同余的存在性，还能计数此类同余的数量。我们的方法在剩余不可约和剩余可约情形下均适用，可证明艾森斯坦级数与尖点形式之间同余的多个新实例（同时独立重新推导马祖尔的经典结果以及兰格-韦克的最新结果）。值得注意的是，我们的方法无需计算艾森斯坦级数的常数项、无需伽罗瓦形变理论与$\boldsymbol{\textsf{R=T}}$定理，也无需雅克比-朗兰兹对应即可证明这些结果。
英文摘要
We introduce a new method for studying mod-$\ell$ congruences between eigenforms through the modular representation theory of $\mathrm{PGL}_2(\mathbb{F}_p)$. When $p\equiv \pm 1 \pmod{\ell}$, we use this theory to construct and describe extra structures on spaces of modular forms with $Γ_0(p^2)$-level at $p$ and a fixed mod-$\ell$ Galois representation. The structural results we obtain can be viewed as a refinement of classical level-raising theorems since they not only allow us to prove the existence of congruences, but also to count the number of such congruences. Our methods work equally well in the residually irreducible and residually reducible cases, allowing us to prove several new instances of congruences between Eisenstein series and cuspforms (as well as independently rederiving classical results of Mazur and more recent results of Lang--Wake). Notably, our approach proves these results without computing constant terms of Eisenstein series, without Galois deformation theory and $R=\mathbb{T}$ theorems, and without the Jacquet--Langlands correspondence.
Comments46 pages
