超越内窥镜法的对称平方表示：简单迹公式情形
Beyond endoscopy for the symmetric square representation: The simple trace formula case
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中文总结 AI 辅助
研究在特定分歧情形下对称平方表示，通过对测试函数加条件推导出渐近公式，利用该公式极限检测二面角形式，证明过程涉及泊松求和、克洛斯特曼和计算及留数分析等。
中文摘要 AI 辅助
本世纪初，朗兰兹引入了一种名为“超越内窥镜法”的策略来攻克泛函性原理。阿尔图格研究了\(\mathbb{Q}\)上\(\mathsf{GL}_2\)在非分歧情形下的标准表示。我们考虑在\(S = \{\infty, q_1, \dots, q_r\}\)（其中\(2 \in S\)）处有分歧的情形，并在对测试函数添加一些额外条件以使迹公式简单的情况下，推导出对称平方表示的渐近公式。一般来说该极限非零，我们可以通过使用这种类似于文卡特什论文中的迹公式极限形式来检测二面角形式。证明涉及第二次泊松求和、变换后的克洛斯特曼和及相应级数的计算，通过留数分析给出主项的渐近公式，并使用技术分析处理误差项。
英文摘要
At the beginning of this century, Langlands introduced a strategy known as \emph{Beyond Endoscopy} to attack the principle of functoriality. Altuğ studied $\mathsf{GL}_2$ over $\mathbb Q$ in the unramified setting for the standard representation. We consider the case with ramification at $S=\{\infty,q_1,\dots,q_r\}$ with $2\in S$ and derive an asymptotic formula for the symmetric square representation adding some additional conditions on the test function so that the trace formula is simple. The limit is nonzero in general and we may detect the dihedral forms by using such limit form of the trace formula which is similar to Venkatesh's thesis. The proof involves a second Poisson summation, computation of the transformed Kloosterman sum and the corresponding series, and giving an asymptotic formula for the main term by residue analysis and using technical analysis to deal with the error term.
