AI 中文总结
该研究完成了有理数域上分歧为5的GL₂的超越内窥镜相关工作，证明了迹公式项的渐近公式为o(X)及迹公式极限形式的恒等式，为相关领域提供了关键理论结果。
AI 中文摘要
我们完成了朗兰兹提出的超越内窥镜（Beyond Endoscopy）框架下，有理数域ℚ上分歧情形的GL₂相关研究工作。针对标准表示，当在集合S={∞,q₁,…,qᵣ}（其中2∈S）的所有位置上取任意光滑测试函数，对n<X求和时，我们证明了迹公式各项的渐近公式为o(X)。我们直接证明了有理数域上GL₂的迹公式极限形式（limit form of the trace formula）的恒等式，该证明运用了Arthur关于加权轨道积分傅里叶变换的结果，改写了涉及互换算子的项，随后将其展开结果与Arthur-Herb-Sally及Hoffmann的实情形结果、Arthur定义下非阿基米德情形的直接计算结果进行比较。
英文摘要
We complete our work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the ramified setting for \emph{Beyond Endoscopy} proposed by Langlands. We prove that the asymptotic formula for each term of the trace formula when summing over $n<X$ with arbitrary smooth test functions at places in $S=\{\infty,q_1,\dots q_r\}$ with $2\in S$, for the standard representation, is $o(X)$. We prove an identity with a variable $X$, called the \emph{limit form of the trace formula} for $\mathsf{GL}_2$ over $\mathbb{Q}$, directly. The proof uses Arthur's result on the Fourier transform of weighted orbital integrals to rewrite the term involving intertwining operators, and then compares the expansion with the results of the real case due to Arthur-Herb-Sally and Hoffmann, and the nonarchimedean case by direct computation using Arthur's definition.
