Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 4: contribution of non-elliptic parts
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910239787319296 |
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| author | Cheng, Yuhao |
| author_facet | Cheng, Yuhao |
| contents | We continue our work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the ramified setting for \emph{Beyond Endoscopy}. We establish asymptotic formulas for each term of the trace formula when summing over $n<X$, using arbitrary smooth test functions at the places in $S=\{\infty,q_1,\dots, q_r\}$ where $2\in S$, for the standard representation, up to an error of $o(X)$. This yields an identity depending on a parameter $X$, leading to certain identities that can be regarded as a limit form of the trace formula for $\mathsf{GL}_2$ over $\mathbb{Q}$. On the spectral side, we employ the contour shift method and the Riemann-Lebesgue lemma. On the geometric side, both the identity part and the unipotent part contribute $o(X)$. The elliptic part was reduced to the hyperbolic part in a previous paper. Finally, using hyperbolic Poisson summation, we relate the hyperbolic part back to the spectral side and determine its contribution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_20719 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 4: contribution of non-elliptic parts Cheng, Yuhao Number Theory Representation Theory 11F70, 11F72 We continue our work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the ramified setting for \emph{Beyond Endoscopy}. We establish asymptotic formulas for each term of the trace formula when summing over $n<X$, using arbitrary smooth test functions at the places in $S=\{\infty,q_1,\dots, q_r\}$ where $2\in S$, for the standard representation, up to an error of $o(X)$. This yields an identity depending on a parameter $X$, leading to certain identities that can be regarded as a limit form of the trace formula for $\mathsf{GL}_2$ over $\mathbb{Q}$. On the spectral side, we employ the contour shift method and the Riemann-Lebesgue lemma. On the geometric side, both the identity part and the unipotent part contribute $o(X)$. The elliptic part was reduced to the hyperbolic part in a previous paper. Finally, using hyperbolic Poisson summation, we relate the hyperbolic part back to the spectral side and determine its contribution. |
| title | Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 4: contribution of non-elliptic parts |
| topic | Number Theory Representation Theory 11F70, 11F72 |
| url | https://arxiv.org/abs/2605.20719 |