The number of automorphic representations of $\mathrm{GL}_2$ with exceptional eigenvalues
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916130877079552 |
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| author | Choi, Dohoon Lee, Min Lee, Youngmin Lim, Subong |
| author_facet | Choi, Dohoon Lee, Min Lee, Youngmin Lim, Subong |
| contents | We obtain an upper bound for the dimension of the cuspidal automorphic forms for $\mathrm{GL}_2$ over a number field, whose archimedean local representations are not tempered. More precisely, we prove the following result.
Let $F$ be a number field and $\mathbb{A}_{F}$ be the ring of adeles of $F$. Let $\mathcal{O}_{F}$ be the ring of integers of $F$. Let $\mathfrak{X}_{F,\mathrm{ex}}$ be the set of irreducible cuspidal automorphic representations $π$ of $\mathrm{GL}_2(\mathbb{A}_{F})$ with the trivial central character such that for each archimedean place $v$ of $F$, the local representation of $π$ at $v$ is an unramified principal series and is not tempered. For an ideal $J$ of $\mathcal{O}_{F}$, let $\mathrm{K}_{0}(J)$ be the subgroup of $\mathrm{GL}_2(\mathbb{A}_{F})$ corresponding to $Γ_0(J) \subset \mathrm{SL}_2(\mathcal{O}_F)$. Let $r_1$ be the number of real embeddings of $F$ and $r_2$ be the number of conjugate pairs of complex embeddings of $F$. Using the Arthur-Selberg trace formula, we have \begin{equation*}
\sum_{π\in \mathfrak{X}_{F,\mathrm{ex}}} \dim π^{\mathrm{K}_0(J)}
\ll_{F} \frac{[\mathrm{SL}_2(\mathcal{O}_{F}) : Γ_0(J)]}{(\log (N_{F/\mathbb{Q}}(J)))^{2r_1+3r_2}} \quad \text{ as } \quad |N_{F/\mathbb{Q}}(J)|\to \infty.
\end{equation*} From this result, we obtain the result on an upper bound for the number of Hecke-Maass cusp forms of weight $0$ on $Γ_0(N)$ which do not satisfy the Selberg eigenvalue conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11761 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The number of automorphic representations of $\mathrm{GL}_2$ with exceptional eigenvalues Choi, Dohoon Lee, Min Lee, Youngmin Lim, Subong Number Theory 11F72 (Primary), 11F12 (Secondary) We obtain an upper bound for the dimension of the cuspidal automorphic forms for $\mathrm{GL}_2$ over a number field, whose archimedean local representations are not tempered. More precisely, we prove the following result. Let $F$ be a number field and $\mathbb{A}_{F}$ be the ring of adeles of $F$. Let $\mathcal{O}_{F}$ be the ring of integers of $F$. Let $\mathfrak{X}_{F,\mathrm{ex}}$ be the set of irreducible cuspidal automorphic representations $π$ of $\mathrm{GL}_2(\mathbb{A}_{F})$ with the trivial central character such that for each archimedean place $v$ of $F$, the local representation of $π$ at $v$ is an unramified principal series and is not tempered. For an ideal $J$ of $\mathcal{O}_{F}$, let $\mathrm{K}_{0}(J)$ be the subgroup of $\mathrm{GL}_2(\mathbb{A}_{F})$ corresponding to $Γ_0(J) \subset \mathrm{SL}_2(\mathcal{O}_F)$. Let $r_1$ be the number of real embeddings of $F$ and $r_2$ be the number of conjugate pairs of complex embeddings of $F$. Using the Arthur-Selberg trace formula, we have \begin{equation*} \sum_{π\in \mathfrak{X}_{F,\mathrm{ex}}} \dim π^{\mathrm{K}_0(J)} \ll_{F} \frac{[\mathrm{SL}_2(\mathcal{O}_{F}) : Γ_0(J)]}{(\log (N_{F/\mathbb{Q}}(J)))^{2r_1+3r_2}} \quad \text{ as } \quad |N_{F/\mathbb{Q}}(J)|\to \infty. \end{equation*} From this result, we obtain the result on an upper bound for the number of Hecke-Maass cusp forms of weight $0$ on $Γ_0(N)$ which do not satisfy the Selberg eigenvalue conjecture. |
| title | The number of automorphic representations of $\mathrm{GL}_2$ with exceptional eigenvalues |
| topic | Number Theory 11F72 (Primary), 11F12 (Secondary) |
| url | https://arxiv.org/abs/2402.11761 |