Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2)
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911593257762816 |
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| author | Tsang, Kin Ming |
| author_facet | Tsang, Kin Ming |
| contents | We consider a variant of the strong multiplicity one theorem. Let $π_{1}$ and $π_{2}$ be two unitary cuspidal automorphic representations for $\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(π_{1}) \right\rvert > \left\lvert a_{v}(π_{2}) \right\rvert$, where $a_{v}(π_{i})$ is the trace of Langlands conjugacy class of $π_{i}$ at $v$. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(π_{1})\right\rvert \neq \left\lvert a_{v}(π_{2}) \right\rvert$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01643 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2) Tsang, Kin Ming Number Theory 11F30 (Primary), 11F41 (Secondary) We consider a variant of the strong multiplicity one theorem. Let $π_{1}$ and $π_{2}$ be two unitary cuspidal automorphic representations for $\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(π_{1}) \right\rvert > \left\lvert a_{v}(π_{2}) \right\rvert$, where $a_{v}(π_{i})$ is the trace of Langlands conjugacy class of $π_{i}$ at $v$. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(π_{1})\right\rvert \neq \left\lvert a_{v}(π_{2}) \right\rvert$. |
| title | Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2) |
| topic | Number Theory 11F30 (Primary), 11F41 (Secondary) |
| url | https://arxiv.org/abs/2408.01643 |