On C-Algebraic and C-Arithmetic Automorphic Representations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916366841282560 |
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| author | La Rosa, Alfio Fabio |
| author_facet | La Rosa, Alfio Fabio |
| contents | In the first part of the article, we consider the conjecture of K. Buzzard and T. Gee proposing that every C-algebraic automorphic representation is C-arithmetic, and we show that it can be reduced to the the analogous statement for cuspidal C-algebraic automorphic representations. We also show that if every cuspidal C-algebraic automorphic representations is C-arithmetic, then every L-algebraic automorphic representation is L-arithmetic. In the second part, we restrict to the simpler setting of anisotropic groups. We prove a criterion which shows that, under certain conditions, the trace formula can be used to establish that an automorphic representation is C-arithmetic. This criterion does not require the existence of a geometric model for the non-Archimedean part of the representation: it only depends on the possibility of isolating a finite family of automorphic representations containing the given one, and on a certain arithmetic condition on the Archimedean orbital integrals. As an application, we give a new proof that regular discrete series are C-arithmetic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On C-Algebraic and C-Arithmetic Automorphic Representations La Rosa, Alfio Fabio Number Theory Representation Theory In the first part of the article, we consider the conjecture of K. Buzzard and T. Gee proposing that every C-algebraic automorphic representation is C-arithmetic, and we show that it can be reduced to the the analogous statement for cuspidal C-algebraic automorphic representations. We also show that if every cuspidal C-algebraic automorphic representations is C-arithmetic, then every L-algebraic automorphic representation is L-arithmetic. In the second part, we restrict to the simpler setting of anisotropic groups. We prove a criterion which shows that, under certain conditions, the trace formula can be used to establish that an automorphic representation is C-arithmetic. This criterion does not require the existence of a geometric model for the non-Archimedean part of the representation: it only depends on the possibility of isolating a finite family of automorphic representations containing the given one, and on a certain arithmetic condition on the Archimedean orbital integrals. As an application, we give a new proof that regular discrete series are C-arithmetic. |
| title | On C-Algebraic and C-Arithmetic Automorphic Representations |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2401.16174 |