Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields
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| Format: | Preprint |
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2025
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| author | Grobner, Harald Harris, Michael Jie, Lin |
| author_facet | Grobner, Harald Harris, Michael Jie, Lin |
| contents | The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. As an application of our main result, we establish Deligne's conjecture for a class of CM-automorphic motives, which we construct in this paper. Our proof uses the results of our recent joint work with Raghuram in combination with the Ichino--Ikeda--Neal-Harris (IINH) formula for unitary groups -- which is now a theorem -- and an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of $\GL(n)\times\GL(m)$ over CM fields. By reinterpreting these critical values in terms of automorphic periods of holomorphic automorphic forms on unitary groups, we show that the automorphic periods of holomorphic forms can be factored as products of coherent cohomological forms, compatibly with a motivic factorization predicted by the Tate conjecture. All of these results are stated under a certain regularity condition and an hypothesis of rationality on archimedean zeta-integrals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_02303 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields Grobner, Harald Harris, Michael Jie, Lin Number Theory Representation Theory 11F67 (primary), 11F70, 11G18, 11R39, 22E55 The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. As an application of our main result, we establish Deligne's conjecture for a class of CM-automorphic motives, which we construct in this paper. Our proof uses the results of our recent joint work with Raghuram in combination with the Ichino--Ikeda--Neal-Harris (IINH) formula for unitary groups -- which is now a theorem -- and an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of $\GL(n)\times\GL(m)$ over CM fields. By reinterpreting these critical values in terms of automorphic periods of holomorphic automorphic forms on unitary groups, we show that the automorphic periods of holomorphic forms can be factored as products of coherent cohomological forms, compatibly with a motivic factorization predicted by the Tate conjecture. All of these results are stated under a certain regularity condition and an hypothesis of rationality on archimedean zeta-integrals. |
| title | Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields |
| topic | Number Theory Representation Theory 11F67 (primary), 11F70, 11G18, 11R39, 22E55 |
| url | https://arxiv.org/abs/2509.02303 |