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PreprintCC BY 4.0arXiv

A divisibility of automorphic periods for the real quadratic base change of GL3

  • Tristan Ricoul
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Abstract

We prove a p-adic divisibility between the automorphic periods of a cuspidal automorphic representation π of GL3(ℚ) and the periods of its Arthur-Clozel base change to a real quadratic field E. As a corollary, we establish a divisibility predicted by the Bloch-Kato conjecture for the adjoint motive of π twisted by an even quadratic character. This generalizes earlier works of Tilouine-Urban and Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods for conjugate self-dual cuspidal automorphic representations of GL3(E), defined within the middle degree of the cuspidal cohomology instead of the top or bottom degrees. Moreover, we prove an à la Hida adjoint L-value formula for GL3(E) that relates these newly defined middle-degree periods to the usual top and bottom automorphic periods.

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Cite

BibTeXRIS
BibTeX
@misc{ricoul2026divisibility,
  title = {{A divisibility of automorphic periods for the real quadratic base change of $\mathrmGL_3$}},
  author = {Tristan Ricoul},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26477},
  eprint = {2609.26477},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26477},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A divisibility of automorphic periods for the real quadratic base change of GL_3
AU  - Tristan Ricoul
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26477
UR  - https://arxiv.org/abs/2609.26477
AB  - We prove a p-adic divisibility between the automorphic periods of a cuspidal automorphic representation π of GL_3(ℚ) and the periods of its Arthur-Clozel base change to a real quadratic field E. As a corollary, we establish a divisibility predicted by the Bloch-Kato conjecture for the adjoint motive of π twisted by an even quadratic character. This generalizes earlier works of Tilouine-Urban and Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods for conjugate self-dual cuspidal automorphic representations of GL_3(E), defined within the middle degree of the cuspidal cohomology instead of the top or bottom degrees. Moreover, we prove an à la Hida adjoint L-value formula for GL_3(E) that relates these newly defined middle-degree periods to the usual top and bottom automorphic periods.
ER  -