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PreprintCC BY 4.0arXiv

Stable base change from unitary groups in three variables and integral relation of automorphic periods

  • Tristan Ricoul
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Abstract

Let E be a real quadratic field and let UE be the quasi-split unitary group in three variables associated with E. We prove a p-adic divisibility between the automorphic periods of a cuspidal automorphic representation πU of UE and the periods of its Rogawski stable base change to GL3(E). This generalizes earlier works of Tilouine-Urban and Hida in the case of GL2. The divisibility we prove involves a new kind of automorphic periods for self-conjugate cuspidal automorphic representations of GL3(E), defined within the middle degree of the cuspidal cohomology rather than the top or bottom degrees. Moreover, we prove an à la Hida adjoint L-value formula for GL3(E), relating these newly defined middle-degree periods to the usual top and bottom automorphic periods. Finally, we also prove a similar formula for UE, which is the first instance of such a result for quasi-split unitary groups.

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Cite

BibTeXRIS
BibTeX
@misc{ricoul2026stable,
  title = {{Stable base change from unitary groups in three variables and integral relation of automorphic periods}},
  author = {Tristan Ricoul},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26500},
  eprint = {2609.26500},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26500},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Stable base change from unitary groups in three variables and integral relation of automorphic periods
AU  - Tristan Ricoul
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26500
UR  - https://arxiv.org/abs/2609.26500
AB  - Let E be a real quadratic field and let U_E be the quasi-split unitary group in three variables associated with E. We prove a p-adic divisibility between the automorphic periods of a cuspidal automorphic representation π_U of U_E and the periods of its Rogawski stable base change to GL_3(E). This generalizes earlier works of Tilouine-Urban and Hida in the case of GL_2. The divisibility we prove involves a new kind of automorphic periods for self-conjugate cuspidal automorphic representations of GL_3(E), defined within the middle degree of the cuspidal cohomology rather than the top or bottom degrees. Moreover, we prove an à la Hida adjoint L-value formula for GL_3(E), relating these newly defined middle-degree periods to the usual top and bottom automorphic periods. Finally, we also prove a similar formula for U_E, which is the first instance of such a result for quasi-split unitary groups.
ER  -