Conductors and Quadratic base change

@misc{cherukara2026conductors,
  title = {{Conductors and Quadratic base change}},
  author = {Siddharth Ramakrishnan Cherukara},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32104},
  eprint = {2609.32104},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32104},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Conductors and Quadratic base change
AU  - Siddharth Ramakrishnan Cherukara
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32104
UR  - https://arxiv.org/abs/2609.32104
AB  - Let F be a totally real field and K/F a totally real quadratic extension. For a Hilbert cusp newform f over F with base change f_K to K, we determine the exact level of f_K in terms of the level of f, using only local representation theory: for each place p of F we compute how the conductor of the local component of f changes under quadratic base change. The tame case (odd residue characteristic) follows from existing base change theory, but the case p=2 requires a finer ε-factor analysis that does not appear in the literature. We give a complete treatment of the dyadic imprimitive representations and, more delicately, the exceptional supercuspidal representations, for which no closed-form base change formula was previously known; in particular we determine, in terms of local root numbers, the base change of every exceptional supercuspidal representation of GL(2,ℚ_2) with trivial central character and conductor $3.$ This yields an explicit formula for the level of f_K, specialized here to F=ℚ, along with a partial converse.
ER  -