Counting number fields with symplectic Galois group

Abstract

Let n≥ 1 and let ℓ be an odd prime. Let G=PGSp_2n(𝔽_ℓ) or GSp_2n(𝔽_ℓ), and fix a faithful transitive permutation representation π:G⟶ S_d. We study degree-d number fields whose Galois closures have Galois group G and whose associated permutation representation is π. For σ∈ S_d, write ind(σ) for its permutation index, namely $\operatorname{ind}(σ) = d-\#\{\text{orbits of σ on {1,…,d}}\}$. If τ denotes a symplectic transvection, or its image in the projective symplectic group, we prove that the number of such fields with absolute discriminant at most X is bounded below by a constant multiple of X^{1/(2n ind(π(τ)))}. For the natural vector and projective actions, the exponents we obtain are asymptotically 1/(4n) of those predicted by the weak form of Malle's conjecture as ℓ→∞. The fields are constructed from the mod-ℓ Galois representations attached to the Jacobians of a one-parameter family of hyperelliptic curves. The proof combines large symplectic monodromy for this family with a squarefree sieve.

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BibTeX RIS

BibTeX

@misc{ray2026counting,
  title = {{Counting number fields with symplectic Galois group}},
  author = {Anwesh Ray},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23093},
  eprint = {2609.23093},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23093},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Counting number fields with symplectic Galois group
AU  - Anwesh Ray
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23093
UR  - https://arxiv.org/abs/2609.23093
AB  - Let n≥ 1 and let ℓ be an odd prime. Let G=PGSp_2n(𝔽_ℓ) or GSp_2n(𝔽_ℓ), and fix a faithful transitive permutation representation π:G⟶ S_d. We study degree-d number fields whose Galois closures have Galois group G and whose associated permutation representation is π. For σ∈ S_d, write ind(σ) for its permutation index, namely $\operatorname{ind}(σ) = d-\#\{\text{orbits of σ on {1,…,d}}\}$. If τ denotes a symplectic transvection, or its image in the projective symplectic group, we prove that the number of such fields with absolute discriminant at most X is bounded below by a constant multiple of X^1/(2n ind(π(τ))). For the natural vector and projective actions, the exponents we obtain are asymptotically 1/(4n) of those predicted by the weak form of Malle's conjecture as ℓ→∞. The fields are constructed from the mod-ℓ Galois representations attached to the Jacobians of a one-parameter family of hyperelliptic curves. The proof combines large symplectic monodromy for this family with a squarefree sieve.
ER  -

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