A Lang-Trotter Problem for Non-Geometric Quadratic Inductions
Abstract
Let K/ℚ be an imaginary quadratic extension and p an odd prime. Write ρ=IndGKGℚχ, where E/ℚp is a finite extension and χ:GK→𝒪E× is a continuous character. For a fixed r∈ℤ∖{0}, let πρ,r(X) denote the number of rational primes ℓ≤ X such that ρ is unramified at ℓ and trρ(Frobℓ)=r. Let a,b be the two weights of χ at p. We prove that if (a,b)∉ℚ2, then πρ,r(X)≪ρ,r,εXε for every ε>0, while if (a,b)∈ℚ2∖ℤ2, then only finitely many such primes occur. These bounds are substantially sparser than the classical CM Lang--Trotter scale. The main input in the non-rational case is a rigidity theorem for algebraic curves in the p-adic analytic trace locus, combined with rigid-analytic Pila--Wilkie counting; the rational non-integral case is treated by a local ramification argument.
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BibTeX
@misc{yuan2026langtrotter,
title = {{A Lang-Trotter Problem for Non-Geometric Quadratic Inductions}},
author = {Haoyang Yuan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08201},
eprint = {2609.08201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08201},
}
RIS
TY - UNPB
TI - A Lang-Trotter Problem for Non-Geometric Quadratic Inductions
AU - Haoyang Yuan
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08201
UR - https://arxiv.org/abs/2609.08201
AB - Let K/ℚ be an imaginary quadratic extension and p an odd prime. Write ρ=Ind_G_K^G_ℚχ, where E/ℚ_p is a finite extension and χ:G_K→𝒪_E^× is a continuous character. For a fixed r∈ℤ∖{0}, let π_ρ,r(X) denote the number of rational primes ℓ≤ X such that ρ is unramified at ℓ and trρ(Frob_ℓ)=r. Let a,b be the two weights of χ at p. We prove that if (a,b)∉ℚ^2, then π_ρ,r(X)≪_ρ,r,εX^ε for every ε>0, while if (a,b)∈ℚ^2∖ℤ^2, then only finitely many such primes occur. These bounds are substantially sparser than the classical CM Lang--Trotter scale. The main input in the non-rational case is a rigidity theorem for algebraic curves in the p-adic analytic trace locus, combined with rigid-analytic Pila--Wilkie counting; the rational non-integral case is treated by a local ramification argument.
ER -