PreprintCC BY 4.0arXiv
Cyclicity of Drinfeld modules modulo primes in arithmetic progressions
Abstract
Let A=𝔽q[T] and K=𝔽q(T), and let ϕ be a Drinfeld A-module over K of rank r≥ 2. We study the reduction of ϕ at primes p ∈ A lying in a prescribed arithmetic progression, for which ϕ(𝔽p) has at most r−1 cyclic components. We obtain asymptotic formulas for the number of such primes of a given degree, with error bounds uniform in the modulus of the arithmetic progression. Furthermore, when EndK̅(ϕ)=A, we sharpen the error term and, under additional hypotheses, establish a necessary and sufficient condition for positivity of the resulting density.
Related
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BibTeX
@misc{jha2026cyclicity,
title = {{Cyclicity of Drinfeld modules modulo primes in arithmetic progressions}},
author = {Prassanna Nand Jha and Arnab Saha and Akshaa Vatwani},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03607},
eprint = {2611.03607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03607},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cyclicity of Drinfeld modules modulo primes in arithmetic progressions
AU - Prassanna Nand Jha
AU - Arnab Saha
AU - Akshaa Vatwani
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03607
UR - https://arxiv.org/abs/2611.03607
AB - Let A=𝔽_q[T] and K=𝔽_q(T), and let ϕ be a Drinfeld A-module over K of rank r≥ 2. We study the reduction of ϕ at primes p ∈ A lying in a prescribed arithmetic progression, for which ϕ(𝔽_p) has at most r−1 cyclic components. We obtain asymptotic formulas for the number of such primes of a given degree, with error bounds uniform in the modulus of the arithmetic progression. Furthermore, when End_K̅(ϕ)=A, we sharpen the error term and, under additional hypotheses, establish a necessary and sufficient condition for positivity of the resulting density.
ER -