Mathematical proof note · Version 1.0 · October 18, 2026
The Exact Asymptotic Constant for Abelian Automorphism Groups over Finite Fields
National Mobile Communications Research Laboratory and the Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China
Abstract
Automorphism groups of algebraic curves provide coordinate symmetries for algebraic-geometry codes, and their orders determine part of the geometric cost of imposing those symmetries. This note determines the exact asymptotic constant for the largest abelian automorphism groups over every fixed finite field. The proof relates inertia-group orders to mean character conductors and combines the resulting local corrections with a joint estimate for the class number and low-degree places of the quotient curve. This approach gives a uniform genus lower bound that accommodates wild ramification and varying quotient genus. Classical Carlitz cyclotomic function fields attain the bound asymptotically, establishing that the extremal constant equals two in every characteristic. Sequences approaching this constant have quotient genus negligible relative to the logarithm of the group order, while their mean conductor degree is asymptotic to that logarithm. For geometric cyclic evaluation codes, the result yields a uniform rate bound under a linear divisor-degree constraint and a lower bound on the divisor degree required for positive rate.
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Keywords
abelian automorphism groups; algebraic-geometry codes; conductors; finite fields; genus bounds.
中文关键词：阿贝尔自同构群；代数几何码；导子；有限域；亏格界。
Suggested citation
S. Yang, "The Exact Asymptotic Constant for Abelian Automorphism Groups over Finite Fields," mathematical proof note, version 1.0, Oct. 18, 2026. [Online]. Available: https://github.com/stalin1943/Coding-Theory-Proof-Notes/blob/main/abelian-automorphism-constant-two/proof.pdf