Bounds for the relative class number problem for function fields

TL;DR

The paper develops explicit, finite-visibility bounds for extensions of function fields in terms of their relative class numbers, enabling a finite computational classification for each fixed m. It introduces the Prym variety framework as the main tool, derives sharp bounds that depend on the base field’s size q and the geometry of the extension (constant vs purely geometric), and reduces the m-case to a finite search via explicit class field computations and exhaustive Magma-based checks. The authors solve the relative class number 2 problem for q>2, listing all admissible (q,g,g′) configurations and identifying a single noncyclic cubic exception, with a detailed computational roadmap for q≤5. Collectively, the results transform the relative class number problem into a tractable, tabulated finite computation, leveraging both deep bounds (Weil-type for Prym varieties, point-count bounds) and explicit curve/Galois-data databases. The work provides a concrete, implementable pathway to classify extensions by their relative class number and advances the understanding of how Prym varieties govern arithmetic of function field extensions.

Abstract

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the relative class number two problem in all cases where the base field has constant field not equal to $\mathbb{F}_2$.

Figures (1)

• Figure 1: Positivity of the function defined by inequality (\ref{['eq:positive']}).

Theorems & Definitions (24)

• Theorem 1.1
• proof
• Theorem 1.2
• Lemma 2.1: Exponential lower bounds
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4