Weierstrass Pure Gaps From a Quotient of the Hermitian Curve

TL;DR

This paper characterizes pure gaps at multiple places on quotients of Hermitian curves using Kummer extensions, extending previous work and providing explicit counts and examples.

Contribution

It offers a new arithmetic characterization of pure gaps on quotient Hermitian curves, generalizing prior results and explicitly calculating their cardinalities.

Findings

Explicit formulas for the number of gaps and pure gaps at pairs of places.

Extension of Matthews' results to quotient Hermitian curves.

Concrete examples illustrating the theoretical results.

Abstract

In this paper, by employing the results over Kummer extensions, we give an arithmetic characterization of pure gaps at many totally ramified places over the quotients of Hermitian curves, including the well-studied Hermitian curves as special cases. The cardinality of these pure gaps is explicitly investigated. In particular, the numbers of gaps and pure gaps at a pair of distinct places are determined precisely, which can be regarded as an extension of the previous work by Matthews (2001) considered Hermitian curves. Additionally, some concrete examples are provided to illustrate our results.