On Weierstra{\ss} semigroups at one and two points and their corresponding Poincar\'e series

TL;DR

This paper introduces and studies Poincaré series linked to Weierstraß semigroups at one and two points on algebraic curves over finite fields, analyzing their properties and functional equations.

Contribution

It provides a new framework for understanding Poincaré series associated with Weierstraß semigroups at multiple points on algebraic curves.

Findings

Defined Poincaré series for Weierstraß semigroups at one and two points.

Derived functional equations for these series in affine complete intersection cases.

Enhanced understanding of algebraic curves over finite fields through semigroup analysis.

Abstract

The aim of this paper is to introduce and investigate the Poincar\'e series associated with the Weierstra{\ss} semigroup of one and two rational points at a (not necessarily irreducible) non-singular projective algebraic curve defined over a finite field, as well as to describe their functional equations in the case of an affine complete intersection.