1. Normalization in Weighted Projective Spaces: Weighted GCDs and Weighted Heights
L. Beshaj, J. Mezinaj
Abstract
Weighted projective spaces arise naturally in arithmetic geometry and moduli problems, yet their computational treatment requires careful handling of normalization and scaling. In particular, the notion of a weighted greatest common divisor (weighted gcd) plays a central role in defining canonical representatives of points and ensuring well-defined arithmetic invariants.
This talk presents algorithmic methods for computing weighted gcds and weighted heights in weighted projective spaces. We discuss normalization procedures, structural properties of weighted heights, and their applications to arithmetic computations and rational point enumeration. Emphasis will be placed on explicit algorithms and their implementation within a computational framework for weighted projective geometry.
References
L. Beshaj, T. Shaska, Weighted greatest common divisors and weighted heights, J. Number Theory 213 (2020), 319–346.
2. Weighted Projective Curves
E. Zhupa, E. Shaska
Abstract
Curves in weighted projective spaces generalize classical projective curves while preserving compatibility with graded coordinate structures. These curves naturally occur in moduli constructions and in explicit models of algebraic varieties with prescribed symmetries and singularities.
This talk develops the geometry of weighted projective curves from a computational perspective. Topics include weighted homogeneous equations, singularity detection, normalization, and genus computations. The goal is to present a systematic framework for implementing weighted curves in symbolic algebra systems and to highlight their role as foundational objects for arithmetic and coding-theoretic applications.
References
(To be announced.)
3. Rational Points of Weighted Curves over Finite Fields
S. Salami, T. Shaska
Abstract
The study of rational points over finite fields lies at the heart of arithmetic geometry and coding theory. Weighted curves introduce additional structural complexity due to their grading and singularity behavior, making point counting and zeta-function computations both subtle and computationally rich.
This talk presents methods for enumerating rational points on weighted curves defined over finite fields. We discuss algorithmic approaches, experimental data, and structural patterns arising in specific families. Connections to weighted hypersurfaces and applications to code construction are also explored.
References
S. Salami, T. Shaska, Rational points of weighted hypersurfaces over finite fields, arXiv:2511.12812.
4. A Python Package for Rational Cubics
E. Shaska
Abstract
Rational cubic curves play a central role in classical algebraic geometry and explicit arithmetic constructions. Their computational treatment provides a testing ground for broader symbolic and arithmetic algorithms on projective varieties.
This talk presents a Python-based computational framework for rational cubic curves, focusing on explicit parametrizations, rational function manipulations, and arithmetic invariants. The implementation serves both as a standalone computational tool and as a prototype module within a larger software ecosystem for projective and weighted projective geometry.
References
E. Badr, E. Shaska, T. Shaska, Rational Functions on the Projective Line from a Computational Viewpoint, arXiv:2503.10835.
5. A Database of Binary Septics and Applications to Galois Theory and Coding
J. Mezinaj
Abstract
Binary septic forms encode rich arithmetic and Galois-theoretic information. Systematic computation and storage of their invariants enable large-scale experimental investigations into Galois groups, arithmetic structures, and applications to coding theory.
This talk describes the construction of a structured database of binary septics, including invariant computation, classification data, and arithmetic properties. We discuss database architecture, algorithmic challenges, and applications to Galois theory and the construction of algebraic codes derived from structured polynomial families.
References
J. Mezinaj, From Polynomials to Databases: Arithmetic Structures in Galois Theory, Albanian J. Math. 19 (2025), No. 1, 03–36.
https://albanian-j-math.com/archives/2025-1.pdf
6. A Database for Genus 2 Curves in the Weighted Projective Space P(2,4,6,10)
A. Aurand
Abstract
The weighted projective space P(2,4,6,10)\mathbb{P}(2,4,6,10)P(2,4,6,10) provides a natural coordinate model for the moduli space of genus 2 curves via weighted invariants. This perspective enables both theoretical analysis and large-scale computational exploration of genus 2 families.
This talk presents the design and implementation of a computational database of genus 2 curves embedded in P(2,4,6,10)\mathbb{P}(2,4,6,10)P(2,4,6,10). We discuss invariant computation, arithmetic data over finite fields, and structural patterns emerging from experimental datasets. Applications include connections to moduli theory, arithmetic statistics, and potential interfaces with machine learning approaches to moduli spaces.
References
E. Shaska, T. Shaska, Machine learning for moduli space of genus two curves and an application to isogeny based cryptography, J. Algebraic Combinatorics 61 (2025), 23.
7. Constructing Graded Quantum Codes
S. Griemert
Abstract
Graded algebraic structures provide a natural extension of classical algebraic geometry code constructions. When combined with CSS-type methods, they offer a systematic pathway to constructing quantum error-correcting codes with structured parameters.
This talk develops the theoretical and computational framework for constructing graded quantum codes derived from weighted algebraic geometry. Emphasis is placed on explicit constructions, stabilizer conditions, and parameter analysis, with an eye toward practical implementation within a computational algebra system.
References
https://arxiv.org/abs/2508.07542
8. QWAG: A Python Package for Quantum Weighted Algebraic Codes
T. Shaska
Abstract
Quantum Weighted Algebraic Geometry (QWAG) codes extend classical algebraic geometry codes into the quantum setting using weighted curves and CSS-type constructions. Their implementation requires a coherent integration of arithmetic geometry, linear algebra over finite fields, and stabilizer formalism.
This talk introduces QWAG, a Python software package for constructing and experimenting with quantum weighted algebraic codes. We describe the underlying architecture, code construction algorithms, and computational experiments illustrating new families of quantum codes arising from weighted geometric data.
References
T. Shaska, QWAG: A Python Package for Quantum Weighted Algebraic Codes, arXiv:2508.07542.