Core Concepts

There exists a uniform upper bound of 18 on the number of shared points between two projectively equivalent sets of roots of unity in P¹, excluding trivial cases where the intersection is infinite.

Abstract

To Another Language

from source content

arxiv.org

Hubbard, D. (2024). Roots of unity and projective equivalence. arXiv:2410.17412v1 [math.AG].

This paper aims to provide an elementary proof for the existence of a uniform upper bound on the number of shared points between two projectively equivalent sets of roots of unity in P¹, a result initially proven by Fu (2022) using a different method.

Key Insights Distilled From

by David Hubbar... at **arxiv.org** 10-24-2024

Deeper Inquiries

While the methods employed in the paper draw inspiration from techniques used to study torsion points on elliptic curves, directly generalizing them presents significant challenges. Here's why:
Underlying Group Structure: The paper heavily relies on the structure of the multiplicative group of roots of unity (which is isomorphic to the torsion subgroup of the algebraic torus Gm). Elliptic curves, on the other hand, have a more complex group structure. While both involve abelian groups, the presence of a non-trivial group law on elliptic curves adds complexity.
Convex Geometry Techniques: The use of convex geometry, particularly Newton polygons and mixed volumes, is effective in analyzing the intersections of varieties defined by Laurent polynomials. These techniques are well-suited for the multiplicative structure of Gm. However, the equations defining elliptic curves and their torsion points are typically described using Weierstrass polynomials, which are not Laurent polynomials. This makes the direct application of these convex geometry tools problematic.
Field of Definition: The paper leverages the fact that cyclotomic fields (fields generated by roots of unity) are abelian extensions of the rational numbers. This property allows for the use of Galois theory arguments. In the case of elliptic curves, the fields generated by torsion points can be significantly more complicated, often non-abelian extensions. This makes it difficult to apply similar Galois-theoretic techniques.
Possible Adaptations and Future Directions:
Arakelov Theory: Instead of direct generalization, one might explore adaptations using Arakelov theory. This theory provides tools to study heights of points on arithmetic varieties, which could be relevant for analyzing the intersection of torsion points on elliptic curves.
Model Theory: Model-theoretic approaches, particularly those related to o-minimality, have shown promise in studying unlikely intersections, including those involving torsion points on elliptic curves. Exploring these techniques could offer alternative avenues for generalization.

It's certainly possible. While the paper provides a more elementary proof with a bound of 18, a more elegant and direct path to the tight bound of 14 might exist, potentially leveraging deeper aspects of cyclotomic fields. Here are some avenues to consider:
Galois Module Structure: Cyclotomic fields possess a rich Galois module structure. A deeper understanding of how the Galois action interacts with the specific projective transformations involved could lead to sharper constraints on the intersection points.
Class Field Theory: Class field theory provides a powerful framework for studying abelian extensions of number fields, which cyclotomic fields are a prime example of. Exploring connections between the projective equivalence of roots of unity and the arithmetic of cyclotomic fields via class field theory might reveal a more elegant proof.
Analytic Number Theory: Cyclotomic fields are intimately connected to analytic number theory, particularly through Dirichlet characters and L-functions. It's conceivable that techniques from analytic number theory could provide sharper estimates on the number of intersection points.

While the specific result in the paper might not have immediate direct applications in cryptography or coding theory, the underlying theme of understanding the distribution and relationships between roots of unity under certain transformations has relevance to these fields.
Cryptography:
Lattice-Based Cryptography: Roots of unity, particularly those associated with cyclotomic fields, play a role in lattice-based cryptography. Understanding the projective equivalence of these roots could have implications for the design and analysis of lattice-based cryptographic schemes.
Pairing-Based Cryptography: Bilinear pairings on elliptic curves, often used in pairing-based cryptography, involve torsion points. While the paper focuses on Gm, the broader theme of studying torsion points under transformations could have indirect relevance to the security analysis of pairing-based systems.
Coding Theory:
Cyclic Codes: Cyclic codes, an important class of error-correcting codes, are closely related to polynomials over finite fields, where roots of unity play a fundamental role. Understanding the behavior of roots of unity under transformations could have implications for the construction and properties of cyclic codes.
Lattice Codes: Similar to lattice-based cryptography, lattice codes used in communications also rely on the structure of lattices, where roots of unity can be used to define lattice points. The paper's focus on projective equivalence might inspire investigations into analogous concepts in the context of lattice codes.

0