What question did this study set out to answer?

This work aims to explore the conditions under which integers are reachable through torsion points on cyclic covers.

May 16, 2026

Torsion points of small order on cyclic covers of ℚ 1 . II

BBBoris M. BekkerSt Petersburg University YZYuri G. ZarhinPennsylvania State University

Key Points

This work aims to explore the conditions under which integers are reachable through torsion points on cyclic covers.
Analyzed torsion points on the Jacobian of smooth projective models of affine curves.
Established criteria for an integer to be reachable based on properties of polynomials and infinite conditions.
Examined specific cases including hyperelliptic conditions.
Showed that if an integer is reachable, certain polynomial degrees or infinite conditions must hold.
Confirmed that reachability is equivalent under specific infinite conditions or when certain degrees are finite.
Determined specific parameters under which reachability occurs in hyperelliptic cases.

Abstract

Let Formula: see text be an integer, Formula: see text a perfect field such that Formula: see text, Formula: see text an integer prime to Formula: see text, Formula: see text a degree Formula: see text monic polynomial without repeated roots, and Formula: see text a smooth projective model of the affine curve Formula: see text. Let Formula: see text be the Jacobian of the Formula: see text-curve Formula: see text. We identify Formula: see text with its canonical image in Formula: see text (such that the infinite point of Formula: see text goes to the zero of the group law on Formula: see text). We say that an integer Formula: see text is Formula: see text-reachable over Formula: see text if there exists a polynomial Formula: see text as above such that Formula: see text contains a torsion point of order Formula: see text. Earlier we proved that if Formula: see text is Formula: see text-reachable, then either Formula: see text or Formula: see text (in addition, both Formula: see text and Formula: see text are Formula: see text-reachable). In the present paper we prove the following. If Formula: see text and if Formula: see text is Formula: see text-reachable over Formula: see text, then either Formula: see text or Formula: see text. If either Formula: see text, or Formula: see text is infinite and Formula: see text, then Formula: see text is Formula: see text-reachable if and only if Formula: see text. If Formula: see text, then Formula: see text is Formula: see text-reachable if and only if Formula: see text. If Formula: see text (the hyperelliptic case) and Formula: see text, then Formula: see text is Formula: see text-reachable if Formula: see text and Formula: see text. (The case when Formula: see text was done earlier by E.V. Flynn.)

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