Ailon–Rudnick — torsion points and common power divisors

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conjecture_a: The paper conjectures infinitely many coprime power pairs when the bases are independent and their first powers minus one are coprime.

conjecture_b: A primitive nonsingular integer matrix with an independent eigenvalue pair is conjectured to have infinitely many primitive powers.

cyclotomic_units: Every unit in a prime cyclotomic field is a root of unity times a real unit, the external input to the primitive-matrix construction.

lang_torsion_theorem: An irreducible curve in the two-dimensional complex torus has finitely many torsion points unless it is a torsion translate of a subtorus.

local_matrix_bounds: Uniform valuation bounds for Jordan powers, including constant eigenvalues, make the multiplicity step in Theorem 3 explicit.

matrix_content: Basic gcd, basis-invariance and specialization facts needed for the paper's integer and polynomial matrix arguments.

proposition_4: A hyperbolic two-by-two unimodular integer matrix has power-minus-identity content bounded below by a constant times the k/2 power of the absolute value of its expanding eigenvalue.

theorem_1: Independent nonconstant polynomials have uniformly bounded common power divisors, with nontrivial gcd confined to finitely many divisibility classes.

theorem_2: Multiplication by a nonreal unit in a prime cyclotomic field gives a primitive matrix at every positive exponent not divisible by that prime.

theorem_3: A nontrivial Jordan block or two independent eigenvalues force uniformly bounded content of polynomial matrix powers minus the identity.

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Citation. Nir Ailon and Zéev Rudnick, Torsion points on curves and common divisors of $a^k-1$ and $b^k-1$, Acta Arithmetica 113 (2004), no. 1, 31–38. DOI: 10.4064/aa113-1-3. The publisher's record confirms the authors, volume, pages and DOI; accessed.

Source version

The canonical eight-page PDF is the published typesetting, obtained from Rudnick's author-hosted copy on 2026-09-05. Its printed pages are 31–38, corresponding to PDF pages 1–8. The final page records receipt on 4 July 2002 and revision on 15 December 2002. The retained file is 124,252 bytes. The file's text layer carries no copyright or license line; the publisher's record offers the PDF "Free download under CC-BY license", a Creative Commons Attribution license with no version or URL named (https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/113/1/83111/torsion-points-on-curves-and-common-divisors-of-a-k-1-and-b-k-1, read 2026-10-02); the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

No separate earlier manuscript or Ailon's 2001 M.Sc. thesis is retained here. In particular, the footnote to Proposition 4 says that the published referee-suggested proof replaced a more complicated original proof; only the published norm argument is reconstructed.

Results and methods

The paper proves polynomial analogs and an integer matrix construction; it does not prove its general integer conjectures. Polynomial gcds below are monic. For matrices, the gcd means the gcd of all entries of $A-I$, as made precise in the complete matrix-content deductions.

• Theorem 1 proves that independent nonconstant $f,g\in\mathbb C[t]$ have one fixed polynomial divisible by every $\gcd(f^k-1,g^k-1)$. If the gcd at $k=1$ is one, the bad exponents form a finite union of proper divisibility classes. The direct proof uses an exact external torsion-point theorem, followed by a complete multiplicity and exponent argument.
• Proposition 4 gives $\gcd(A^k-I)\ge c_A|\varepsilon|^{k/2}$ for hyperbolic $A\in\operatorname{SL}_2(\mathbb Z)$ and its expanding eigenvalue $\varepsilon$. Its distinct complete proof uses a quadratic field norm and an integral linear combination of the entries.
• Theorem 2 proves primitivity of multiplication by $u^k$ when $u$ is a nonreal unit in $\mathbb Q(\zeta_p)$, $p>3$ is prime and $p\nmid k$. The proof uses an explicit external unit decomposition, then a complete coefficient-symmetry argument in an integral basis.
• Theorem 3 proves bounded polynomial matrix content when there is a nontrivial Jordan block or an independent eigenvalue pair. Its two cases are treated separately; the diagonalizable case also uses the external torsion-point theorem.

All four numbered proofs and the two elementary dependency pages are reconstructed at these stated external-input boundaries. Neither external theorem is proved here. The two conjecture pages record statements and elementary relationships, not proofs of the conjectures: Conjecture A concerns scalar integer powers and Conjecture B concerns integer matrices.

Proof qualifications

Multiplicative independence is taken in the full group-theoretic sense: $x^u y^v=1$ with integer exponents forces $u=v=0$. For the original integer bases other than $0,\pm1$, and for nonconstant polynomials, this agrees with the introduction's comparison of positive powers.

The printed determinant factorization in Section 5 misidentifies the diagonal entries as those of $B-I$ rather than $B$. Proposition 4 keeps absolute values for a negative expanding eigenvalue, and Conjecture A's $b=-a$ example records the parity condition already forced by initial coprimality.

The Bugeaud–Corvaja–Zannier theorem is the subexponential integer-gcd background cited in the introduction. It does not turn the polynomial proof or a subexponential upper bound into an integer gcd-one result.

Bears on. For $a=2,b=3$, Conjecture A is the infinitely-often coprimality subquestion of Problem 820. The paper gives related background for the common-divisor threshold in Problem 770. It does not settle the remaining estimates in either problem, and this source compilation makes no determination of the conjectures' current status.