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Math-119 · mathematics · Number theory

Torsion Conjecture for Abelian Varieties of Dimension at Least Two

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

For an abelian variety $A/K$, its rational torsion subgroup is \[ A(K)_{\rm tors}=\{P\in A(K):mP=0\text{ for some integer }m\ge1\}. \] For each fixed number field $K$ and dimension $g\ge2$, the conjecture asks for a finite $B(g,K)$ with $\#A(K)_{\rm tors}\le B(g,K)$ for every $g$-dimensional $A/K$. The bound cannot depend on height, endomorphism ring, polarization, or reduction type. It need not be effective, and uniformity over all fields of a fixed degree is not appended.

Formulation and concept references: [S1].

Short English statement

Is rational torsion uniformly bounded across all abelian varieties of a fixed dimension over a fixed number field?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-119-A1 · No claim recorded · LLM-generated · gpt 6 astra pro ·

gpt 6 astra pro attempt on Math-119

Torsion Conjecture for Abelian Varieties of Dimension at Least Two

Definitions and mathematical statement

For an abelian variety $A/K$, its rational torsion subgroup is \[ A(K)_{\rm tors}=\{P\in A(K):mP=0\text{ for some integer }m\ge1\}. \] For each fixed number field $K$ and dimension $g\ge2$, the conjecture asks for a finite $B(g,K)$ with $\#A(K)_{\rm tors}\le B(g,K)$ for every $g$-dimensional $A/K$. The bound cannot depend on height, endomorphism ring, polarization, or reduction type. It need not be effective, and uniformity over all fields of a fixed degree is not appended.

Formulation and concept references: [S1].

Short English statement

Is rational torsion uniformly bounded across all abelian varieties of a fixed dimension over a fixed number field?

Research attempt

Current frontier.
The number-field and function-field claims of the supplied 2026 manuscript must be separated. Its Theorem 1.3 concerns characteristic-zero function fields with a trace condition; Theorem 1.8 retains reduction/height quantities, and Theorem 1.9 assumes its generalized Szpiro Conjecture 1.7 for the number-field conclusion in the semistable case. None is an unconditional proof for all $A/K$. These are the manuscript's stated results, not an independent verification of its proofs. [S1] A 2026 geometric approach to $\ell$-primary torsion concerns fibres of an abelian scheme over a curve, not all primes and all varieties at once. [E2]
Attack route.
Combine several good reductions while retaining prime-primary information. Uniformly small good places would yield a bound without height estimates. We derive an exact certificate and then disprove that proposed place-selection lemma. Two alternatives are to pass to a principally polarized variety or through an isogeny; their costs are different and must be tracked.
Attempt: a multi-place divisibility certificate.
Let $T=A(K)_{\rm tors}$ and let $V$ be a finite set of good places with at least two distinct residue characteristics. Write $p_v=\operatorname{char}k_v$, $q_v=\#k_v$ and $a_v=\#A(k_v)$. Then \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-38-3a-63-65-72-74-69-66-69-63-61-74-65} |T|\mid D_V:=\prod_{\ell\ {\rm prime}} \ell^{\min_{v\in V,\ p_v\ne\ell}v_\ell(a_v)}. \end{equation} Each minimum is nonempty, and only finitely many exponents are nonzero. At a good place the reduction kernel is a formal group with additive residue-field successive quotients, so it has no prime-to-$p_v$ torsion. Thus $T[\ell^\infty]$ injects into $A(k_v)$ if $\ell\ne p_v$; taking valuations proves the formula. The local and finite-field ingredients are reviewed in Clark--Xarles, Sections 3.1--3.2. [E1]

For two places of characteristics $p\ne q$, with point counts $a,b$, \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-38-3a-74-77-6f} D_{v,w}=\gcd(a,b)\, p^{\max(0,v_p(b)-v_p(a))}\, q^{\max(0,v_q(a)-v_q(b))}. \end{equation} In particular \[ |T|\le ab\le\left((1+\sqrt{q_v})(1+\sqrt{q_w})\right)^{2g}. \] The last step is the Weil point bound at good reduction, not a bound on a bad special fibre. [E1] The residue-characteristic corrections in (30006748:two) cannot simply be dropped by taking a greatest common divisor. For $E:y^2=x(x-1)(x-2)$ and $A=E^g$, the companion program computes odd-prime point counts, applies the certificate, and checks divisibility by $4^g$ from the visible rational two-torsion subgroup. It is an exact finite certificate computation, not an assertion of exact torsion for every input.

A small-prime escape construction.
The proposed uniform lemma "two good places of different characteristics have norm at most $Q(g,K)$" fails already over ${\mathbb{Q}}$. Given $B\ge3$, take \[ t=2\prod_{\substack{3\le p\le B\\p\ {\rm prime}}}p, \qquad E_t:y^2=x(x-1)(x-t). \] Its invariants are \[ c_4=16(t^2-t+1),\qquad \Delta=16t^2(t-1)^2. \] For every odd $p\le B$, $v_p(c_4)=0$ and $v_p(\Delta)>0$. The nonintegral $j=c_4^3/\Delta$ excludes good reduction. The integral equation is minimal there since a nonminimal scaling would require $v_p(c_4)\ge4$; its reduction is nodal. Thus $E_t^g$ has bad reduction at every such prime, also evident from its nontrivial direct-sum inertia action. Even if $2$ were good, it would not supply two different small residue characteristics. No large rational torsion is asserted for these examples.

There is a local obstruction to weakening good reduction to semistability. For a Tate curve over ${\mathbb{Q}}_p$ with parameter $q=p^m$, uniformization gives $E_q({\mathbb{Q}}_p)={\mathbb{Q}}_p^\times/q^{{\mathbb{Z}}}$. The point represented by $p$ has order exactly $m$, because $p^r\in p^{m{\mathbb{Z}}}$ precisely when $m\mid r$. Arbitrary semistable curves over a fixed local field therefore have no uniform torsion bound. The split-multiplicative distinction is essential in the local results. [E1] These are local points, not rational points over a fixed number field, and do not contradict the catalogue target.

Two transfers with different costs.
Zarhin's construction $Z(A)=(A\times A^\vee)^4$ has dimension $8g$ and a principal polarization over $K$; the line-bundle construction is in the source's Appendix B. [S1] Since $A$ embeds as a factor over $K$, a uniform bound for principally polarized varieties in dimension $8g$ would imply the target in dimension $g$, without a degree-of-polarization loss. That change of dimension is allowed by the quantifiers.

In contrast, for a $K$-isogeny $\varphi:A\to B$ of degree $d$, \[ |A(K)_{\rm tors}|\le d\,|B(K)_{\rm tors}|. \] The kernel has at most $d$ rational points and the image of torsion is torsion. Decomposition up to isogeny into elliptic factors is therefore not a uniform argument unless the isogeny degree is controlled. The factor embedding in Zarhin's construction avoids this particular loss.

Stress test.
The certificate handles all primary parts; it does not assume injectivity on the residue-characteristic primary subgroup. Taking more places gives curve-dependent improvements but no uniform theorem without a uniform selection mechanism. The Legendre and Tate examples disprove two specific local shortcuts, not the global assertion. Bounds for each fixed $\ell$ would still require a uniform exclusion of sufficiently large torsion primes before they could bound the whole torsion group. No such exclusion has been obtained.
Outcome.
Outcome: Exploratory approach with unresolved gap. The attempt proves an exact finite-reduction certificate and explains a polarization-independent transfer, together with explicit failures of two local strategies. These elementary deductions are not claimed to be new in the literature. A genuinely global uniform constraint remains necessary when all small places have bad reduction; the conjectural height/reduction estimates in the source are not proved here.

Sources

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