Integral torsion points on abelian varieties over function fields

Abstract: We prove an analogue, over global function fields, of a conjecture due to Su-Ion Ih concerning the non-Zariski density of torsion points on abelian varieties that are integral with respect to a given non-special divisor. Along the way, we establish a Tate--Voloch type theorem for abelian varieties over completions of global function fields, which allows us to obtain a logarithmic equidistribution result for Galois orbits of torsion points.

• The paper proves a function field analogue of Ih's conjecture, demonstrating that the set of torsion points that are integral with respect to a divisor is not Zariski dense.
• Two key results include the establishment of a Tate–Voloch type theorem for abelian varieties over characteristic p, and a logarithmic equidistribution theorem for Galois orbits of prime-to-p torsion points.
• The paper identifies the missing link for extending Ih's conjecture to number fields, highlighting the need for an archimedean logarithmic equidistribution theorem for torsion orbits.

Overview

This paper, by Robin de Jong, Nicole Looper, and Farbod Shokrieh (2601.19757), proves a function field analogue of Ih's conjecture concerning the non-Zariski density of torsion points that are integral with respect to a divisor. The result is obtained as the culmination of two technically distinct components: first, a Tate–Voloch type theorem for abelian varieties over completions of global function fields in characteristic $p$, proved by adapting Corpet's ultrafield-based argument; second, a logarithmic equidistribution theorem for Galois orbits of prime-to-$p$ torsion points, derived from the equidistribution results of Faber and Gubler combined with the Tate–Voloch input. The paper also articulates explicitly how its proof strategy transfers to the number field case and identifies precisely which ingredient is still missing there.

Ih's conjecture and the main theorem

Ih's conjecture, formulated over number fields, asserts that for an abelian variety $A$ over a number field $K$, a nonzero effective divisor $D$ on $A_{\overline{K}}$ with at least one non-special irreducible component, and a finite set $S$ of places containing the archimedean ones, the set of torsion points that are $S$-integral with respect to $D$ (for some model) is not Zariski dense. The conjecture remains open in general; prior to this work it was known only for elliptic curves, where Baker–Ih–Rumely gave a proof. The conjecture was motivated by an analogy: it should relate to Manin–Mumford (proved by Raynaud) as Lang's conjecture relates to Mordell–Lang.

The main theorem establishes the analogue over global function fields:

Main Theorem. Let $K$ be a field of transcendence degree one over a finite field $p$0 of characteristic $p$1, let $p$2 be an abelian variety, let $p$3 be a nonzero effective divisor on $p$4 with at least one component not $p$5-special, and fix a model $p$6. Then the set of torsion points of $p$7 of prime-to-$p$8 order that are $p$9-integral with respect to $A$0 and $A$1 is not Zariski dense.

Two qualifications deserve emphasis. First, the notion of "special" must be adjusted from the classical definition (translates of abelian subvarieties by torsion points): in positive characteristic one must allow constant subvarieties arising via the $A$2-trace, following Yamaki and Xie–Yuan. Second, the statement concerns only prime-to-$A$3 torsion. This restriction is not cosmetic: the set of prime-to-$A$4 torsion points is itself Zariski dense, so nothing is lost in density terms, but restricting to it allows the authors to exploit Galois-theoretic methods and avoid inseparability pathologies. Whether the full statement including $A$5-power torsion holds is left open here.

A Tate–Voloch type theorem in characteristic $A$6

The first part of the paper proves that any closed subvariety $A$7 over a global function field $A$8 is unapproximable, in the $A$9-adic topology, by torsion points of order prime to $K$0: there exists $K$1 such that every such torsion point either lies in $K$2 or has distance at least $K$3 from $K$4. This is the direct analogue, over completions of function fields, of the Tate–Voloch conjecture proved by Scanlon over $K$5 using model theory.

The distance functions used are those of Voloch, defined scheme-theoretically via reduction modulo ideals $K$6 of the valuation ring; these are equivalent to Silverman's local height functions via $K$7, enjoy ultrametric properties, Galois equivariance, and functoriality under morphisms. The proof follows Corpet's thesis closely but with simplifications, notably exploiting the fact that the "satellite variety" constructed from a polynomial $K$8 and $K$9 is special.

The architecture of the argument proceeds through several reductions:

• Discreteness of torsion. The authors prove that torsion in $D$0 is discrete when the residue field is finite — Mattuck's theorem transplanted to characteristic $D$1 — by showing the kernel of reduction carries only $D$2-power torsion (via the formal group being pro-$D$3) and that multiplication-by-$D$4 strictly contracts near the origin.
• Special subvarieties. If $D$5 is $D$6-special, then all torsion points outside $D$7 have distance exactly $D$8 from $D$9. This uses the fact that the trace map induces an isomorphism on torsion and that constant subvarieties cannot be approximated by non-constant torsion.
• Unramified torsion. For torsion in $A_{\overline{K}}$0 of prime-to-$A_{\overline{K}}$1 order, the Weil polynomial of the special fiber of the Néron model yields a polynomial $A_{\overline{K}}$2 without roots of unity among its complex roots such that $A_{\overline{K}}$3 annihilates all such torsion, where $A_{\overline{K}}$4 is Frobenius. The satellite construction then produces a $A_{\overline{K}}$5-invariant subvariety $A_{\overline{K}}$6 which, since no weight-zero factors exist, is special by a dichotomy theorem deduced from Pink–Rössler; discreteness of torsion on special varieties closes the argument.
• Arbitrary prime-to-$A_{\overline{K}}$7 torsion. The key observation is that all prime-to-$A_{\overline{K}}$8 torsion lies in the maximal tamely ramified extension $A_{\overline{K}}$9, and that inertia acts unipotently of index two ($S$0, from SGA 7). An induction on $S$1 combines this with the unramified case: choosing $S$2 with $S$3 in the ideal generated by $S$4 and $S$5 forces the multiples $S$6 into the unramified extension along an ultrafilter, contradicting the already-proved unramified case.

An implication worth noting: this Tate–Voloch theorem is exactly what licenses truncating local height functions in the equidistribution argument below, since it provides a uniform upper bound for local heights of torsion orbits away from $S$7.

Normalized canonical local heights

The second part develops the height-theoretic machinery. Building on de Jong–Shokrieh's framework of canonical metrics and Berkovich skeleta, the authors use normalized canonical local heights

$S$8

which are intrinsic to the divisor $S$9 rather than to the choice of section or rigidification. At places of good reduction, the normalized height equals the local intersection multiplicity $S$0, and hence vanishes identically on points integral with respect to the Néron model. This vanishing property is the linchpin connecting integrality to heights.

The paper extends the local-to-global formula to points defined only over $S$1 by averaging over Galois orbits: the Néron–Tate height of $S$2 equals the sum over places of orbit-averaged normalized local heights minus a correction term involving integrals of $S$3 against Haar measure on the skeleton. Both sums are finite because of the good-reduction formula.

For divisors, the paper derives a Mahler measure-type formula: for an ample integral divisor $S$4 with associated line bundle $S$5 and symmetric bundle $S$6,

$S$7

where $S$8. The proof combines Chambert-Loir–Thuillier's recursive arithmetic intersection formula with Gubler's computation that the top self-intersection of the canonical measure is $S$9, together with the vanishing of the total self-intersection of $D$0 on $D$1 (a consequence of Gubler's criterion relating dense small points to vanishing height). Combined with Yamaki's geometric Bogomolov conjecture for divisors, this yields: if $D$2 for ample symmetric $D$3, then $D$4 is $D$5-special.

Logarithmic equidistribution

The Faber–Gubler equidistribution theorem states that for a generic net of torsion points, orbit averages of continuous test functions converge to integrals against the Haar measure $D$6 on the skeleton. Local height functions, however, are singular along $D$7 and do not extend continuously, so the theorem does not apply directly. The paper bridges this gap:

Logarithmic equidistribution. For a generic net $D$8 of prime-to-$D$9 torsion points avoiding $K$0, and any local height $K$1 for $K$2, $K$3

The proof truncates $K$4 at a uniform bound $K$5 supplied by the Tate–Voloch theorem (which guarantees local heights of torsion points off $K$6 are bounded above), applies ordinary equidistribution to the continuous function $K$7, and observes that both sides are unaffected by the truncation. This is where the two halves of the paper meet: the analytic proximity control of Part I is precisely what makes the logarithmic refinement of Part II possible.

Proof of the main theorem

The final section reduces to the case where $K$8 is an ample integral divisor defined over $K$9, $p$00 has good reduction outside $p$01, and integrality is taken with respect to the Néron model. The reduction passes to the quotient by the identity component of the stabilizer of $p$02, which makes $p$03 ample while preserving non-specialness; integrality of images under the quotient map at good-reduction places follows from the Serre–Tate good-reduction criterion applied to the stabilizer subgroup.

The contrapositive argument is clean. Suppose a generic net $p$04 of $p$05-integral prime-to-$p$06 torsion points exists. Each $p$07 has Néron–Tate height zero, so the orbit-averaged local-to-global formula gives

$p$08

At places outside $p$09, integrality plus good reduction forces each normalized local height to vanish. At places inside $p$10, logarithmic equidistribution identifies each limit with $p$11, which is zero by the normalization. Hence $p$12, and by the geometric Bogomolov conjecture for divisors, $p$13 is $p$14-special — contradicting the hypothesis. The conclusion is that $p$15-integral prime-to-$p$16 torsion is not Zariski dense whenever $p$17 has a non-special component.

Relation to the number field case

The paper states plainly that its proof supplies a template for Ih's conjecture over number fields. In that setting, Zhang's equidistribution theorem replaces Faber–Gubler, and Zhang's Bogomolov-type corollary replaces Yamaki; Scanlon's theorem handles the non-archimedean logarithmic equidistribution. The single missing ingredient, identified explicitly by the authors, is a suitable logarithmic equidistribution result for Galois orbits of torsion points at archimedean places. This is a concrete, well-posed open problem rather than a vague programmatic gap.

Limitations and open questions

Several restrictions bound the scope of the results. The main theorem addresses only prime-to-$p$18 torsion; extending to all torsion would require controlling $p$19-power torsion, where the formal group methods used here behave differently and inseparability issues arise. The Tate–Voloch theorem is stated for abelian varieties over completions of global function fields, not for semi-abelian varieties generally, whereas Scanlon's characteristic-zero result covers semi-abelian varieties over $p$20. The logarithmic equidistribution theorem requires the generic-net hypothesis and applies to divisors; whether analogous statements hold for higher-codimension subvarieties, or for nets converging to proper subvarieties, is not addressed. Finally, the archimedean logarithmic equidistribution needed to complete the number field case of Ih's conjecture remains open, as noted above.

Conclusion

The paper proves Ih's conjecture over global function fields for prime-to-$p$21 torsion, via a combination of a new Tate–Voloch type theorem in characteristic $p$22 (obtained by adapting Corpet's ultrafield method and simplifying it through the specialness of satellite varieties) and a logarithmic refinement of Faber–Gubler equidistribution enabled by the normalized canonical local heights of de Jong–Shokrieh. Beyond the function field result, the paper isolates with precision what separates the current state of knowledge from Ih's conjecture over number fields: the absence of an archimedean logarithmic equidistribution theorem for torsion orbits.