p-adic elliptic polylogarithms and cubic Chabauty

Abstract: The Chabauty--Coleman--Kim method, under favourable circumstances, describes the set of integral points of a hyperelliptic curve inside the $p$-adic zeroes of certain transcendental functions. For an elliptic curve of Mordell--Weil rank one, the Chabauty--Coleman--Kim set in depth 2 is given by the zeroes of a (finite union of) quadratic polynomial(s) in the $p$-adic logarithm of the elliptic curve and the local $p$-adic height at $p$. Here, we give an explicit formula for a finite set containing the Chabauty--Coleman--Kim set in depth 3 for an elliptic curve of rank at most 2 under an assumption on non-vanishing of a special value of a $p$-adic $L$-function. The finite set is given by the zeroes of a polynomial in $p$-adic elliptic polylogarithms. We use these formulas to verify new instances of Kim's conjecture.

• The paper delivers an explicit computation method for integral points on elliptic curves using determinants of matrices formed by p-adic elliptic polylogarithms and iterated integrals.
• It employs non-abelian Chabauty techniques at depth 3, integrating p-adic Hodge theory with algorithmic Coleman integration to validate Kim’s conjecture in practical cases.
• The approach bridges theoretical progress in p-adic L-functions with effective computation, offering a blueprint for enumerating integral points on elliptic curves.

$p$-adic Elliptic Polylogarithms and Cubic Chabauty: An Expert Overview

Introduction and Context

This work addresses explicit computation in Kim's non-abelian Chabauty program, particularly at depth 3 for integral points on elliptic curves over $\mathbb{Q}$. The methods comprise the confluence of non-abelian $p$-adic Hodge theory, iterated Coleman integration, and the emerging theory of $p$-adic elliptic polylogarithms. For $E/\mathbb{Q}$ an elliptic curve with rank $r\leq 2$, the authors provide an explicit finite set containing the integral points, given as the $p$-adic zeros of polynomials in $p$-adic elliptic polylogarithms. The argument is conditional upon the nonvanishing of a special value of the $p$-adic $L$-function associated to $\mathbb{Q}$0.

This bridges, in a highly constructive way, the breakthroughs of Kim on the non-abelian Chabauty method using non-abelian fundamental groups and the explicit theory of iterated $\mathbb{Q}$1-adic integrals pioneered by Bannai, Kings, Kobayashi, Tsuji, et al. The results are positioned as direct, algorithmic confirmations of Kim's conjecture that for large enough $\mathbb{Q}$2, $\mathbb{Q}$3 for a curve $\mathbb{Q}$4. For elliptic curves minus the origin, the case $\mathbb{Q}$5 (cubic Chabauty) is the technical focus.

Non-Abelian Chabauty and Polylogarithmic Quotients

The classical Chabauty-Coleman technique ensures finiteness of $\mathbb{Q}$6 for curves $\mathbb{Q}$7 of genus $\mathbb{Q}$8 with $\mathbb{Q}$9. Depth 2 (quadratic Chabauty) and above require non-abelian generalizations, where Selmer varieties of unipotent fundamental groups and non-abelian Galois cohomology feature crucially.

For elliptic curves $p$0 (with the origin removed), the controlling object in depth $p$1 is the maximal $p$2-step unipotent quotient of the pro-$p$3 metabelian (étale or de Rham) fundamental group. The Lie algebraic structure allows identification of the depth $p$4 quotient with a combination of symmetric powers of $p$5 and explicit extension classes in Galois cohomology. Beilinson-Levin's elliptic polylogarithm sheaf formalism enables transfer of this structure into an explicit context suitable for algorithmic computation.

$p$6-adic Elliptic Polylogarithms and Explicit Formulae

The main technical advance is the explicit computation of the non-abelian Albanese map (Kim's unipotent Albanese map) in terms of $p$7-adic elliptic polylogarithms. The zero locus that contains $p$8 is given by the vanishing of determinants of matrices whose entries are $p$9-adic polylogarithms and their iterated integrals, evaluated at suitable rational points.

When $p$0, under the assumption that a certain $p$1-adic $p$2-value is nonzero (parallel to the analytic rank conjecture context), the cubic Chabauty set corresponds to the zeros of a degree-3 (or higher) polynomial in $p$3-adic polylogarithmic iterated integrals, generalizing the quadratic setting where only the $p$4-adic logarithm and the $p$5-adic height appeared. Formulas are given for the cubic Chabauty locus via determinants involving

$p$6

and further regularized triple integrals, in addition to terms involving the $p$7-adic discriminant $p$8.

Algorithmic and Computational Aspects

A considerable section is devoted to concrete algorithms for computations:

• Triple Coleman Integrals: The authors use Kedlaya's and Tuitman's techniques for Frobenius pullback and expansion of differential forms to express triple integrals recursively in terms of lower iterated integrals and explicit shuffle relations.
• Constructing Annulators: By choosing integral points $p$9 with suitable independence properties, one constructs determinant expressions whose zeros cut out the cubic Chabauty set.
• Comparison with Goncharov–Levin's Divisorial Theory: The determinantal relations directly correspond, up to isogeny and specialization to the real completion, to those used by Goncharov and Levin in expressing $E/\mathbb{Q}$0 as a value of (real) elliptic trilogarithms at special divisors.

Strong Results and Explicit Examples

The method is exemplified with explicit computations in several genus 1, rank 0, 1, 2 situations:

• For some curves, mock rational points (which are rational over algebraic extensions but not over $E/\mathbb{Q}$1) persist in depth 2 and are nearly, but not fully, eliminated in depth 3, confirming Kim’s conjecture for those cases.
• For specific high-rank examples (e.g., 433.a1 over $E/\mathbb{Q}$2), the method precisely returns the set of $E/\mathbb{Q}$3-integral points, yielding the first such computational confirmation for a rank 2 curve at depth 3.
• The zero loci of the determinant expressions are explicitly computed and shown, in examples, to tightly control $E/\mathbb{Q}$4, in particular, often coinciding with the actual set of integral points.

Theoretical and Practical Implications

Theoretically, the work provides a blueprint for extending non-abelian Chabauty methods to higher depths in explicit form. It gives credence—on concrete computational grounds—to the optimistic case that for genus 1 (and potentially higher) curves, the Kim set at depth 3 or slightly above will often coincide with the integral/rational points, subject to expected non-vanishing conditions resembling the BSD conjecture. The explicit nature of the matrix equations implies that the polylogarithmic quotient is the effective barrier to a full description in the non-abelian Chabauty–Kim filtration.

Practically, these results suggest that, given sufficient computational resources (notably efficient $E/\mathbb{Q}$5-adic integration algorithms and $E/\mathbb{Q}$6-adic $E/\mathbb{Q}$7-function machinery), one can algorithmically enumerate integral points for families of genus 1 curves outside the reach of classical descent or Chabauty. The polylogarithmic approach is constructive rather than merely producing bounds.

Future directions include generalization to higher rank and genus, extension to non-hyperelliptic curves, and further integration with arithmetic statistics models. On the theoretical side, deeper understanding of the connection between special $E/\mathbb{Q}$8-adic $E/\mathbb{Q}$9-values, polylogarithms, and Selmer varieties could inform the non-abelian Birch–Swinnerton–Dyer heuristics.

Conclusion

This paper advances the explicit theory of non-abelian Chabauty by harnessing $r\leq 2$0-adic elliptic polylogarithms to describe cubic Chabauty loci on elliptic curves. Through explicit, computable determinantal expressions in polylogarithmic variables, the method provides both theoretical insight and concrete algorithms for determining integral points on elliptic curves of rank up to two. The approach further reinforces the deep and subtle connections between higher arithmetic geometry, $r\leq 2$1-adic analysis, and special values of motivic $r\leq 2$2-functions.