Quadratic Chabauty Locus

• Quadratic Chabauty Locus is defined as the zero set of a p-adic Coleman function that cuts out S-integral points on hyperbolic curves through explicit quadratic relations.
• It leverages Kim’s non-abelian Chabauty method by computing a bilinear form using p-adic logarithms and dilogarithms to reveal underlying arithmetic structures.
• The approach integrates K-theoretic reductions and motivic cohomology, transforming non-abelian cohomological data into explicitly computable analytic equations.

The Quadratic Chabauty locus is a central object in modern non-abelian Diophantine geometry, describing the set of $p$-adic points on an algebraic curve cut out by explicit quadratic relations in the context of Kim’s non-abelian Chabauty–Kim method. For the thrice-punctured projective line $X = \mathbb{P}^1 \setminus \{0,1,\infty\}$, the locus is realized as the vanishing set of a bilinear (quadratic) Coleman function arising from the depth-2 unipotent $p$-adic Hodge morphism, with coefficients determined by $p$-adic logarithms and dilogarithms. This construction generalizes to higher-dimensional and modular settings, providing a powerful framework for controlling $S$-integral or rational points on hyperbolic curves.

1. Non-Abelian Chabauty–Kim Theory and the Role of the Depth-2 Morphism

Kim’s non-abelian Chabauty program replaces the classical abelian Chabauty–Coleman approach with Galois-theoretically defined towers of Selmer varieties attached to increasingly deep unipotent quotients $\pi_1^{(n)}$ of the étale fundamental group. For $X = \mathbb{P}^1 \setminus \{0,1,\infty\}$ and finite $S$, the integer $n$ defines the depth. The depth-2 global $p$-adic Hodge morphism

$$h_2: \text{Sel}_2(G_T, \pi_1^{(2)}) \to \mathrm{Alb}_2$$

has three coordinates, with the first two arising linearly from the abelianization (i.e., classical Chabauty level) and the third—$h_{1,3}$—encoding novel quadratic information. This third coordinate is a bilinear map

${}_{(Q)} h_{1,3}: (Q^S) \times (Q^S) \to Q_p,$

where $Q^S$ is a $Q$-vector space with one basis element per $q\in S$ (the "2|S|" variables stem from two abelian factors in the cohomological description). This quadratic form is at the heart of the explicit quadratic Chabauty locus.

2. Bilinear (Quadratic) Polynomial: Coefficient Structure and Computational Formula

The explicit description of $h_{1,3}$ is given by a bilinear polynomial: $${}_{(Q)} h_{1,3}(x, y) = \sum_{1\leq i, j\leq s} h_{1,3}^{(q_i, q_j)} \, x_i y_j,$$ where each $h_{1,3}^{(q, q')}$ is computed via decompositions in Milnor $K$-theory, informed by the vanishing of $K_2(\mathbb{Q}) \otimes \mathbb{Q}$. Explicitly,

$${}_{(Q)} h_{1,3}^{(q, q')} = \sum_k s_k \big( \log(q)(u_k) \log(q)(v_k) \big) - \sum_l d_l \mathrm{Li}(t_l),$$

where $s_k$, $d_l$ are rational numbers obtained from reduction algorithms, and $u_k,v_k,t_l$ are selected to control size with respect to $q,q'$. $\log$ is the $p$-adic logarithm, and $\mathrm{Li}$ denotes the $p$-adic dilogarithm. Under the de Rham realization, $\rho_2^A(x \otimes y)$ encodes both the symmetric product of logarithms and this bilinear form.

3. Quadratic Chabauty Locus as a Zero Set: Cutting out Integral Points

The image of the global Selmer variety under $h_2$ is typically lower-dimensional than the full de Rham fundamental group. This enables the definition of a nonzero Coleman function—explicitly, the pullback via the unipotent Albanese map of the coordinate function provided by $h_{1,3}$—whose vanishing locus in the $p$-adic analytic space of $X$ is the quadratic Chabauty locus: $\{\text{%%%%31%%%%-adic points on %%%%32%%%% where the Coleman function vanishes}\}.$ Integral points map, under the unipotent Kummer map, to $(t, 1-t)$, so the locus is defined by

${}_{(Q)} h_{1,3}(t, 1-t) = -\mathrm{Li}(t),$

and the vanishing condition becomes

$2\mathrm{Li}(t) - \log(t)\log(1-t) = 0,$

which is a quadratic relation in the abelianized coordinates. Thus, the integral points lie precisely in the zero set of an explicit quadratic polynomial in $2|S|$ variables.

4. Cohomological and K-theoretic Underpinnings

The construction of the coefficients of the quadratic form leverages Milnor $K$-theory and Tate's theorem ($K_2(\mathbb{Q}) \otimes \mathbb{Q} = 0$). Every tensor $q \otimes q'$ is reduced to sums of tensors of units and $t_l \otimes (1-t_l)$. These reductions feed into the computation of the $p$-adic logarithms and dilogarithms entering the explicit formula.

The bilinear form exhibits a "twisted antisymmetry" property: $${}_{(Q)} h_{1,3}(u, v) + {}_{(Q)} h_{1,3}(v, u) = (\log u) (\log v).$$ This reflects the decomposition of the quadratic piece into an antisymmetric dilogarithmic term and a symmetric logarithmic self-product, further tying the polynomial to $K$-theoretic structures.

5. Function-theoretic and Motivic Interpretation

In the motivic Chabauty–Kim framework, the Selmer variety parametrizes non-abelian cohomology classes associated to mixed Tate motives over $\mathbb{Z}[S^{-1}]$. The unipotent $p$-adic Hodge morphism maps these classes into the de Rham fundamental group, with the vanishing of the regular function (Coleman function) exactly characterizing the image. The depth-2 novelty is precisely the quadratic coordinate $h_{1,3}$, which introduces new motivic and functional constraints beyond those seen in abelian Chabauty.

6. Explicit Geometric and Computational Realizations

In practice, as demonstrated in the case of $X = \mathbb{P}^1 \setminus \{0,1,\infty\}$, the quadratic Chabauty locus is realized as the zero set of a single explicit quadratic polynomial involving $p$-adic logarithms and dilogarithms, computable through the reduction algorithm described in the main theorem. The function is purely $p$-adic analytic and serves directly as a cutting tool for $S$-integral points. In explicit computational examples—including rings of the form $\mathbb{Z}[1/2]$ or the ring of integers of real quadratic fields—a single Coleman function suffices to cut out all integral points.

7. Conceptual and Diophantine Implications

The explicit quadratic nature of the Chabauty locus provides a functional machinery for detecting non-density of the image of the global Selmer variety in the de Rham fundamental group, and hence for establishing finiteness statements for $S$-integral or rational points (akin to Siegel's theorem). In the broader Chabauty–Kim program, this clarifies how non-abelian cohomological invariants descend to explicit analytic equations, fulfilling conjectural expectations that such analytic loci coincide with sets of integral points. The approach also demonstrates that in certain settings, the quadratic locus gives a complete, uniform description across several base rings.

Table: Key Structural Ingredients of the Quadratic Chabauty Locus

Object/Concept	Description	Formula/Role
$h_{1,3}$	Quadratic coordinate in $p$-adic Hodge morphism	$h_{1,3}(x, y) = \sum_{i,j} c_{i,j}x_iy_j$
Coefficients $c_{i,j}$	In terms of log/dilog via K-theory reduction	$$c_{i,j} = \sum s_k \log(u_k)\log(v_k) - \sum d_l \mathrm{Li}(t_l)$$
Quadratic Chabauty locus	Zero set of Coleman function in $p$-adic coordinates	$2\mathrm{Li}(t) - \log(t)\log(1-t) = 0$

Summary

The quadratic Chabauty locus in Kim’s explicit Chabauty–Kim theory for the thrice-punctured line is characterized as the explicit vanishing locus of a $p$-adic analytic bilinear polynomial—the third coordinate of the unipotent $p$-adic Hodge morphism at depth two. Its coefficients are given algorithmically by $p$-adic logarithms and dilogarithms arising from $K$-theoretic reductions. The locus “cuts out” exactly the $S$-integral points, transforming motivic and cohomological information into computable analytic equations that govern finiteness of integral solutions for $X = \mathbb{P}^1 \setminus \{0,1,\infty\}$, and providing a model for higher-dimensional generalizations in the Chabauty–Kim program (Dan-Cohen et al., 2012).