Quadratic Chabauty Algorithm Overview

• Quadratic Chabauty is a depth-two, nonabelian extension of classical Chabauty that leverages p-adic heights and iterated Coleman integrals to bound rational points on curves.
• It incorporates the Jacobian's Néron–Severi group and algebraic correspondences to relax traditional rank constraints, extending finiteness criteria for rational or integral points.
• The algorithm combines explicit residue-disk computations, modular forms techniques, and height pairings to offer practical strategies for determining rational points on complex curves.

Quadratic Chabauty is the depth-two, first genuinely nonabelian layer of the Chabauty–Kim method for determining rational or integral points on curves. For a curve $X/\mathbf Q$ of genus $g\ge 2$, a prime $p$ or $\ell$ of good reduction, and Jacobian $J$, it replaces the linear Coleman-integral constraints of classical Chabauty by functions built from $p$-adic heights, iterated Coleman integrals, and algebraic correspondences. In its standard form it seeks a finite $p$-adic set containing $X(\mathbf Q)$ or $X(\mathbf Z)$, often in the borderline regime $\operatorname{rank}J(\mathbf Q)=g$, and it occupies the depth-$g\ge 2$0 stage in Kim’s descending sequence $g\ge 2$1 (Siksek, 2017, Balakrishnan et al., 2013, Balakrishnan et al., 2021).

1. Conceptual position within Chabauty–Kim

Kim’s nonabelian program attaches to a hyperbolic curve a tower of local sets cut out by Selmer conditions. At depth $g\ge 2$2, one recovers classical Chabauty–Coleman. At depth $g\ge 2$3, one reaches quadratic Chabauty, the first stage where nonabelian extensions enter explicitly through mixed Galois representations, $g\ge 2$4-adic heights, and iterated Coleman integrals. For elliptic curves, later work makes this hierarchy explicit as “depth 1 = classical Chabauty–Coleman; depth 2 = quadratic Chabauty; depth 3 = cubic Chabauty” (Balakrishnan et al., 22 Apr 2026).

The term “quadratic Chabauty algorithm” does not denote a single uniform black-box procedure. Some papers establish finiteness criteria, some give explicit formulas for local or global pieces, and others assemble full computations of rational points. This distinction is explicit in the literature: Siksek’s note on modular curves is structural rather than implementational, whereas later work on hyperelliptic, non-hyperelliptic, and modular examples develops residue-disk computations, Hodge filtrations, Frobenius structures, and sieving procedures (Siksek, 2017, Balakrishnan et al., 2017, Arul et al., 2022).

A complementary reformulation replaces the original fundamental-group viewpoint by canonical $g\ge 2$5-adic heights coming from $g\ge 2$6-adic adelic metrics on line bundles. In that perspective, the quadratic Chabauty function arises by pulling back a canonical height on $g\ge 2$7 along an Abel–Jacobi embedding $g\ge 2$8 for a line bundle $g\ge 2$9 with $p$0. This yields the same type of locally analytic function on $p$1, while avoiding $p$2-adic Hodge theory and arithmetic fundamental groups as the starting point (Besser et al., 2021).

2. Finiteness criteria and the role of algebraic cycles

Classical Chabauty gives finiteness of $p$3 under the inequality

$p$4

Quadratic Chabauty relaxes this by incorporating the Néron–Severi group. The Balakrishnan–Dogra criterion, as quoted and used repeatedly in later work, states that $p$5 is finite provided

$p$6

If $p$7, this gives no formal improvement; if $p$8, each extra Néron–Severi class increases the allowable Mordell–Weil rank by one (Siksek, 2017).

This numerical gain is geometric. The identification

$p$9

shows that Rosati-fixed endomorphisms, equivalently algebraic correspondences symmetric under the Rosati involution, provide the extra depth-two directions. In explicit depth-two quotients of the unipotent fundamental group, such correspondences appear as “nice correspondences” $\ell$0, often through a nontrivial class in $\ell$1 (Siksek, 2017, Arul et al., 2022).

A refined criterion due to Dogra and Le Fourn replaces the full Jacobian by a quotient. If $\ell$2 admits an isogeny $\ell$3 with $\ell$4, one defines a Chow–Heegner map

$\ell$5

and proves that $\ell$6 is finite if

$\ell$7

This is stronger than the naive quotient analogue $\ell$8, because not every Néron–Severi class on $\ell$9 contributes a usable depth-two function on $J$0 (Dogra et al., 2019).

Over arbitrary number fields, a geometric version replaces the Jacobian by a torsor $J$1 built from the Poincaré biextension and $J$2 Néron–Severi directions. The associated dimension condition is

$J$3

equivalently

$J$4

with $J$5, $J$6, and $J$7. This gives a conditional effective bound once the relevant quotient algebra is finite-dimensional mod $J$8 (Čoupek et al., 2021).

3. Core analytic and arithmetic architecture

In explicit implementations, the global $J$9-adic height decomposes as

$p$0

The local term at the distinguished prime $p$1 is locally analytic; the non-$p$2 terms have finite image on local points and therefore contribute a finite set of constants. The standard quadratic Chabauty function is

$p$3

or, with a chosen correspondence $p$4, $p$5. Rational points satisfy $p$6, where $p$7 is a finite set assembled from the bad-prime local images. The computational task is therefore residue-disk root finding for finitely many analytic equations $p$8, $p$9 (Balakrishnan et al., 2021).

At $p$0, the local height is expressed through the filtered $p$1-module attached to a mixed extension $p$2. In the non-hyperelliptic and modular literature, the essential object is $p$3, or equivalently the pullback of a universal unipotent vector bundle with connection $p$4. One computes a Hodge filtration trivialization and a Frobenius trivialization, represented by block matrices, and inserts them into an explicit formula for $p$5. This is the technical source of the iterated Coleman integrals, Frobenius equations, and local analytic power series that define the depth-two equations on residue disks (Balakrishnan et al., 2017, Arul et al., 2022).

Away from $p$6, the decisive issue is bad reduction. For hyperelliptic curves and odd primes $p$7, recent work computes local heights $p$8 by replacing direct intersection theory on regular models with a combinatorial formula on the Berkovich reduction graph. The normalized local height factors through the skeleton as a piecewise polynomial function whose Laplacian is determined by the action of the correspondence $p$9 on graph homology and by vertex traces on residue components. This provides an effective algorithm for nontrivial local heights at bad primes, and thereby enlarges the practical range of quadratic Chabauty beyond cases where all non-$X(\mathbf Q)$0 local contributions vanish (Betts et al., 2024).

The computational workflow described for modular curves is representative. One searches for rational points of bounded height, computes a de Rham basis, computes Frobenius on rigid or de Rham cohomology, uses Eichler–Shimura to obtain Hecke operators, constructs classes $X(\mathbf Q)$1, computes the local analytic functions $X(\mathbf Q)$2, determines the global bilinear height pairing $X(\mathbf Q)$3 either from known curve points or from Jacobian points via Coleman–Gross heights, forms the finite bad-prime sets $X(\mathbf Q)$4, and solves the equations $X(\mathbf Q)$5 residue-disk by residue-disk. The output is usually a finite $X(\mathbf Q)$6-adic superset of the rational points; a Mordell–Weil sieve is then used to eliminate spurious candidates (Balakrishnan et al., 2021).

4. Explicit incarnations on elliptic and hyperelliptic curves

The first explicit quadratic Chabauty constructions were developed for odd-degree hyperelliptic curves $X(\mathbf Q)$7 with $X(\mathbf Q)$8 in the critical case $X(\mathbf Q)$9. In that setting, one writes

$X(\mathbf Z)$0

where $X(\mathbf Z)$1 are Coleman integrals of a basis of holomorphic differentials and $X(\mathbf Z)$2 is the local $X(\mathbf Z)$3-adic height, expressible as an iterated Coleman integral. For $X(\mathbf Z)$4-integral points, $X(\mathbf Z)$5 takes values in a finite computable set $X(\mathbf Z)$6 coming from bad-prime local heights. This yields the first explicit depth-two algorithms in the rank-equals-genus regime (Balakrishnan et al., 2013).

That hyperelliptic method was then combined with the Mordell–Weil sieve to give a certifying algorithm for integral points on odd-degree hyperelliptic curves with $X(\mathbf Z)$7. The practical advance was not the function $X(\mathbf Z)$8 alone but the integration of several quadratic Chabauty primes, congruence information in $X(\mathbf Z)$9, and sieve primes chosen so that the candidate classes could be eliminated effectively. The resulting computations included genus-$\operatorname{rank}J(\mathbf Q)=g$0 examples well beyond the range of classical Chabauty (Balakrishnan et al., 2015).

For elliptic curves, depth two has especially explicit formulas. In rank $\operatorname{rank}J(\mathbf Q)=g$1, $\operatorname{rank}J(\mathbf Q)=g$2 is described by the simultaneous conditions

$\operatorname{rank}J(\mathbf Q)=g$3

for $\operatorname{rank}J(\mathbf Q)=g$4 ranging over a finite product of bad-prime local height sets. In rank $\operatorname{rank}J(\mathbf Q)=g$5, one obtains

$\operatorname{rank}J(\mathbf Q)=g$6

or equivalently the determinant formula

$\operatorname{rank}J(\mathbf Q)=g$7

on classes with $\operatorname{rank}J(\mathbf Q)=g$8. A substantial computational refinement replaces the double Coleman integral $\operatorname{rank}J(\mathbf Q)=g$9 by the $g\ge 2$00-adic sigma function and division polynomials, using

$g\ge 2$01

This makes the elliptic quadratic Chabauty equations more practical to evaluate (Bianchi, 2019, Balakrishnan et al., 22 Apr 2026).

A variant for even-degree hyperelliptic curves replaces the genuinely quadratic height by a linear functional. If $g\ge 2$02 has two rational points at infinity $g\ge 2$03, the fixed divisor $g\ge 2$04 defines a linear map

$g\ge 2$05

Under the usual quadratic Chabauty assumptions, this linear functional can be expressed in holomorphic Coleman integrals, giving

$g\ge 2$06

with $g\ge 2$07 for a finite computable set $g\ge 2$08. This “linear quadratic Chabauty” is both narrower in scope and computationally simpler than earlier depth-two methods (Gajović et al., 2023).

5. Modular curves as a preferred domain

Modular curves are unusually favorable for quadratic Chabauty because their Jacobians carry abundant endomorphisms. Siksek showed that for a modular curve $g\ge 2$09 of genus at least $g\ge 2$10, the Jacobian $g\ge 2$11 always has $g\ge 2$12. More precisely, if

$g\ge 2$13

is the decomposition into $g\ge 2$14-simple factors attached to weight-$g\ge 2$15 cuspidal eigenforms with Hecke fields $g\ge 2$16, then

$g\ge 2$17

where $g\ge 2$18 are totally real and the others are CM. Consequently, all modular curves of genus at least $g\ge 2$19 satisfy a quadratic Chabauty condition strictly weaker than the classical rank bound (Siksek, 2017).

Dogra and Le Fourn strengthened the modular picture by proving finiteness of $g\ge 2$20 for the prime-level quotients $g\ge 2$21 and $g\ge 2$22 via suitable quotients $g\ge 2$23 of their Jacobians. Their argument uses modular forms $g\ge 2$24 with $g\ge 2$25, a Kolyvagin–Logachev type rank-one theorem giving $g\ge 2$26, and Chow–Heegner points to show that the quotient-theoretic kernel $g\ge 2$27 is large enough. For prime $g\ge 2$28 and genus at least $g\ge 2$29, this yields finiteness of the depth-two Chabauty–Kim set for all $g\ge 2$30 in the $g\ge 2$31 case, and similarly for $g\ge 2$32 when a rational base point exists (Dogra et al., 2019).

The first explicit non-hyperelliptic quadratic Chabauty algorithm was carried out on the split Cartan modular curve $g\ge 2$33, completing the determination of its rational points. The method computes the filtered $g\ge 2$34-module $g\ge 2$35 without hyperelliptic symmetry, using explicit universal connections, Hodge filtrations, Frobenius equations, and determinant functions attached to two independent Tate classes. This made depth-two Chabauty workable on a plane quartic model (Balakrishnan et al., 2017).

Subsequent modular implementations systematized the method for curves of genus $g\ge 2$36, $g\ge 2$37, and $g\ge 2$38. The modular-specific simplification is the use of Hecke correspondences: Frobenius is computed on de Rham or rigid cohomology, Eichler–Shimura recovers the Hecke action, and powers of a Hecke operator $g\ge 2$39 generate classes in $g\ge 2$40. The resulting quadratic Chabauty functions were combined with Jacobian height computations and Mordell–Weil sieving to determine rational points on several Atkin–Lehner quotients, the exceptional modular curve $g\ge 2$41, and the genus-$g\ge 2$42 non-split Cartan curve $g\ge 2$43 (Balakrishnan et al., 2021).

A concrete genus-$g\ge 2$44 case is $g\ge 2$45, where $g\ge 2$46, so classical Chabauty fails. The computation uses a $g\ge 2$47-adic model chosen so that all relevant points lie in affine non-Weierstrass disks, constructs a Hecke correspondence $g\ge 2$48 from $g\ge 2$49, computes the local height through $g\ge 2$50, reconstructs the global height from six known rational points, forms

$g\ge 2$51

and finds that $g\ge 2$52 vanishes at the six rational points and at 22 additional $g\ge 2$53-adic points. A Mordell–Weil sieve at $g\ge 2$54 eliminates the extras, proving that the known six points are all rational points (Arul et al., 2022).

More recently, one of the main local steps for prime-level Atkin–Lehner quotients $g\ge 2$55 has been recast in a model-free modular-forms language. Instead of using an explicit plane model and Tuitman-style Frobenius merely to recover the Hecke action, the Hodge filtration on the universal connection $g\ge 2$56 is computed directly from $g\ge 2$57-expansions of weight-$g\ge 2$58 cusp forms and weakly holomorphic modular forms. The algorithm outputs the filtration matrix $g\ge 2$59, hence the quantities $g\ge 2$60 and $g\ge 2$61 entering the local height formula, and scales to the genus-$g\ge 2$62 curve $g\ge 2$63 (Rendell, 2 Sep 2025).

6. Variants, extensions, and limitations

Quadratic Chabauty is intrinsically modular: it produces a finite $g\ge 2$64-adic superset of rational or integral points, not necessarily the exact global set. Extra $g\ge 2$65-adic zeros are therefore typical rather than exceptional. This is explicit in the hyperelliptic, modular, and elliptic literature: the depth-two equations often leave residual $g\ge 2$66-adic candidates that are removed only after a Mordell–Weil sieve, point recognition, or additional global input (Balakrishnan et al., 2015, Balakrishnan et al., 2021, Arul et al., 2022).

Several papers isolate one component of the full pipeline rather than the entire algorithm. Siksek’s note establishes applicability on modular curves; the adelic-metric approach reconstructs the quadratic Chabauty function without beginning from fundamental groups; the model-free modular-forms algorithm computes the Hodge filtration but not the full end-to-end procedure; and the hyperelliptic bad-prime local-height algorithm supplies the non-$g\ge 2$67 arithmetic contribution that earlier implementations often avoided by working only in vanishing cases. This suggests that “the quadratic Chabauty algorithm” is best understood as a layered architecture whose pieces are still being generalized and optimized (Siksek, 2017, Besser et al., 2021, Betts et al., 2024, Rendell, 2 Sep 2025).

The method extends beyond $g\ge 2$68, but the extension is not uniform. Over imaginary quadratic fields, rank-$g\ge 2$69 elliptic curves can be treated using cyclotomic and anticyclotomic heights, yielding two locally analytic functions $g\ge 2$70 and $g\ge 2$71 on $g\ge 2$72. The finite candidate set is then cut out by the conditions $g\ge 2$73 and $g\ge 2$74, after which a sieve compares reduction and logarithmic information at two split primes. This gives the first complete quadratic Chabauty determination of integral points on a rank-$g\ge 2$75 elliptic curve over an imaginary quadratic field that is not a base change (Jha, 2023).

A plausible implication is that depth two is often the practical frontier rather than the theoretical endpoint. When quadratic Chabauty still leaves mock points, higher depth becomes relevant. Recent work on cubic Chabauty for punctured elliptic curves makes this explicit: in rank $g\ge 2$76, depth $g\ge 2$77 is governed by a quadratic relation between the local $g\ge 2$78-adic height and the elliptic logarithm, while depth $g\ge 2$79 replaces that relation by determinant or resultant equations in $g\ge 2$80-adic elliptic polylogarithms such as

$g\ge 2$81

In this sense, quadratic Chabauty is both a mature computational method and the depth-two prototype for higher nonabelian Chabauty algorithms (Balakrishnan et al., 22 Apr 2026).