Effective Chabauty

One of the deepest and most important results in number theory is the Mordell Conjecture, proved by Faltings (and independently by Vojta shortly thereafter). It asserts that if X / {\mathbf Q} is an algebraic curve of genus at least 2, then the set X({\mathbf Q}) of rational points on X is finite. At present, we do not know any effective algorithm (in theory or in practice) to compute the finite set X({\mathbf Q}). The techniques of Faltings and Vojta lead in principle to an upper bound for the number of rational points on X, but the bound obtained is far from sharp and is difficult to write down explicitly. In his influential paper Effective Chabauty, Robert Coleman combined his theory of p-adic integration with an old idea of Chabauty and showed that it led to a simple explicit upper bound for the size of X({\mathbf Q}) provided that the Mordell-Weil rank of the Jacobian of X is not too large.  (For a memorial tribute to Coleman, who passed away on March 24, 2014, see this blog post.)

More specifically, around 1940 Claude Chabauty had the idea that in order to prove that X({\mathbf Q}) is finite, one could try to show that X({\mathbf Q}_p) \cap \overline{J({\mathbf Q})} is finite, where \overline{J({\mathbf Q})} is the p-adic closure of J({\mathbf Q}) in J({\mathbf Q}_p) for some prime number p. Under the assumption that r < g, Chabauty proved that this strategy actually works! Coleman made this argument effective, proving the following important result:

Theorem (Coleman): Let X / {\mathbf Q} be an algebraic curve of genus at least 2. Assume that r := {\rm rank} J({\mathbf Q}) < g.  Let p>2g be a prime at which X has good reduction. Then \# X({\mathbf Q}) \leq \# \bar{X}({\mathbf F}_p) + 2g-2.

This result was extended to curves with bad reduction at p by Lorenzini and Tucker, following a suggestion of McCallum (when p > 2g, see the remarks before 1.11 in Lorenzini-Tucker).  A new proof of this bound was subsequently given by McCallum and Poonen. In this refinement, \bar{X}({\mathbf F}_p) is replaced by the set \bar{\mathfrak X}^{\rm sm}({\mathbf F}_p) of smooth points on the special fiber of the minimal proper regular model for X over {\mathbf Z}_p. The strategy of proof is similar to Coleman’s original approach, but there are some extra technical details involving relative dualizing sheaves.

Michael Stoll proved that if r < g-1 in the setting of Coleman’s original theorem then one can do even better: the 2g-2 appearing in Coleman’s result can be replaced by 2r.   This was originally a question of Lorenzini-Tucker (see the remarks after 1.11 in their paper).

The “fiber product” of these ideas was worked out recently in an elegant paper by Eric Katz and David Zureick-Brown. They prove the following definitive version of the Chabauty-Coleman bound:

Theorem (Katz, Zureick-Brown): Let X / {\mathbf Q} be an algebraic curve of genus at least 2. Assume that r := {\rm rank} J({\mathbf Q}) < g.  Let p>2g be a prime number, and let \mathfrak X be a proper regular model for X over {\mathbf Z}_p.  Then \# X({\mathbf Q}) \leq \# \bar{\mathfrak X}^{\rm sm}({\mathbf F}_p) + 2r.

Katz and Zureick-Brown give an example in their paper where r < g-1, the above bound is sharp, and no previously known bound is sharp.

Interestingly, the proof of the theorem of Katz and Zureick-Brown relies in a crucial way on the Riemann-Roch theorem for graphs which Norine and I proved in 2007. In order to explain how Riemann-Roch for graphs enters the picture, I will first summarize the combined ideas of Chabauty, Coleman, Lorenzini-Tucker, and McCallum-Poonen. (Here I will follow closely the wonderfully lucid article by McCallum and Poonen).

Let X / {\mathbf Q} be an algebraic curve and let J be its Jacobian. There is a canonical bilinear pairing J({\mathbf Q}_p) \times H^0(J_{{\mathbf Q}_p}, \Omega^1) \to {\mathbf Q}_p, denoted \langle Q, \omega_J \rangle \mapsto \int^Q_0 \omega_J, which is uniquely characterized by the following two properties:

(1) For fixed \omega_J, the map \eta_J : J({\mathbf Q}_p) \to {\mathbf Q}_p given by Q \mapsto \int^Q_0 \omega_J is a group homomorphism.
(2) On some open subgroup U of J({\mathbf Q}_p), one can compute \int^Q_0 \omega_J for Q \in U by formally integrating power series in suitable local coordinates.

Now for Q,Q' \in X({\mathbf Q}_p) and \omega_X \in H^0(X_{{\mathbf Q}_p},\Omega^1), we define \int^{Q'}_Q \omega_X := \int^{[Q'-Q]}_0 \omega_J, where \omega_J is the differential corresponding to \omega_X under the canonical isomorphism between H^0(J_{{\mathbf Q}_p}, \Omega^1) and H^0(X_{{\mathbf Q}_p}, \Omega^1).

If Q,Q' \in X({\mathbf Q}_p) have the same reduction in some proper regular model {\mathfrak{X}} for X, then \int^{Q'}_Q \omega_X can be calculated by expanding \omega_X in a power series with respect to a local parameter on X and formally integrating.

One deduces from this formalism that if we fix P \in X({\mathbf Q}) and define

V_{\rm chab} := \{ \omega \in H^0(X_{{\mathbf Q}_p}, \Omega^1) \; | \; \int^Q_P \omega = 0 {\rm \; for \; all \;} Q \in X({\mathbf Q}) \},

then {\rm dim} V_{\rm chab} \geq g-r. In particular, if r < g then {\rm dim} V_{\rm chab} > 0.

Fix a nonzero differential \omega \in V_{\rm chab}.  Since {\mathfrak{X}} is a regular model, each point of X({\mathbf Q}) reduces to a point of \bar{\mathfrak X}^{\rm sm}({\mathbf F}_p).  The set of points of X({\mathbf Q}_p) reducing to a given point of \bar{\mathfrak X}^{\rm sm}({\mathbf F}_p) (which one calls a (nonsingular) residue class) naturally has the structure of an open p-adic disc.

Fix a residue class D and a point P \in D, and let T be a local parameter on X at P.
On the disc D, \int_P \omega can be expressed as a power series w(T) \in {\mathbf Q}_p[[T]] by
integrating a formal power series expansion \omega(T) for \omega relative to the local parameter T term-by-term (with constant term 0).  Assuming that p > 2g, Coleman observed that the Newton polygon of w(T)/T coincides with the Newton polygon of \omega(T); the point is that the denominators which crop up when one integrates \omega(T) do not change the shape of the Newton polygon as long as p is large enough.

By assumption, if P \in X({\mathbf Q}) then \int^Q_P \omega = 0 for all Q \in X({\mathbf Q}).  Accordingly, we can bound the number of rational points on X by partitioning X({\mathbf Q}) into residue classes and bounding the number of zeros of w(T) on each on each residue class. The Newton polygon argument above, together with Hensel’s Lemma, shows that if D is the residue class of \bar{Q} \in \bar{\mathfrak X}^{\rm sm}({\mathbf F}_p) then

\# \left( D \cap X({\mathbf Q}) \right) \leq 1 + {\rm ord}_{\bar{Q}}(\bar{\omega}),

where \bar{\omega} is obtained by reducing \omega (suitably rescaled so that its reduction is well-defined and nonzero) to the component containing \bar{Q}.

Summing over all residue classes and using the fact that \sum_{\bar{Q} \in \bar{\mathfrak X}^{\rm sm}({\mathbf F}_p)} {\rm ord}_{\bar{Q}}(\bar{\omega}) \leq 2g-2 (which follows from the Riemann-Roch theorem when \bar{\mathfrak X} is smooth), we obtain the Lorenzini-Tucker and McCallum-Poonen bound \# X({\mathbf Q}) \leq \# \bar{\mathfrak X}^{\rm sm}({\mathbf F}_p) + 2g-2.  (Coleman’s original bound was the special case of this inequality when X has good reduction at p.)

Michael Stoll noticed that when X has good reduction at p, one can improve 2g-2 to 2r in Coleman’s original bound by a suitable use of Clifford’s inequality. The key idea is that instead of choosing just one differential \omega, as we did in the argument above, we should choose the best possible differential for each residue class. Doing so and following the above chain of reasoning gives us the bound \# X({\mathbf Q}) \leq \sum_{\bar{Q} \in \bar{{\mathfrak{X}}}^{\rm sm}({\mathbf F}_p)} \left( 1 + n_{\bar{Q}} \right), where n_{\bar{Q}}= \min_{\omega \in V_{\rm chab}} {\rm ord}_{\bar{Q}}(\bar{\omega}).

Let D_{\rm chab} = \sum_{\bar{Q} \in \bar{{\mathfrak{X}}}^{\rm sm}({\mathbf F}_p)} n_{\bar{Q}} (\bar{Q}) \in {\rm Div}(\bar{{\mathfrak{X}}}) and set d = {\rm deg}(D_{\rm chab}).  Then D_{\rm chab} and K_{\bar X} - D_{\rm chab} are both linearly equivalent to effective divisors, so Clifford’s inequality implies that r(K_{\bar X} - D_{\rm chab}) = h^0(K_{\bar X} - D_{\rm chab}) - 1 \leq \frac{1}{2} (2g-2-d).  On the other hand, by the semicontinuity of h^0, we have h^0(K_{\bar X} - D_{\rm chab}) \geq {\rm dim} V_{\rm chab} \geq g-r.  Combining these two inequalities gives g-r-1 \leq \frac{1}{2}(2g-2-d) and therefore d \leq 2r as desired.  We have thus proved:

Theorem (Stoll): Let X / {\mathbf Q} be an algebraic curve of genus at least 2. Assume that r := {\rm rank} J({\mathbf Q}) < g.  Let p>2g be a prime at which X has good reduction. Then \# X({\mathbf Q}) \leq \# \bar{X}({\mathbf F}_p) + 2r.

Now, what to do if X has bad reduction at p?  Well, first of all, Katz and Zureick-Brown prove a lemma to the effect that one can reduce to the case where \bar {\mathfrak{X}} is semistable; the point is that extending scalars only makes the degree of D_{\rm chab} bigger.  However, it is well-known that Clifford’s inequality fails in general for singular curves, even semistable ones, so another idea is needed.  In May 2011, I suggested to David Zureick-Brown that one could potentially use Clifford’s inequality for graphs (proved in my paper with Norine) to carry out Stoll’s idea in the case of totally degenerate semistable reduction.  It turns out that this works!  We will explain the idea, and then briefly indicate how Katz and Zureick-Brown handle the more general case of mixed bad reduction.

Assume for the moment that {\mathfrak{X}} is a proper regular semistable model for X which is totally degenerate, i.e., every irreducible component of \bar {\mathfrak{X}} is a smooth rational curve.  Equivalently, if G is the dual graph of \bar {\mathfrak{X}} (whose vertices correspond to the irreducible components and edges correspond to crossings between these components), then X is totally degenerate if and only if the genus of G equals the genus of X.

There is a divisor \bar{D}_{\rm chab} on G which records which irreducible component of \bar {\mathfrak{X}} a given point lies on.  Clifford’s theorem for graphs implies that r(K_G - \bar{D}_{\rm chab}) \leq \frac{1}{2}(2g-2-d), and a variant of the Specialization Lemma from this paper (the analogue in this context of semicontinuity) implies that r(\bar{D}_{\rm chab}) \geq \dim V_{\rm chab} - 1 \geq g-r-1.  Combining these inequalities yields d \leq 2r as before.

The general case, where {\mathfrak{X}} is a proper regular semistable model for X which is not assumed to be totally degenerate, follows similarly by replacing Clifford’s inequality for graphs with an inequality which can be deduced formally from the Riemann-Roch theorem for metrized complexes of curves which Amini and I recently proved. (Alternatively, one can use the earlier Riemann-Roch theorem for vertex-weighted graphs due to Amini and Caporaso.)  One also need a suitable extension of the Specialization Lemma to this context, c.f. Theorem 5.11 in Amini-Baker.  For more details, see the paper of Katz and Zureick-Brown, Section 5.5 in my paper with Amini, or these slides by David Zureick-Brown.

Concluding remarks:

1. The p-adic integration pairing on X discussed above differs in the bad reduction case from more general theories of p-adic integration, e.g. the theory constructed recently by Berkovich.  Indeed, the latter is multi-valued and has non-trivial periods while the above pairing is uniquely defined.  However, the pairing discussed here (which uses the Jacobian of X to kill off polyvalency) seems to be better suited at the present time for computations and for applications of the above sort.

2. The Riemann-Roch theorem for metrized complexes of curves alluded to above includes as special cases both Riemann-Roch for graphs and Riemann-Roch for algebraic curves.  I plan to write a future blog post explaining some other applications of the theory of metrized complexes of curves, which include a generalization of the Eisenbud-Harris theory of limit linear series to semistable curves not of compact type.

3. The Chabauty-Coleman bounds discussed in this post all have analogues for number fields other than {\mathbf Q}, and (in a slightly messier form) for primes which are less than 2g.

4. Michael Stoll has an exciting recent preprint in which he uses singular residue classes (which look p-adic analytically like open annuli rather than open discs) on semistable curves to refine the method of Chabauty-Coleman, showing that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g-3.  (If K = Q, an explicit bound is 8 (r + 4) (g - 1) + {\rm max} \{1, 4r \} g.)  Integrals of holomorphic 1-forms on annuli generally include a p-adic logarithm term (coming from the integral of dT/T) which is hard to control, and there is also a hard-to-control error term coming from comparing the two different kinds of integrals mentioned in Remark 1 above, but Stoll makes the clever observation that if r \leq g-3 then one can use three independent 1-forms in the space V_{\rm chab} to cancel out these two kinds of problem terms.

5. One can also apply the Chabauty-Coleman method to symmetric powers of curves, obtaining bounds for the number of rational points on an algebraic curve X / {\mathbf Q} which are defined over a number field of degree at most d.  See for example this paper by Samir Siksek, as well as forthcoming work of Bjorn Poonen’s Ph.D. student Jennifer Park (which makes crucial use of my Georgia Tech colleague Joe Rabinoff‘s thesis work on tropical analytic geometry).  I have a particular fondness for this topic; when I participated in the Arizona Winter School as a graduate student in 1999, our student group (which included Manjul Bhargava) computed all the quadratic points on a specific genus 3 hyperelliptic curve.  And my first published paper applied the Coleman-Chabauty method to symmetric powers of a modular curve.

6. Bjorn Poonen and Michael Stoll have a recent preprint in which they prove that for g \geq 3, a positive fraction of hyperelliptic curves of odd degree 2g+1 over {\mathbf Q} have only one rational point, the point at infinity. They also prove a lower bound on this fraction that tends to 1 as the genus tends to infinity.  Their method combines a refinement of the Chabauty-Coleman method (based on an old idea of McCallum) with the Bhargava-Gross equidistribution theorem for nonzero 2-Selmer group elements.

One thought on “Effective Chabauty

  1. A few comments that came up in Banff, but that didn’t make it to my slides:

    The rank < genus condition is restrictive, but not too restrictive — one expects that the average rank is 1/2 (i.e. 100% of the time the rank is either 0 or 1). The rank is 1 infinitely often as well. (E.g. Jac X could certainly be a product of rank 1 elliptic curves, or X could have a large number of points.) There is some progress: Bhargava and Gross prove that the average rank is at most 1.5 (http://www.math.harvard.edu/~gross/preprints/stable23.pdf).

    In practice too, when a collection of curves arises in the solution to some problem, almost all, but not usually literally all, of them have rank < g. For instance, Smart gave a list of genus 2 curves with good reduction outside of two. There are around 400 of these; all have rank at most 2, and only a few have rank equal to 2.

    Its also unclear whether ranks are unbounded. The record for elliptic curves is 28; I'm not sure what the record is for higher genus curves, but Michael Stoll found a genus 2 curve with simple Jacobian of rank 20.


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