Classical Chabauty
\(C\) be a smooth algebraic curve over \(\mathbb{Q}\) with genus \(g\ge 2\). Let \(J\) be the Jacobian of \(C\) which is a principally polarised abelian variety over \(\mathbb{Q}\) of dimension \(g\). This Jacobian \(J\) is built upon \(\mathrm{Pic}^0(C)\), the group of degree zero divisors on \(C\) modulo principal divisors.
Foundations
Embed Into Jacobian. For any rational point \(b\in C(\mathbb{Q})\) we can fix an embedding
\[\iota_b : C \to J, \quad P\mapsto [P-b]\]
where \([P-b]\) denotes the class of the divisor \(P-b\) in the Jacobian \(J\).
Good Reduction. If \(p\) is a prime at which \(C\) has good reduction, which requires there exists a smooth projective curve \(C_p/\mathbb{Z}_{(p)}\) such that \(C_p\otimes \mathbb{Q}\cong C\), we can fix the model up to \(\mathbb{Z}_{(p)}\)-isomorphism.
Reduction Map. Let \(p\) be good reduction and \(C\) is proper (or consider projective curves to get the idea), every point \(P:\mathrm{Spec}(\mathbb{Q}_p)\to C\) extends to a unique morphism \(\mathrm{Spec}(\mathbb{Z}_p)\to C\), from which we can get a \(\mathbb{F}_p\)-point on \(C_p\), \(\mathrm{Spec}(\mathbb{F}_p)\to \mathrm{Spec}(\mathbb{Z}_p)\to C\). This defines a reduction mod \(p\) map
\[r_p : C(\mathbb{Q}_p) \to C(\mathbb{F}_p), \quad P\mapsto \overline{P}\]
which is a group homomorphism.
Residue disks. For any point in reduction \(x\in C(\mathbb{F}_p)\), we can associate a \(p\)-adic residue disk \(D_x\subset C(\mathbb{Q}_p)\), which is the inverse image \(r_p^{-1}(x)\) under the reduction map \(r_p:C(\mathbb{Q}_p)\to C(\mathbb{F}_p)\).
Take a trivial example \(C=\mathbb{P}^1=\{[x:y]\}\), we can represent
\[\mathbb{P}^1(\mathbb{F}_p)=\mathbb{F}_p\cup\{\infty\}\]
\[\begin{align*} \mathbb{P}^1(\mathbb{Q}_p) &=\{[x:y]\in\mathbb{P}^1(\mathbb{Z}_p)\mid \text{at least one of $x,y$ is in $\mathbb{Z}_p^\times$}\} \\ &=\{[1:y']\mid y'\in \mathbb{Z}_p\}\sqcup \{[x':1]\mid x'\in p\mathbb{Z}_p\} \end{align*}\]
upon such representation, the reduction map is
\[r_p : \mathbb{P}^1(\mathbb{Q}_p)\to \mathbb{P}^1(\mathbb{F}_p), \quad [x:y]\mapsto [\overline{x}:\overline{y}].\]
We see that the residue disks are represented by
\[D_a=\{[1:y']\mid y'\in a+p\mathbb{Z}_p\}, \quad D_\infty=\{[x':1]\mid x'\in p\mathbb{Z}_p\}.\]
Each of which is ‘isomorphic’ to \(p\mathbb{Z}_p\).
Idea Of Chabauty
We can embed rational points \(C(\mathbb{Q})\) into \(\mathbb{Q}_p\) points \(C(\mathbb{Q}_p)\), this works for Jacobian as well. If we can say something for the \(\mathbb{Q}_p\) points, we can say something for the rational points. The diagram is
This naturally means \(C(\mathbb{Q})\hookrightarrow C(\mathbb{Q}_p)\times_{J(\mathbb{Q}_p)}J(\mathbb{Q})\), which is the set theoretic intersection \(\iota(C(\mathbb{Q}_p))\cap J(\mathbb{Q})\) inside \(J(\mathbb{Q}_p)\). Here some interesting things happen:
\(J(\mathbb{Q}_p)\) is \(p\)-adic analytic and is expected to be of dimension \(g\), just like \(J(\mathbb{C})\). In fact by \(p\)-adic Lie group theory, it should be of the form
\[J(\mathbb{Q}_p)\cong \mathbb{Z}_p^g\oplus \{\text{finte abelian group}\}.\]
\(J(\mathbb{Q})\) has rank \(r\) that might be different from \(g\). If \(r<g\), the image is expected to be ‘sparse’ in \(J(\mathbb{Q}_p)\), contained in a \(p\)-adic \(r\)-dimensional subset: choose \(r\) generators \(P_1,\dots,P_r\) of \(J(\mathbb{Q})\)
\[\bigoplus_{i=1}^r \mathbb{Z}P_i \subset \bigoplus_{i=1}^r \mathbb{Z}_p P_i \subset J(\mathbb{Q}_p).\]
We see that it is obviously contained in a \(p\)-adic manifold of dimension \(r\). And there exists at least one non-trivial linear functional \(l:J(\mathbb{Q}_p)\to \mathbb{Q}_p\) that vanishes on the image.
\(\iota(C(\mathbb{Q}_p))\) is a \(p\)-adic curve and should be thought as \(1\)-dimensional.
This means, if \(r<g\), the intersection is unlikely event and should be finite! Thus producing a bound on the number of rational points.
Concretely, the argument goes as follows
Theorem (Chabauty). If \(r<g\), then \(C(\mathbb{Q})\) is finite.
Proof.
Since \(r<g\), we can take a non-trivial linear functional \(l: J(\mathbb{Q}_p)\to \mathbb{Q}_p\) (a group homomorphism) that vanishes on the image of \(J(\mathbb{Q})\).
\[l[J(\mathbb{Q})]=0.\]
Pullback \(\iota_p^*l : C(\mathbb{Q}_p)\to \mathbb{Q}_p\) we get a function (transcendental!) that vanishes on \(J(\mathbb{Q})\cap \iota_p [C(\mathbb{Q}_p)]\), therefore also vanishes on \(C(\mathbb{Q})\).
\[\iota_p^*l[C(\mathbb{Q})]=0.\]
On each residue disk \(C_x\subset C(\mathbb{Q}_p)\) for \(x\in C(\mathbb{F}_p)\), the function \(\iota_p^*l\) can be expanded as a \(p\)-adic power series. The \(p\)-adic logarithm map
\[\log : J(\mathbb{Q}_p)\to T_0J(\mathbb{Q}_p) = \mathbb{Q}_p^g\]
has the property that \(\log\circ \iota_p : C(\mathbb{Q}_p)\to \mathbb{Q}_p^g\) is transcendental on each disk, therefore the algebraic image \(\iota_p[C(\mathbb{Q}_p)]\) is not contained in any hyperplane of \(J(\mathbb{Q}_p)\), therefore \(\iota_p^*l\) is not identically zero on any residue disk.
A convergent \(p\)-adic power series on a residue disk \(C_x\cong D(x,1/p)\) necessarily have finite many zeros in \(C_x\) (just like a holomorphic function on a closed disk in \(\mathbb{C}\)). Therefore the set of rational points \(C(\mathbb{Q})\) is finite, and if we can bound the number of zeros on each disk, we can bound the number of rational points.
\[\#C(\mathbb{Q})\le \sum_{x\in C(\mathbb{F}_p)} \#\left\{\text{zeros of $\iota_p^*l$ on $C_x$}\right\}.\]
Random Thoughts.
Is there a general intersection theory we can use, instead of pulling back to curves?
Can we use different primes at the same time? Does that make things stronger?
Quadratic Chabauty
If we go further in the diagram,
Here
\(\kappa,\kappa_p\) are Kummer maps, which are injective and compatible with the Galois action.
\(T_pJ\) is the Tate module and we can denote \(V_pJ=T_pJ\otimes_{\mathbb{Z}_p}\mathbb{Q}_p\), you can call it the Tate vector space.
We want to replace \(V_pJ\) by some non-abelian algebraic group, pick any unipotent group \(U\) over \(\mathbb{Q}_p\) endowed with Galois action
\[G_T=\mathrm{Gal}(\mathbb{Q}^{\mathrm{ur}(\{p\}^c)}/\mathbb{Q})\mathrel{\circlearrowright}U(\mathbb{Q}_p).\]
and a surjection \(U\to V_pJ=(\mathbb{G}_a)^g\) that is an algebraic group compatible with the Galois action.
We have a similar diagram with the Jacobian stuff replaced by \({\mathrm{Sel}}(U)\xrightarrow{\mathrm{loc}_p} H^1(G_{\mathbb{Q}_p},U(\mathbb{Q}_p))\),
Kim showed that \({\mathrm{Sel}}(U)\) and \(H^1(G_{\mathbb{Q}_p},U)\) is the set of \(\mathbb{Q}_p\) points of some affine schemes of finite type over \(\mathbb{Q}_p\) and \(\mathrm{loc}_p\) is an algebraic map.
If the following magic holds
\(\log_p\) is not dominant, i.e. it is rare or non-dense. This can be satisfied if you have
\[\dim {\mathrm{Sel}}(U) < \dim H^1(G_{\mathbb{Q}_p},U),\]
this is called the quadratic Chabauty condition. It can be expressed as
\[ r < g + \rho - 1\]
where \(\rho = \mathrm{rank}\,\mathrm{NS}(J)\) is the Néron-Severi rank of the Jacobian.
\(\kappa_p\) is transcendental.
Then we could expect that the intersection to be finite! OwO