Author: Eiko

Time: 2025-02-23 09:45:10 - 2025-02-23 09:45:10 (UTC)

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

  $${\iota }_b:C\to J,P\mapsto [P-b]$$

  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

  $$r_p:C({\mathbb{Q}}_p)\to C({\mathbb{F}}_p),P\mapsto \bar{P}$$

  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

  $${\mathbb{P}}^1({\mathbb{F}}_p)={\mathbb{F}}_p\cup \{\mathrm{\infty }\}$$

  $$\begin{array}{rl}{\mathbb{P}}^1({\mathbb{Q}}_p)& =\{[x:y]\in {\mathbb{P}}^1({\mathbb{Z}}_p)\mid \text{at least one of }x,y\text{ is in }{\mathbb{Z}}_p^{\times }\}\\ & =\{[1:y^{\prime }]\mid y^{\prime }\in {\mathbb{Z}}_p\}\bigsqcup \{[x^{\prime }:1]\mid x^{\prime }\in p{\mathbb{Z}}_p\}\end{array}$$

  upon such representation, the reduction map is

  $$r_p:{\mathbb{P}}^1({\mathbb{Q}}_p)\to {\mathbb{P}}^1({\mathbb{F}}_p),[x:y]\mapsto [\bar{x}:\bar{y}].$$

  We see that the residue disks are represented by

  $$D_a=\{[1:y^{\prime }]\mid y^{\prime }\in a+p{\mathbb{Z}}_p\},D_{\mathrm{\infty }}=\{[x^{\prime }:1]\mid x^{\prime }\in p{\mathbb{Z}}_p\}.$$

  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 $J(\mathbb{Q})$

  $$\underset{i=1}{\overset{r}{⨁}}\mathbb{Z}{P}_i\subset \underset{i=1}{\overset{r}{⨁}}{\mathbb{Z}}_p{P}_i\subset J({\mathbb{Q}}_p).$$

  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.

1. 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 $J(\mathbb{Q})$.

   $$l[J(\mathbb{Q})]=0.$$

2. Pullback ${\iota }_p^{\ast }l:C({\mathbb{Q}}_p)\to {\mathbb{Q}}_p$ we get a function (transcendental!) that vanishes on $J(\mathbb{Q})\cap {\iota }_p[C({\mathbb{Q}}_p)]$, therefore also vanishes on $C(\mathbb{Q})$.

   $${\iota }_p^{\ast }l[C(\mathbb{Q})]=0.$$

3. On each residue disk $C_x\subset C({\mathbb{Q}}_p)$ for $x\in C({\mathbb{F}}_p)$, the function ${\iota }_p^{\ast }l$ can be expanded as a $p$-adic power series. The $p$-adic logarithm map

   $$\log :J({\mathbb{Q}}_p)\to T_{0}J({\mathbb{Q}}_p)={\mathbb{Q}}_p^g$$

   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 ${\iota }_p^{\ast }l$ is not identically zero on any residue disk.

4. 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.

   $$\mathrm{\#}C(\mathbb{Q})\le \sum _{x\in C({\mathbb{F}}_p)}\mathrm{\#}\left\{\text{zeros of }{\iota }_p^{\ast }l\text{ on }{C}_x\right\}.$$

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.

• $T_{p}J$ is the Tate module and we can denote $V_{p}J=T_{p}J{\otimes }_{{\mathbb{Z}}_p}{\mathbb{Q}}_p$, you can call it the Tate vector space.

• We want to replace $V_{p}J$ by some non-abelian algebraic group, pick any unipotent group $U$ over ${\mathbb{Q}}_p$ endowed with Galois action

  $$G_T=\mathrm{Gal}({\mathbb{Q}}^{\mathrm{ur}(\{p{\}}^c)}/\mathbb{Q})\circlearrowright U({\mathbb{Q}}_p).$$

  and a surjection $U\to V_{p}J=({\mathbb{G}}_a{)}^g$ that is an algebraic group compatible with the Galois action.

• We have a similar diagram with the Jacobian stuff replaced by $\mathrm{Sel}(U)\stackrel{{\mathrm{loc}}_p}{\to }{H}^1(G_{{\mathbb{Q}}_p},U({\mathbb{Q}}_p))$,

  

• 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

Technical Details