# Integral points on curves

## Summary

• We describe a new method for finding integral solutions to some equations in 2 variables
• The method finds a $p$-adic function taking a prescribed finite set of values on these points.
• Uses the quadraticity of the $p$-adic height pairing (Quadratic Chabauty)
• Uses existing techniques to rule out "false solutions"

## Overview

• What are diophantine equations?
• Rational points on curves = 1-dimensional equations: The genus trichotomy
• In comes $p$-adic numbers: Chabauty's method.
• Height pairings and Quadratic Chabauty.
• Mordell-Weil Sieve
• Examples

## Diophantine equations

• Finding solutions to polynomial equations in several variables in rationals or integers
• One of the oldest mathematical problems
• Famous example: Fermat's last Theorem $x^n+y^n=z^n$.
• Can't hope to do too well: Matiyasevich's theorem (even in 9 variables)

## Rational points on curve

• Curves $\sim$ 1 equation in 2 variables. E.g., $C:y^2=x^5+x+2$
• $C(L)=$ set of solutions of this equation with coordinates in the field $L$.
• genus $g(C)=$ number of "holes" in the surface of complex solutions

### Genus $0$ case

• Rational solutions are found easily

### Genus 1 case = Elliptic curves

• $E$ elliptic curve over a field $L$
• Typical examples: $y^2=x^3+ax+b$, $a,b\in L$
• $E(L)$ is an abelian group
• Mordell Weil Theorem: For $E/\mathbb{Q}$, $E(\mathbb{Q})$ is of finite rank.
• Finding generators is not trivial.
• Birch and Swinnerton-Dyer conjecture: Relates the rank to some analytic data.
• Arithmetic Statistics (Bhargava) - On average most elliptic curves have small rank.

### Genus $>1$

• Mordell's conjecture (Faltings 1983) : If $g(C)>1$ then $C(\mathbb{Q})$ is finite.
• Drawbacks:
1. Bound is not effective
2. Bound is on the size of solutions. Could be huge but only a few solutions.

## Chabauty's method

• In 1941 Chabauty proved finiteness of $C(\mathbb{Q})$ in some cases.
• His method was made effective, with really good bounds, by Coleman.
• Lots of numerical work using this method.

### The Jacobian variety $J$ of $C$

• $J$ is an abelian variety, $J(F)$ is an abelian group for any field $F$.
• $J(F)=$ divisors of degree $0$ on $C$, $\sum n_i (P_i)$ defined over $F$, modulo divisors of rational functions on $C$ defined over $F$.
• If $P_0\in C(\mathbb{Q})$ we have a map $C\to J$ given by $P\mapsto (P)-(P_0)$.
• $J$ has dimension $g$, typically many equations in many variables.

### The Jacobian over the complex numbers

• $J(\mathbb{C})$ is a complex torus $\mathbb{C}^g/L$ where $L$ has rank $2g$.
• More canonically
• $\mathbb{C}^g$ is the dual of $\Omega^1(C)$, $L$ is the image of $H_1(C,\mathbb{Z})$
• $C\to J$ is given by $P\mapsto \int_{P_0}^P$, path indeterminacy contained in $L$.
• The exponential map $\mathbb{C}^g \to J(\mathbb{C})$ is surjective.
• Integration of $1$ forms along a path only interacts with the homology. To get a handle on the fundamental group one needs Chen's iterated integrals

$\int_{P_0}^P \omega_1\circ \omega_2 \circ \cdots$

### The $p$-adic numbers

• The ring $\mathbb{Z}_p$ and the field $\mathbb{Q}_p$ are the completions of $\mathbb{Z}$ and $\mathbb{Q}$ with respect to the absolute value

$|p^n a/b|= p^{-n}$, $p$ prime to $a$ and $b$.

• $p$-adic integers to a finite precision $k$ are just elements of $\mathbb{Z}/p^k$

### The logarithm map

• $J(\mathbb{Q}_p)$ is a $p$-adic Lie group
• Its Lie algebra $\mathbb{L}$ has dimension $g$.
• There is a logarithm homomorphism $\log: J(\mathbb{Q}_p) \to \mathbb{L}$
• Near $0$ it is an inverse to the exponential map
• $\log(Q) = \log(nQ)/n$ with $nQ$ near $0$.

### Chabauty's theorem

• Mordell-Weil implies $J(\mathbb{Q})$ is a finitely generated abelian group.
• Let $r=$ rank of $J(\mathbb{Q})$.
• Chabauty's Theorem : If $g>r$ then $C(\mathbb{Q})$ is finite.

### Proof of Chabauty's theorem

• Let $Q_1,\ldots, Q_r$ be a basis of $J(\mathbb{Q})$.
• By dimensions $\exists$ a functional $\alpha$ on $\mathbb{L}$ vanishing on $\log(Q_1),\ldots,\log(Q_r)$.
• so $\alpha\circ\log$ vanishes on $J(\mathbb{Q})$.
• In particular, $\ell(P):= \alpha(\log((P)-(P_0)))$ vanishes on $C(\mathbb{Q})$.
• $\ell$ is locally analytic, hence has a finite number of zeros.

### Coleman's contribution

• Coleman (1985) interpreted $\alpha$ as a holomorphic form $\omega$ on $C$ and $\ell=\ell_\omega$ as the $p$-adic Coleman integral $\int_{P_0}^P \omega$.
• As a function of $P$ it is a primitive of $\omega$.
• Theorem (Coleman) : If $p>2g$ is a prime of good reduction and $r \lt g$, then the number of points in $C(\mathbb{Q})$ is at most $p+1+2g\sqrt{p} +2g-2$

### Kim's non-abelian Chabauty

• If $r\ge g$ Chabauty fails completely.
• Kim: Replace Coleman integrals with Coleman iterated integrals

• e.g.,

$\int (\omega \times \int \eta)$

• Each integral involves a choice of constant of integration.
• Coleman integrals are locally analytic.
• Computable in many cases (Balakrishnan, de Jeu and Escriva).
• $J$ is replaced by a "Selmer variety".

## Statement of main results

### Hyperelliptic curve $C$

• Easy equation $y^2=Q(x)$.
• More general (required for minimality) $y^2+R(x)y= Q(x)\;,\; 2\deg R\le \deg Q$
• $\deg Q = 2g+1$

### The main Theorem

Theorem : Let $C$ be a hyperelliptic curve $y^2=Q(x)$, $Q$ with integral coefficients. Suppose that Chabauty's method does not apply to $C$ and that $r=g$. Then there exists explicit Coleman integral $I$ and a finite set of values $T$ such that $I$ obtains on each integral point of $C$ a value in $T$.

### Classical height pairings on elliptic curves

• $E/\mathbb{Q}$ elliptic, $P=(x,y)\in E(\mathbb{Q})$
• $H(P) = H(x)$, $H(a/b)$ with $a,b$ coprime is $\max(|a|,|b|)$.
• $h=\log(H)$ is quadratic up to a bounded function.
• Tate: Canonical height $\hat{h}(P) = \lim_{n\to \infty} \frac{h(2^n P)}{4^n}$
• $\hat{h}$ quadratic $\to \langle~,~\rangle : E(\mathbb{Q})\times E(\mathbb{Q}) \to \mathbb{R}$.
• Birch-Swinnerton-Dyer conjecture:

\begin{equation*} L^*(E,1) = \frac{\Omega_E\cdot \operatorname{Reg}_E\cdot \prod c_p\cdot |\text{Sha}(E)|}{|E(\mathbb{Q})_{tor}|^2 } \end{equation*}

where $\operatorname{Reg}_E$ is the determinant of the height pairing.

• Extend to Abelian varieties $A$ over a number field $F$.

### $p$-adic height pairings on curves - Coleman Gross (1985)

• Needs the following:
• $C/F$ smooth complete curve, good reduction above $p$.
• Some auxiliary data.
• $x,y\in Div_0(C)$ with disjoint supports: $h(x,y) = \sum_{v<\infty} h_v(x,y)\in \mathbb{Q}_p$
• Local $h_v$ extend to $x,y\in Div_0(C_v)$
• Individual $h_v$ don't factor via $J$, but $h$ does.

### Proof of the main theorem

• Either Chabauty's method applies, or the functionals $\psi_i=\alpha_{\omega_i}\circ \log$ form a basis to $\operatorname{Hom}(J(\mathbb{Q}),\mathbb{Q}_p)$.
• $\psi_i\cdot \psi_j$ form a basis to the vector space of $\mathbb{Q}_p$ valued quadratic forms on $J(\mathbb{Q})$.
• $h$ is also quadratic, so we have $a_{ij}$ with $h+\sum a_{ij} \psi_i\cdot\psi_j=0$
• For $P\in C(\mathbb{Q})$ this means $h((P)-(0),(P)-(0))+\sum a_{ij} \int_0^P \omega_i \int_0^P \omega_j=0$.
• $h((P)-(0),(P)-(0))-H(P) = \sum_{q\ne p} h_q((P)-(0),(P)-(0))$ with $H(P)=h_p((P)-(0),(P)-(0))$
• If $C$ has good reduction at $q$, $h_q=0$ by integrality.
• Otherwise, depends on which component of special fiber $P$ hits.

## The Mordell-Weil Sieve

### The problem: eliminating "false positives"

• When solving $I(P)\in T$, many solutions will be "false positive"
• A true solution is easily recognized: solves the equation
• A false solution does not solve the equation in reasonably sized integers
• How to prove it is really false?

### The Mordell-Weil Sieve

• A standard technique for proving non-existence of solutions, adapted to the problem
• Suppose $Q_1,\ldots Q_g$ a basis for $J(\mathbb{Q})$, $(P)-(P_0) = \sum n_i Q_i$, assuming $P$ is a true solution
• Our data gives a good $p$-adic approximation to the $n_i$
• For an auxiliary prime $l$ s.t. $p$ divides the order of $J(\mathbb{Z}/l)$ this limits the location of $(P)-(P_0)$ mod $l$.
• If this location does not come from the curve we win

## Examples

Here is an example due to Jen and Steffen