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

• Linear or quadratic equations
• 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 \(T\).

p-adic height pairings

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