- 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"

- 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

- 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)

- 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

- Linear or quadratic equations
- Rational solutions are found easily

- \(E\) 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.

`Mordell's conjecture (Faltings 1983)`

: If \(g(C)>1\) then \(C(\mathbb{Q})\) is finite.- Drawbacks:
- Bound is not effective
- Bound is on the size of solutions. Could be huge but only a few solutions.

- 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.

- \(J\) 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.

- \(J(\mathbb{C})\) 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 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 \(\mathbb{Z}/p^k\)

- \(J(\mathbb{Q}_p)\) 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 \(0\).

- 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.

- 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 (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 \(p+1+2g\sqrt{p} +2g-2\)

- 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".

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

`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 \(T\).

- \(E/\mathbb{Q}\) 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 \(F\).

- 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.

- 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.

- 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?

- 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

Here is an example due to Jen and Steffen