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New Algebraic Points on Curves

Maleeha Khawaja, Samir Siksek

TL;DR

This work investigates how L-new points on a hyperbolic curve behave as L runs through degree-n extensions of the base field, proposing a statistical Diophantine stability conjecture that C(L)_new should be empty for 100% of fields in natural discriminant-ordered families. The authors develop a multi-pronged approach: a single-source reduction that forces most primitive degree-n points to lie in fibers of a degree-n map to P^1, discriminant-density analyses for polynomials defining L_t, and powerful results on cubic fields from Bhargava–Taniguchi–Thorne, complemented by Vojta’s theorem and unit-equation analyses. They verify the conjecture for n = 2 and certain hyperelliptic modular curves X0(N) (N ≠ 37) and for n = 3 on X0(23), X0(29), X0(31), X0(64), plus an analogue for the unit equation. Collectively, the results provide statistically compelling evidence that new points over random degree-n fields are rare, supporting a probabilistic view of Diophantine stability and offering concrete verifications on central families of curves.

Abstract

Let $C$ be a smooth projective absolutely irreducible curve of genus at least 2, defined over the rationals. For a number field $L$, we define the set of $L$-new points on $C$ to be $C(L)_{new} = \{P \in C(L) : \mathbb{Q}(P)=L\}$; this is the set of points on $C$ defined over $L$ but not any strictly smaller field. Let $n$ be at least 2. We conjecture that $C(L)_{new}$ is empty for 100 percent of degree $n$ number fields $L$ when ordered by absolute discriminant. For degrees $n=2$, $3$, we give sufficient criteria for our conjecture to hold in terms of an explicit model for $C$. For general $n$ we prove a theorem that harmonises with the conjecture. In particular, we verify our conjecture for $n=2$ and $C=X_0(N)$ for the $18$ values $N \ne 37$ such that $X_0(N)$ is hyperelliptic, and also for $n=3$ and $C=X_0(23)$, $X_0(29)$, $X_0(31)$, $X_0(64)$. Moreover, we prove the analogue of our conjecture for the unit equation, again with $n=3$.

New Algebraic Points on Curves

TL;DR

This work investigates how L-new points on a hyperbolic curve behave as L runs through degree-n extensions of the base field, proposing a statistical Diophantine stability conjecture that C(L)_new should be empty for 100% of fields in natural discriminant-ordered families. The authors develop a multi-pronged approach: a single-source reduction that forces most primitive degree-n points to lie in fibers of a degree-n map to P^1, discriminant-density analyses for polynomials defining L_t, and powerful results on cubic fields from Bhargava–Taniguchi–Thorne, complemented by Vojta’s theorem and unit-equation analyses. They verify the conjecture for n = 2 and certain hyperelliptic modular curves X0(N) (N ≠ 37) and for n = 3 on X0(23), X0(29), X0(31), X0(64), plus an analogue for the unit equation. Collectively, the results provide statistically compelling evidence that new points over random degree-n fields are rare, supporting a probabilistic view of Diophantine stability and offering concrete verifications on central families of curves.

Abstract

Let be a smooth projective absolutely irreducible curve of genus at least 2, defined over the rationals. For a number field , we define the set of -new points on to be ; this is the set of points on defined over but not any strictly smaller field. Let be at least 2. We conjecture that is empty for 100 percent of degree number fields when ordered by absolute discriminant. For degrees , , we give sufficient criteria for our conjecture to hold in terms of an explicit model for . For general we prove a theorem that harmonises with the conjecture. In particular, we verify our conjecture for and for the values such that is hyperelliptic, and also for and , , , . Moreover, we prove the analogue of our conjecture for the unit equation, again with .

Paper Structure

This paper contains 8 sections, 19 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Counting integers with certain restricted factorisations
  3. Degree $n$ points from a single source
  4. Proofs of Theorems \ref{['thm:hyp']} and \ref{['thm:primitive']}
  5. Proof of Theorem \ref{['thm:cubic']}
  6. Proof of Theorem \ref{['thm:unit']}
  7. Proof of Theorem \ref{['thm:modular']}
  8. Suggested Future Projects

Key Result

Theorem 4

Let $g \in \mathbb{Q}[T]$ be squarefree, of even degree $d \ge 6$, and write $G=\mathop{\mathrm{Gal}}\nolimits(g)$ for the Galois group of $g$. Suppose there is an element $\sigma \in G$ acting freely on the roots of $g$. Let $C/\mathbb{Q}$ be the hyperelliptic curve defined by $Y^2=g(T)$, and write Then $C(L)_{\mathrm{new}}=\emptyset$ for $100\%$ of quadratic fields $L/\mathbb{Q}$, when ordered b

Theorems & Definitions (36)

  • Definition 2
  • Conjecture 2
  • Conjecture 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 26 more