New Algebraic Points on Curves

TL;DR

This paper investigates the rarity of new algebraic points on curves over number fields, proposing a conjecture that such points are almost always absent for most fields of a given degree, and verifies this in specific cases.

Contribution

It introduces a conjecture about the scarcity of new points on curves over number fields and provides partial proofs and criteria for certain degrees and specific modular curves.

Findings

Conjecture that $C(L)_{new}$ is empty for most degree $n$ fields when ordered by discriminant.

Verified the conjecture for quadratic fields and hyperelliptic modular curves $X_0(N)$ with certain $N$.

Confirmed the conjecture for cubic fields and specific $X_0(N)$ curves, and for the unit equation with $n=3$.

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 $n=3$.