New algebraic points on covers of elliptic curves

Abstract

For a smooth projective curve $C/\mathbb{Q}$ of genus $\geq 2$ and $L/\mathbb{Q}$ an extension, we write $C(L)_{\text{new}}=\{P\in C(L):\mathbb{Q}(P)=L\}$. Recent work of Khawaja and Siksek conjectures that this set is empty for $100\%$ of degree $n$ number fields $L$, when ordered by absolute discriminant. Moreover, they bring evidence towards this conjecture when $C$ is a degree $n$ cover of $\mathbb{P}^1$. We complement their work by proving analogous results for degree $n$ covers $ψ:C\to E$ of elliptic curves $E$. Our main result shows that, under suitable hypotheses, the number of distinct absolute discriminants at most $X$ of primitive degree $n$ fields $L$ with $C(L)_{\text{new}}\neq\varnothing$ is $O(X^{1/2})$ or $O(X/(\log X)^α)$, for some $α>0$. In degrees $2,3,4$ and $5$ we show that these fields have density $0$ among all fields of the same degree (in degree $4$, also among the primitive ones). The novelty is for degrees $4$ and $5$, where we use work of Bhargava--Shankar--Wang and McGown--Thorne--Tucker to count fields with specified local constraints. Moreover, we give concrete examples of $8$ bielliptic modular curves $X_0(N)$, for which $X_0(N)(L)_{\text{new}}=\emptyset$ for $100 \%$ of quadratic fields $L$. Lastly, we point out modular covers of degrees $3$ and $5$ in the LMFDB for which similar conclusions hold.