For every smooth projective curve C/Q with genus g(C)≥ 2 and every integer n≥ 1, let F_n(X)=L/Q:[L:Q]=n, |operatornameDisc(L)|≤ X. Define C(L)_new=P∈ C(L):Q(P)=L. Then displaystyle lim_X→∞frac#L∈F_n(X):C(L)_new≠varnothing#F_n(X)=0; equivalently, C(L)_new=varnothing for 100% of degree-n number fields L, ordered by absolute discriminant.

Khawaja–Siksek conjecture

For every smooth projective curve $C/\mathbb{Q}$ with genus $g(C)\geq 2$ and every integer $n\geq 1$ , let $\mathcal{F}_n(X)=\{L/\mathbb{Q}:[L:\mathbb{Q}]=n,\ |\operatorname{Disc}(L)|\leq X\}$ . Define $C(L)_{\mathrm{new}}=\{P\in C(L):\mathbb{Q}(P)=L\}$ . Then $\displaystyle\lim_{X\to\infty}\frac{\#\{L\in\mathcal{F}_n(X):C(L)_{\mathrm{new}}\neq\varnothing\}}{\#\mathcal{F}_n(X)}=0$ ; equivalently, $C(L)_{\mathrm{new}}=\varnothing$ for $100\%$ of degree- $n$ number fields $L$ , ordered by absolute discriminant.

References

Primary source

arXiv

Additional references

New algebraic points on covers of elliptic curves — arXiv — Diana Mocanu, George C. Turcas

Posted 16 hours ago · edited 15 hours ago

Progress summary

Refreshed 15 hours ago

Claimed progress

The conjecture remains open in general, but recent work gives stronger evidence for it in several low-degree settings.

Khawaja and Siksek conjecture that, for any suitable curve of genus at least $2$ , genuinely new points occur in only a zero-density set of number fields of each fixed degree. The conjecture was formulated in their paper submitted on November 19, 2025.

Known results

Degree $2$ : proved for $18$ hyperelliptic modular curves $X_0(N)$ other than $X_0(37)$ (Khawaja–Siksek, 2025).
Degree $3$ : proved for $X_0(23)$ , $X_0(29)$ , $X_0(31)$ , and $X_0(64)$ (Khawaja–Siksek, 2025).
A corresponding density- $100\%$ result was proved for a unit equation in cubic fields (Khawaja–Siksek, 2025).

October 2026 elliptic-curve extension

Mocanu and Turcas report analogous results for covers of elliptic curves, including quantitative discriminant bounds and density-zero conclusions in degrees $2$ through $5$ . These are substantial partial results, but they do not prove the conjecture for arbitrary curves and extensions.

Current status (as of October 2026): The conjecture is open in general; several modular-curve and unit-equation cases are established, while the elliptic-cover results are claimed partial progress and do not settle the full statement.

Sources

No solutions have been posted yet.