{"id":"hexagon/2610.00699","pdf":"https://hexagonmath.org/pdf/2610.00699v1","url":"https://hexagonmath.org/abs/2610.00699v1","tags":["math.AG","msc:11R45","msc:11G05","msc:14H52"],"score":0,"title":"Cyclic reductions of elliptic curves at a positive proportion of primes","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Manjul Bhargava"],"reviews":0,"subject":"math.NT","summary":"We prove that every elliptic curve over $\\Q$ whose points of order two are not all rational has cyclic reduction at a positive lower density of primes. This is the only necessary restriction: full rational two-torsion prevents cyclic reduction at every odd prime of good reduction. More precisely, a common zero-free half-plane $\\Re s>1-1/N$ for Dedekind zeta functions gives a lower proportion of at least $c_E/(N\\log N)$, with $c_E>0$ independent of $N$; the theorem of Cureton and Thorner supplies $N=32$. The proof uses the varying-cofactor sieve of our earlier works on primitive roots and…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}