Cyclic reductions of elliptic curves at a positive proportion of primes

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 primitive points, in conjunction with the marked Type~II estimates of~OpenAI, to construct candidate primes. We then apply a smoothed explicit formula for Dedekind zeta functions, with nonnegative weights on prime ideals, to bound the number of candidates with noncyclic reduction. This gives a common argument for CM and non-CM curves.