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Cyclic reductions of elliptic curves at a positive proportion of primes

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Manjul Bhargava ↗

Version 1 / Oct 31, 2026 / CC BY 4.0

Abstract

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 ℜs>1−1/N\Re s>1-1/N for Dedekind zeta functions gives a lower proportion of at least cE/(Nlog⁡N)c_E/(N\log N), with cE>0c_E>0 independent of NN; the theorem of Cureton and Thorner supplies N=32N=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.

Provenance statement

Building on the techniques of [1] and [2], this paper shows that every elliptic curve over $\mathbb Q$ without full rational 2-torsion has cyclic reduction modulo a positive proportion of primes. The argument was worked out in handwritten notes, which were then converted into LaTeX, with additional writing and details added by both the author and GPT-6. The deduction of Theorems 1 and 2 from specified analytic and arithmetic inputs was then formalized in Lean with the help of GPT-6. The development proves the construction of the arithmetic progression, that every candidate prime with noncyclic reduction splits completely in some division field $K_\ell$ with $Y<\ell\le x^{\epsilon}$, the bound and subtraction of these exceptional primes, and the final lower proportion $c_E/(N\log N)$ with its specialization at $N=32$. The Cureton–Thorner zero-free hypothesis, candidate-prime estimate (Proposition 6), smoothed prime-ideal bound, prime number theorem, elementary prime-sum estimates, and standard facts about torsion, reduction, and division fields remain explicit hypotheses, and are described explicitly in the write up; their proofs were not formalized here.

Tools used

OpenAI
ChatGPTVersion GPT-6

References

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Version history

  1. v1Submitted by Manjul BhargavaInitial depositCurrentOct 31, 2026