HEXAGONbeta

Rational points on elliptic curves that are primitive modulo 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 17, 2026 / CC BY 4.0

Abstract

In 1977, Lang and Trotter conjectured that, subject to certain necessary arithmetic restrictions, a rational point PP of infinite order on an elliptic curve EE over $\Q$ should generate its reduction modulo infinitely many primes pp---and indeed should do so for a specified positive density of primes. However, to date, even this infinitude was not known unconditionally for any explicit pair (E,P)(E,P). For the curve E:y2=x3−2xE:y^2=x^3-2x and point P=(2,2)P=(2,2), we prove the existence of infinitely many---and indeed a positive lower density of---primes for which the reduction of PP generates $E(\F_p)$. More precisely, we prove that a common zero-free half-plane ℜs>1−1/N\Re s>1-1/N for Dedekind zeta functions gives a lower density of at least c/(Nlog⁡N)c/(N\log N), with c>0c>0 independent of NN. The announced theorem of Cureton and Thorner provides N=32N=32. Our proof combines this input with the marked Type~II estimates of OpenAI, a sieve with varying cofactor, and estimates for division fields and heights. The same method also applies to other CM elliptic curves and points.

Provenance statement

This is a follow-up paper to [1] giving positive-proportion results for the Lang--Trotter primitive-point conjecture - a natural elliptic-curve analogue of Artin's primitive root conjecture. The strategy, construction of the curve and point, the real-cyclotomic division fields, and the additional examples were worked out in handwritten notes, which were then converted into LaTeX, with writing and details added by both the author and GPT-6. The deduction of Theorem 1 from specified analytic and arithmetic assumptions was then formalized in Lean with the help of GPT-6. Specifically, the Cureton–Thorner zero-free region, specified lemmas from the OpenAI preprints (the root-residue count, the character-transform bounds, and the block sieve), a van der Corput derivative test, and the handful of classical analytic and arithmetic inputs listed in the paper remain explicit hypotheses; they were not formalized here. The signed marked Type II estimate itself was proved in the Lean development from the specified analytic inputs, including the passage from prime predecessors to prime successors and the uniformity required for the growing family of dyads; these proofs are also described in detail in the write up. The proof of Theorem 2 follows a path analogous to that of Theorem 1 and so was not separately formalized.

Tools used

OpenAI
ChatGPTVersion GPT-6

References

  1. M. Bhargava, Every nonsquare integer not equal to −1-1 is a primitive root modulo a positive lower density of primes, Hexagon preprint 2610.00228v1, October 2026. Preprint (https://hexagonmath.org/2610.00228v1).hexagon
  2. Y.-M. J. Chen and J. Yu, On primitive points of elliptic curves with complex multiplication, J. Number Theory 114 (2005), no. 1, 66--87. doi:10.1016/j.jnt.2005.02.005 (https://doi.org/10.1016/j.jnt.2005.02.005).DOI
  3. D. Cureton and J. Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions, Hexagon preprint 2610.00200v2, October 2026. Preprint (https://hexagonmath.org/2610.00200v2).hexagon
  4. D. Cureton and J. Thorner, Dedekind Zeta Zero-Free Region, formalization repository, Axiom Math, 2026. Repository and statement (https://github.com/AxiomMath/DedekindZetaZeroFree).link
  5. R. Gupta and M. Ram Murty, Primitive points on elliptic curves, Compositio Math. 58 (1986), no. 1, 13--44. Numdam (https://www.numdam.org/item/CM_1986__58_1_13_0/).link
  6. R. Gupta and M. Ram Murty, Cyclicity and generation of points mod pp on elliptic curves, Invent.\ Math. 101 (1990), no. 1, 225--235. doi:10.1007/BF01231502 (https://doi.org/10.1007/BF01231502).DOI
  7. C. Hooley, On Artin's conjecture, J.\ Reine Angew.\ Math. 225 (1967), 209--220.
  8. N. Jones, F. Pappalardi, and P. Stevenhagen, Locally imprimitive points on elliptic curves, Math.\ Proc.\ Cambridge Philos.\ Soc. 180 (2026), 663--683. doi:10.1017/S0305004125101795 (https://doi.org/10.1017/S0305004125101795).DOI
  9. N. M. Katz, Lang--Trotter revisited, Bull.\ Amer.\ Math.\ Soc.\ (N.S.) 46 (2009), no. 3, 413--457. doi:10.1090/S0273-0979-09-01257-9 (https://doi.org/10.1090/S0273-0979-09-01257-9).DOI
  10. S. Lang and H. Trotter, Primitive points on elliptic curves, Bull. Amer. Math. Soc. 83 (1977), no. 2, 289--292. doi:10.1090/S0002-9904-1977-14310-3 (https://doi.org/10.1090/S0002-9904-1977-14310-3).DOI
Show all 17 referencesShow fewer references
  1. OpenAI, The Poisson--Dirichlet law for prime predecessors, preprint, 24 September 2026, Theorem 3.1 and Sections 3--5; version consulted: repository commit 1581d4cbdd69 . Preprint (https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf).link
  2. OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane ℜ(s)>7/8\Re(s)>7/8, preprint, 30 September 2026. Preprint and accompanying files (https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026).link
  3. OpenAI, Primitive roots for every admissible integer base, preprint, 4 October 2026, Lemma 10.2, Proposition 10.3, and Section 12; version consulted: repository commit 1581d4cbdd69 . Preprint (https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf).link
  4. K. A. Ribet, Dividing rational points on abelian varieties of CM-type, Compositio Math. 33 (1976), 69--74. Author's copy (https://math.berkeley.edu/~ribet/Articles/compositio.pdf).link
  5. O. Robert, On van der Corput's kk-th derivative test for exponential sums, Indag.\ Math. 27 (2016), no. 2, 559--589. Author's manuscript (https://perso.univ-st-etienne.fr/rool6510/robert-2015-indag.pdf).link
  6. J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Publ.\ Math.\ IHES 54 (1981), 123--201. Numdam (https://www.numdam.org/item/PMIHES_1981__54__123_0/).link
  7. J. H. Silverman, The Arithmetic of Elliptic Curves , second edition, Graduate Texts in Mathematics 106 , Springer, 2009.

Version history

  1. v1Submitted by Manjul BhargavaInitial depositCurrentOct 17, 2026