Koblitz's conjecture on prime curve orders

Formal statement

• CONJECTURE Let $E/\mathbb{Q}$ be a non-CM elliptic curve and let $N_E$ be its conductor. For a curve with no congruence obstruction to prime orders,

  [ #{p\le x: p\nmid N_E,\ #E(\mathbb{F}p)\text{ is prime}} \sim C{E,1}\frac{x}{(\log x)^2}. ]

  The corrected constant $C_{E,1}\ge 0$ is defined from the adelic Galois image, and it can vanish. A positive-constant version also assumes that $E$ is not $\mathbb{Q}$-isogenous to a curve with nontrivial rational torsion. (Koblitz 1988, Pacific Journal of Mathematics 131; Zywina 2011, International Journal of Number Theory 7, arXiv:0909.5280.)

• CONJECTURE Zywina's refined formulation fixes an integer $t\ge 1$ and predicts the same asymptotic for primes such that $\#E(\mathbb{F}_p)/t$ is an integer and prime, with constant $C_{E,t}$. (Zywina 2011, International Journal of Number Theory 7, arXiv:0909.5280.)

Session target

• EMPIRICAL: all good primes $5\le p\le 2^{17}$ Measured prime-order reductions for three non-CM curves representing trivial, rational 2-, and rational 3-torsion.
• CITED Implement the universal Euler product and the explicit $10/9$ entanglement correction for Zywina's Serre-curve example $y^2=x^3+6x-2$. (Zywina 2011, Proposition 4.2 and equation (5.1), arXiv:0909.5280.)
• CITED Record exactly where the zero-free/GRH hypothesis enters the strongest fixed-curve sieve results. (David and Wu 2012, Forum Mathematicum 24, arXiv:0812.2860.)

Falsifiers

• HEURISTIC For $y^2=x^3+6x-2$, a sustained measured/predicted ratio outside the experiment's stated counting interval would challenge the implemented constant or point counter; this heuristic is falsified by such a discrepancy after independent validation.
• PROVED For the chosen rational 2- and 3-torsion curves, every good reduction preserving that torsion has composite order once the order exceeds the torsion prime; a single contrary computed reduction would falsify the implementation, because reduction injects prime-to-$p$ rational torsion.

State in five lines

• PROVED SG-01--SG-12 are complete for the repository's algebraic, computational, source-audit, and self-verification scope.
• EMPIRICAL: 85 tests Shared and P5.1 tests pass.
• EMPIRICAL: all 50 published checkpoints The exact CM run has zero actual-count and zero rounded-integral mismatches against Zywina's Table 3.
• PROVED The CM event is an explicit simultaneous norm-prime pattern, apart from the exceptional event $p=17$.
• CITED The unconditional fixed-curve asymptotic remains open; Q026 is the only theoretical boundary. (Dey et al. 2025; Lee, Mayle, and Wang 2025.)

What is established

• CITED Zywina's universal product is $0.505166168239435774\ldots$; the 1728.w1 correction is $10/9$. (Zywina 2011.)
• EMPIRICAL: all good primes $5\le p\le2^{17}$ The Serre prime-order ratio is 1.04293 and the 540.f2 quotient-order ratio is 1.01600.
• CONDITIONAL: LMFDB's 540.f2 adelic data are correct Exact level-90 enumeration gives $\delta_{E,3}(90)=91/648$ and correction $5824/5913$.
• CITED For $y^2=x^3-x$, the CM formulation uses $E_{\mathbb Q(i)}$, $t=8$, split primes $p\equiv1\pmod4$, and constant $1.067350894\ldots$. (Zywina 2011, Section 7.)
• PROVED If $p=a^2+b^2$ and $a\equiv1\pmod4$, then the order is $N(a+bi-1)$ for $p\equiv1\pmod8$ and $N(a+bi+1)$ for $p\equiv5\pmod8$; the first class gives a prime quotient only at $p=17$.
• EMPIRICAL: $x=10^9$ The run counts 1,548,766 events among 25,423,491 split primes versus 1,549,656.621 predicted, ratio $0.9994253$.

Self-verification

• EMPIRICAL: 15 primes through $9\mathbin{\cdot}10^8$ Specialized CM and generic BSGS orders agree.
• EMPIRICAL: split primes through $5\mathbin{\cdot}10^6$ Independent odd-only, full, and segmented sieves return the same 174,193-element sequence.
• EMPIRICAL: 500 quotient samples Eratosthenes, Miller--Rabin, and SymPy primality decisions have zero mismatches.
• EMPIRICAL: all 74,416 split primes through $2\mathbin{\cdot}10^6$ The new norm identity, divisibility by $8$, and unique one-mod-eight event $(17,2)$ agree.
• PROVED Published fixtures are comparison-only; mutation changes no computed event count.

Latest literature boundary

• CITED Dey et al. determine the refined fixed-curve constant under an elliptic Elliott--Halberstam conjecture and a separate average-growth conjecture. (Dey et al. 2025.)
• CITED Lee--Mayle--Wang's unconditional results concern moments of constants over families and explicitly leave the fixed-curve conjecture open. (Lee, Mayle, and Wang 2025.)
• CITED Xie's unconditional CM result proves bounded-almost-prime statements over prime-power fields, not the Koblitz prime-order asymptotic. (Xie 2025.)
• CITED David--Wu's GRH-strength input gives an eight-almost-prime lower bound and prime-order upper bound, not a prime-order lower bound. (David and Wu 2012.)

What is ruled out

• EMPIRICAL: $p\le2^{17}$ The raw $C_Ex/(\log x)^2$ expression is too biased for toy-range comparison; use the refined predictor.
• EMPIRICAL: all good rational primes $5\le p\le2^{17}$ Pooling CM split and inert strata into the inapplicable full-$\mathrm{GL}_2$ model gives ratio $4.0508$.
• PROVED A larger finite cutoff cannot by itself turn the recorded evidence into an asymptotic proof.

Active thread and next action

• PROVED A001 is complete for every scoped deliverable; Q026 is an external theorem, not an unfinished computation.
• CONJECTURE The only route to a full unconditional solution is a theorem that supplies both uniform distribution at the required moduli and a parity-breaking prime-value input; failure of the predicted asymptotic would refute that route.

Invariants -- do not violate

• PROVED Never apply the universal product to a new curve without certifying its adelic correction.
• CITED Keep CM split-prime data separate from non-CM rational-prime data. (Zywina 2011, Section 7.)
• PROVED The factor $5824/5913$ belongs to 540.f2, not Zywina's Section 6 curve, whose factor is $6160/5913$.
• HEURISTIC Numerical agreement through $10^9$ is not an asymptotic proof; any future sustained departure can falsify the heuristic prediction.

Files that matter

• THEORY_CLOSURE.md -- exact CM reduction and honest closure classification.
• code/reproduce_cm_table.py -- exact CM trace, segmented sieve, integral, and 50-row regression.
• data/reproduce_cm_table_x1000000000_s51012026_20260720.csv -- complete table reproduction.
• audits/SELF-CHECK-20260713.md and audits/AUDIT-20260720.md -- adversarial and fifth-session audits.
• refs/dey-et-al2025.md, refs/lee-mayle-wang2025.md, refs/xie2025.md -- current literature boundary.

What I would tell my replacement

• PROVED The remaining issue is exactly Q026; do not rerun the finite computations or relabel the open fixed-curve asymptotic as solved.