REVIEW
Average twin prime conjecture for elliptic curves in arithmetic progressions
Number Theory · Submitted 30 Aug 2026 · 28 references
Pith's verdict: A genuine unconditional average Koblitz result over arithmetic progressions with the Lee-Mayle-Wang constant; the n=1 stress-test misfires because it substitutes the wrong definition of U.
T0 · AI review
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Referee report
The paper
The paper claims
Koblitz conjecture on average for elliptic curves is extended to arithmetic progressions, with an asymptotic formula that matches the predicted constant for moduli up to a fixed power of log x.
The reading
What carries the argument
The proof reduces the global average over the family F to a weighted sum over primes of local counting functions ν(p;P). Using Deuring's theorem, each ν(p;P) is expressed in terms of Hurwitz class numbers, which are then approximated by Dirichlet L-functions evaluated at 1. Truncation of the L-series and application of quadratic reciprocity convert the problem into counting prime pairs (p, p+1−r) with a fixed difference r, subject to congruence conditions arising from n and k. The crucial Proposition 4.3 provides an asymptotic for the weighted prime-pair sums, obtained via a Barban–Davenport–Halberstam type estimate (Proposition 4.1) and an error analysis using Rankin's trick. The constant emerges from an Euler product factorization of the main term, with the factor S(n;k) evaluated explicitly in Section 8 using Hensel's lemma and character sum evaluations.
Extended reading notes
Core claim
The paper establishes that the average over the family F of elliptic curves of π'_E(x;k,n) — the count of primes p ≤ x with p ≡ k mod n for which the reduction E(F_p) has prime order — equals C'_{n,k} x / $log^{2}$ x + O(x / (φ(n) $log^{3}$ x)), where C'_{n,k} is the explicit Euler product given in equation (2). This holds for any coprime 1 ≤ k ≤ n, any fixed C, ε > 0, provided A,B > x^ε and AB > x $log^{{10}}$ x, and n ≪ (log x)^C. The constant C'_{n,k} matches the average of the conjectural constants C'_{E,n,k} computed by Lee, Mayle, and Wang, demonstrating that the arithmetic progression analogue of Koblitz's conjecture holds on average.
Load-bearing premise
The error term in Lemma 3.1 must be uniformly O($x^{3}$ $log^{2}$ x / (φ(n)L)) for all moduli n up to Y/2 = x/(2L) with L = floor(log^M x); if this bound fails for some n in the allowed range, the main term of Theorem 1.4 could be swamped.
Editorial extensions
If this is right
- The uniform control of the modulus n up to (log x)^C shows that the average Koblitz constant varies with n and k in a way that is now fully explicit, enabling refined heuristics for individual curves in arithmetic progressions.
- The result suggests that the Koblitz prime density for individual curves, if it exists, should experience biases modulo n that average out to the product formula C'_{n,k}, providing a testable prediction for numerical experiments.
- The methods yield a similar average result for the function M_E(N;n,k) counting primes p with |E(F_p)| = N and p ≡ k mod n, linking the arithmetic progression problem to the distribution of elliptic curve orders.
- The proof's tracking of the modulus n throughout the analytic argument offers a template for extending other average results (cyclicity, divisibility) to arithmetic progressions with explicit uniformity in n.
Reading between the lines
- If the error term in Lemma 3.1 were not uniformly O(x^3 log^2 x / (φ(n)L)), the main term in Theorem 1.4 could be swamped for some n within the allowed range; this technical condition is the most fragile step but the final restriction n ≪ (log x)^C is chosen precisely to keep it under control.
- The explicit constant C'_{n,k} contains factors that vanish when n has many prime factors dividing k−1 or r(r−2), aligning with the idea that the average density becomes smaller for progressions where divisibility conditions restrict the trace r; this could indicate that individual curves with such n have fewer Koblitz primes.
- The agreement between the analytic average constant and the Galois-theoretic average constant strengthens the expectation that the Zywina refinement (accounting for division field entanglements) is correct on average, though the paper does not test the vanishing cases individually.
Original abstract
In 1988, Koblitz conjectured an asymptotic formula for the number of primes $p \le x$ for which the reduction of an elliptic curve over $\mathbb{Q}$ has prime order. Building on the work of Balog, Cojocaru, and David, who proved this conjecture on average in 2011, we extend the result to primes $p$ lying in arithmetic progressions. Our asymptotic formula holds uniformly for moduli up to a fixed power of $\log x$. We also show that the resulting average constant matches the theoretical constant predicted by Lee, Mayle, and Wang in 2025 using Galois representations.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
In 1988, Koblitz conjectured an asymptotic formula for the number of primes $p \le x$ for which the reduction of an elliptic curve over $\mathbb{Q}$ has prime order. Building on the work of Balog, Cojocaru, and David, who proved this conjecture on average in 2011, we extend the result to primes $p$ lying in arithmetic progressions. Our asymptotic formula holds uniformly for moduli up to a fixed power of $\log x$. We also show that the resulting average constant matches the theoretical constant predicted by Lee, Mayle, and Wang in 2025 using Galois representations.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In 1988, Koblitz conjectured an asymptotic formula for the number of primes $p \le x$ for which the reduction of an elliptic curve over $\mathbb{Q}$ has prime order. | background 1.00 | 0.03 | 0.03 | asserted 0.68 |
| s1 | Building on the work of Balog, Cojocaru, and David, who proved this conjecture on average in 2011, we extend the result to primes $p$ lying in arithmetic progressions. | contribution 0.88 | 0.56 | 0.24 | asserted 1.00 |
| s2 | Our asymptotic formula holds uniformly for moduli up to a fixed power of $\log x$. | result 0.94 | 0.91 | 0.31 | asserted 1.00 |
| s3 | We also show that the resulting average constant matches the theoretical constant predicted by Lee, Mayle, and Wang in 2025 using Galois representations. | result 0.95 | 0.81 | 0.16 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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