On a remark of Serre
Pith reviewed 2026-05-18 22:23 UTC · model grok-4.3
The pith
All q less than 10^70 where the Hasse bound is not achieved for elliptic curves over F_q have been identified.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The authors computationally enumerate every q smaller than 10 to the power of 70 such that there is no elliptic curve over the finite field with q elements that attains the Hasse bound, while also exploring the expected number of such q and analogous conditions for higher genus curves.
What carries the argument
The Hasse bound |N - (q+1)| <= 2 sqrt(q) for the number N of points on an elliptic curve over F_q, with focus on cases where the maximum possible N is not attained.
Load-bearing premise
The search algorithm is complete and has not overlooked any qualifying q below 10^70.
What would settle it
A q less than 10^70 not included in the list but shown by independent computation to have no elliptic curve achieving the Hasse bound would disprove the result.
read the original abstract
Inspired by a remark of Serre, we extend the search for primes $p$ such that the maximum Hasse bound for the number of points on an elliptic curve over $\mathbb{F}_{p^5}$ is not achieved. We then give a list of all $q<10^{70}$ such that the Hasse bound is not achieved over $\mathbb{F}_{q}$. We explore the heuristics for how many such numbers should exist in each case. Finally, look at similar criteria for genus $2$ and $3$ curves.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper, inspired by a remark of Serre, extends the computational search for primes p such that elliptic curves over F_{p^5} fail to attain the maximum Hasse bound, then produces an explicit list of all q < 10^{70} for which the Hasse bound is not achieved over F_q. It also explores heuristics for the density of such q and formulates analogous criteria for genus-2 and genus-3 curves.
Significance. If the enumeration is exhaustive, the explicit list up to 10^{70} supplies a concrete, large-scale dataset that can be used to test heuristics and theoretical predictions about when the Hasse bound is sharp. The heuristics section and the genus-2/3 extensions add modest but useful context for arithmetic geometry.
major comments (2)
- [§3] §3 (Enumeration procedure): the reduction from arbitrary q to a finite set of candidate forms (prime powers or endomorphism-ring cases) is stated at a high level but lacks an explicit necessary-and-sufficient criterion or pseudocode; without this, it is impossible to confirm that every qualifying q below 10^{70} was detected and that no false negatives occurred for composite q.
- [§4] §4 (List of q): the paper asserts completeness for all q < 10^{70}, yet the text does not indicate whether the search was performed directly on composites or reduced via a theorem; any gap in the handling of non-prime-power q would undermine the central claim.
minor comments (2)
- [§5] The heuristics in §5 would benefit from a short table comparing predicted versus observed counts for small ranges of q.
- Notation for the Hasse bound and the quantity a in the trace is introduced inconsistently between the elliptic-curve and higher-genus sections.
Simulated Author's Rebuttal
We thank the referee for the careful reading and the specific comments on the enumeration procedure and the claimed completeness of the list. We address each major comment below and indicate the revisions we will make.
read point-by-point responses
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Referee: [§3] §3 (Enumeration procedure): the reduction from arbitrary q to a finite set of candidate forms (prime powers or endomorphism-ring cases) is stated at a high level but lacks an explicit necessary-and-sufficient criterion or pseudocode; without this, it is impossible to confirm that every qualifying q below 10^{70} was detected and that no false negatives occurred for composite q.
Authors: We agree that the current description in §3 is too high-level for independent verification. The reduction is based on the fact that the Hasse bound fails to be attained precisely when no integer t with |t| equal to the maximal admissible value satisfies the conditions for an elliptic curve over F_q (i.e., the trace equation and the endomorphism-ring constraints from the CM theory). We will add an explicit necessary-and-sufficient criterion in the revised §3, together with pseudocode that enumerates candidate prime powers q = p^k < 10^{70}, computes the maximal possible |t|, and checks whether any admissible t exists. This will make the absence of false negatives verifiable. revision: yes
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Referee: [§4] §4 (List of q): the paper asserts completeness for all q < 10^{70}, yet the text does not indicate whether the search was performed directly on composites or reduced via a theorem; any gap in the handling of non-prime-power q would undermine the central claim.
Authors: Finite fields F_q exist if and only if q is a prime power. Consequently our list contains only prime powers q < 10^{70}; there are no non-prime-power q to consider. The enumeration proceeds by generating all prime powers q = p^k below the bound and applying the criterion of §3. We will insert a clarifying paragraph at the beginning of §4 stating this fact and describing the reduction to prime powers, thereby removing any ambiguity about the scope of the search. revision: yes
Circularity Check
Direct computational enumeration of Hasse-bound exceptions with no self-referential derivation
full rationale
The paper extends a prior search for primes p where the Hasse bound fails to be attained over F_{p^5} and then enumerates all qualifying q below 10^{70} by direct computational search. This constitutes an exhaustive listing rather than a derivation whose central equations or predictions reduce by construction to fitted parameters, self-citations, or ansatzes imported from the authors' own prior work. The result is self-contained against external benchmarks: the list can be independently verified by re-running the search procedure on the stated bound, and no load-bearing step equates a claimed prediction to its own input data or relies on a uniqueness theorem justified only by overlapping authorship.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Hasse's theorem: for an elliptic curve over F_q the number of points N satisfies |N - (q+1)| <= 2 sqrt(q)
discussion (0)
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