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arxiv: 1906.09407 · v1 · pith:WALIB2E2new · submitted 2019-06-22 · 🧮 math.NT

Rank and Bias in Families of Hyperelliptic Curves via Nagao's Conjecture

Trajan Hammonds , Seoyoung Kim , Benjamin Logsdon , \'Alvaro Lozano-Robledo , Steven J. Miller This is my paper

Pith reviewed 2026-05-25 18:28 UTC · model grok-4.3

classification 🧮 math.NT
keywords hyperelliptic curvesJacobian rankNagao conjecturetraces of Frobeniusmoments of L-functionsbiasspecializationgenus g
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The pith

Generalized Nagao conjecture converts first-moment calculations into Jacobian ranks of 4g+2 for hyperelliptic families over Q(T).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the first moments of traces of Frobenius for several families of hyperelliptic curves y squared equals f(x) defined over the rational function field Q(T) of genus g at least 1. Under the assumption that the Jacobian has no subvariety defined over Q together with a generalization of Nagao's conjecture, these moments imply that the rank of the Jacobian over Q(T) equals 4g+2. The resulting infinitely many such curves over Q(T) specialize, by Silverman's theorem, to hyperelliptic curves over Q whose Jacobians have large rank. The authors also observe and prove a bias in the second-moment expansions for a number of these families, matching the form previously established by Michel for elliptic curves.

Core claim

By calculating the first moments A_{X,1}(p) for various families of hyperelliptic curves X over Q(T), the limit as X to infinity of (1/X) sum_{p less than or equal to X} minus A_{X,1}(p) log p equals the rank of J_X(Q(T)), which evaluates to 4g+2. This holds under the no-subvariety assumption and the generalized Nagao conjecture, extending earlier elliptic-curve results and yielding examples whose rank approaches but does not reach Shioda's record of 4g+7.

What carries the argument

The r-th moment A_{X,r}(p) equals (1/p) sum_{t=1 to p} a_{X_t}(p)^r of the trace of Frobenius at the specialization X_t, whose first moment is linked to the Jacobian rank over Q(T) by the generalized Nagao conjecture.

If this is right

  • Infinitely many hyperelliptic curves over Q(T) have Jacobian rank exactly 4g+2.
  • Specialization produces hyperelliptic curves over Q whose Jacobians have rank at least as large as the function-field rank.
  • The second moments satisfy p times A_{X,2} equals p squared plus O of p to the 3/2, matching Michel's elliptic-curve result.
  • The largest non-vanishing lower-order term in the second-moment expansion is on average negative for the families examined.
  • The bias in that lower-order term is proven for a number of the families.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same moment method could be applied to other one-parameter families to produce explicit rank formulas without computing the full L-function.
  • The proven bias may be explained by the same random-matrix heuristics that predict sign biases in elliptic-curve L-functions.
  • If the no-subvariety condition can be verified for families with even larger predicted moments, the construction would yield ranks exceeding 4g+2.
  • Numerical verification of the generalized Nagao limit for these explicit families would give direct evidence for the ranks without relying on the conjecture alone.

Load-bearing premise

The Jacobian of X over Q(T) has no subvariety defined over Q, which together with the unproven generalized Nagao conjecture is required to convert the computed moments into the stated rank.

What would settle it

A concrete counterexample would be a specific family in the paper where the numerical limit of the first-moment sum fails to match the independently computed rank of J_X(Q(T)) for that family.

read the original abstract

Let $\mathcal{X} : y^2 = f(x)$ be a hyperelliptic curve over $\mathbb{Q}(T)$ of genus $g\geq 1$. Assume that the jacobian of $\mathcal{X}$ over $\mathbb{Q}(T)$ has no subvariety defined over $\mathbb{Q}$. Denote by $\mathcal{X}_t$ the specialization of $\mathcal{X}$ to an integer $T=t$, let $a_{\mathcal{X}_t}(p)$ be its trace of Frobenius, and $A_{\mathcal{X},r}(p) = \frac{1}{p}\sum_{t=1}^p a_{\mathcal{X}_t}(p)^r$ its $r$-th moment. The first moment is related to the rank of the jacobian $J_\mathcal{X}\left(\mathbb{Q}(T)\right)$ by a generalization of a conjecture of Nagao: $$\lim_{X \to \infty} \frac{1}{X} \sum_{p \leq X} - A_{\mathcal{X},1}(p) \log p = \operatorname{rank} J_\mathcal{X}(\mathbb{Q}(T)).$$ Generalizing a result of S. Arms, \'A. Lozano-Robledo, and S.J. Miller, we compute first moments for various families resulting in infinitely many hyperelliptic curves over $\mathbb{Q}(T)$ having jacobian of moderately large rank $4g+2$, where $g$ is the genus; by Silverman's specialization theorem, this yields hyperelliptic curves over $\mathbb{Q}$ with large rank jacobian. Note that Shioda has the best record in this directon: he constructed hyperelliptic curves of genus $g$ with jacobian of rank $4g+7$. In the case when $\mathcal{X}$ is an elliptic curve, Michel proved $p\cdot A_{\mathcal{X},2} = p^2 + O\left(p^{3/2}\right)$. For the families studied, we observe the same second moment expansion. Furthermore, we observe the largest lower order term that does not average to zero is on average negative, a bias first noted by S.J. Miller in the elliptic curve case. We prove this bias for a number of families of hyperelliptic curves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript computes the first moments A_{X,1}(p) for families of hyperelliptic curves X: y^2 = f(x) over Q(T) of genus g, showing A_{X,1}(p) = -(4g+2) plus lower-order terms that vanish in the limit. Under a generalized Nagao conjecture equating the limit (1/X) sum_{p≤X} -A_{X,1}(p) log p to the rank, this predicts rank 4g+2 for J_X(Q(T)) assuming no Q-subvariety; specialization then yields high-rank examples over Q. The paper also proves a negative bias in the second-moment lower-order term for several families and observes the Michel-type expansion p A_{X,2}(p) = p^2 + O(p^{3/2}).

Significance. The explicit moment computations and unconditional bias proofs for multiple families extend the elliptic-curve results of Arms-Lozano-Robledo-Miller and Miller to higher genus, providing concrete evidence for moderately large ranks. The specialization step via Silverman's theorem is standard and correctly applied. If the generalized Nagao conjecture holds for these families, the constructions are of interest, though Shioda already achieves 4g+7.

major comments (3)
  1. [Abstract] Abstract: The no-subvariety-over-Q assumption is stated as a hypothesis but receives no verification (e.g., via endomorphism ring computation or Galois action on torsion) for the concrete families that achieve the 4g+2 prediction; without this, the moment limit only bounds the rank of the Q(T)-part of the Jacobian.
  2. [Abstract] Abstract, displayed Nagao limit: The rank claim is obtained by defining rank J_X(Q(T)) to be exactly the displayed limit; the manuscript therefore reduces the rank statement to the moment calculation under the unproven generalized conjecture, and should state the precise form of the conjecture together with any known conditional results.
  3. [Section on bias proofs (near the end of the abstract)] The second-moment bias is proved for a number of families, but the manuscript does not indicate which families receive unconditional proofs versus which only receive numerical observation; this distinction is load-bearing for the claim that the bias is established beyond the elliptic-curve case.
minor comments (2)
  1. [Abstract] The citation to Michel's second-moment result should include the precise reference and the exact statement used.
  2. Notation for A_{X,r}(p) is introduced in the abstract but the normalization (sum over t=1 to p) should be repeated when the moments are computed in the body.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The no-subvariety-over-Q assumption is stated as a hypothesis but receives no verification (e.g., via endomorphism ring computation or Galois action on torsion) for the concrete families that achieve the 4g+2 prediction; without this, the moment limit only bounds the rank of the Q(T)-part of the Jacobian.

    Authors: We agree that the assumption is not verified for the concrete families. The moment computations determine the rank of the Q(T)-part of the Jacobian under the stated hypothesis; the assumption is needed to equate this to the full rank. We will revise the abstract to clarify this distinction and note that explicit verification (e.g., via endomorphism rings) lies outside the scope of the present work. revision: yes

  2. Referee: [Abstract] Abstract, displayed Nagao limit: The rank claim is obtained by defining rank J_X(Q(T)) to be exactly the displayed limit; the manuscript therefore reduces the rank statement to the moment calculation under the unproven generalized conjecture, and should state the precise form of the conjecture together with any known conditional results.

    Authors: The manuscript presents the displayed limit as equal to the rank via the generalized Nagao conjecture. We will revise the abstract to state the conjecture explicitly in its precise form and to reference known conditional results (primarily from the elliptic-curve literature). This will make clear that the rank prediction is conditional on the conjecture. revision: yes

  3. Referee: [Section on bias proofs (near the end of the abstract)] The second-moment bias is proved for a number of families, but the manuscript does not indicate which families receive unconditional proofs versus which only receive numerical observation; this distinction is load-bearing for the claim that the bias is established beyond the elliptic-curve case.

    Authors: We will revise the abstract and the relevant section to explicitly distinguish the families for which the bias is proved unconditionally from those supported only by numerical observation. This will strengthen the presentation of the unconditional results. revision: yes

Circularity Check

0 steps flagged

No circularity detected; moments computed independently, rank via external conjecture

full rationale

The paper states the generalized Nagao conjecture as an external relation equating the limit of the first moment to the rank, and separately assumes no Q-subvariety. It then performs explicit computation of A_{X,1}(p) for concrete families, obtaining A_{X,1}(p) = -(4g+2) + lower-order terms. The rank conclusion follows only by applying the stated conjecture to this independent calculation; no equation inside the paper equates the rank to the moment by definition or by fitting. The cited prior result of Arms-Lozano-Robledo-Miller is invoked only for generalization of the moment method, not as a load-bearing uniqueness theorem or self-referential premise. No self-definitional, fitted-prediction, or ansatz-smuggling steps appear in the provided derivation chain. The derivation is therefore self-contained once the external conjecture and assumption are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central rank claims rest on the unproven generalized Nagao conjecture and the assumption that the Jacobian over Q(T) has no Q-defined subvariety; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Generalized Nagao conjecture: lim (1/X) sum_{p<=X} -A_{X,1}(p) log p equals rank J_X(Q(T))
    Invoked to convert computed first moments into the stated Jacobian ranks
  • domain assumption Jacobian of X over Q(T) has no subvariety defined over Q
    Stated at the beginning of the abstract; required for the specialization argument

pith-pipeline@v0.9.0 · 5988 in / 1588 out tokens · 66846 ms · 2026-05-25T18:28:11.271195+00:00 · methodology

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Reference graph

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