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arxiv: 2606.25153 · v1 · pith:75QYNNS7new · submitted 2026-06-23 · 🧮 math.NT · math.AG

Hyperelliptic Atkin-Lehner quotients of Shimura curves

Eran Assaf , Sachi Hashimoto This is my paper

Pith reviewed 2026-06-25 22:12 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords Shimura curvesAtkin-Lehner quotientshyperelliptic curvesmodular curvesgenus two modelsdiscriminant coprime level
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The pith

The classification of hyperelliptic Atkin-Lehner quotients is extended from modular curves to Shimura curves.

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

The paper works to classify all cases where an Atkin-Lehner quotient of a Shimura curve X_0(D,N) is hyperelliptic, under the condition that the level N is coprime to the discriminant D and the quotient is by a subgroup W of the full Atkin-Lehner group. This extends Ogg's earlier classification when the quotient is trivial and the results of Furumoto and Hasegawa when D equals 1. A sympathetic reader would care because these quotients arise in the study of elliptic curves and their covers, and hyperellipticity gives an explicit geometric structure that simplifies arithmetic questions. As a side result the authors derive equations for some quotients of genus at most two by adapting earlier model-construction techniques.

Core claim

The quotients X_0(D,N)/W of Shimura curves are hyperelliptic precisely in the cases obtained by extending the lists of Ogg for W=1 and of Furumoto-Hasegawa for D=1; the same methods also yield explicit models for several quotients of genus at most two.

What carries the argument

The Atkin-Lehner quotient X_0(D,N)/W of the Shimura curve, whose hyperellipticity is controlled by the genus of the quotient and the fixed-point behavior of the involutions in W.

If this is right

  • The complete list of hyperelliptic quotients now includes all previously known examples together with new families where the discriminant D exceeds 1.
  • Explicit Weierstrass or hyperelliptic equations become available for several quotients of genus at most two.
  • Some open questions of Padurariu and Saia about the existence of such models receive affirmative answers in the Shimura setting.

Where Pith is reading between the lines

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

  • A finished classification would let one enumerate all hyperelliptic Shimura quotients by checking only finitely many small discriminants and levels.
  • The same geometric criteria might apply to quotients by larger groups of automorphisms beyond the Atkin-Lehner ones.

Load-bearing premise

The model-construction techniques developed by Guo and Yang for low-genus quotients carry over without essential change when the underlying curve is a Shimura curve rather than a modular curve.

What would settle it

An explicit Shimura curve quotient X_0(D,N)/W of genus two or less that is hyperelliptic yet whose minimal equation fails to match any of the forms produced by the extended Guo-Yang method.

Figures

Figures reproduced from arXiv: 2606.25153 by Eran Assaf, Sachi Hashimoto.

Figure 1
Figure 1. Figure 1: Final genus distribution of all quotients, separated by status. 8.3. Prime bounds. Some of the techniques above are prime-dependent. It is likely that pushing the prime bounds, for example, by computing more primes on Weil polynomials, would rule out a handful of extra curves. However, these computations are expensive, and unlikely to yield large gains. It is possible that many of the remaining open curves… view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of the tests that determined the 17595 resolved quo￾tients. Bar lengths are log-scaled. 8.4. Special fiber method. The isomorphism described in Proposition 6.2.1 can also be used to rule out hyperellipticity in the following way, as in [FH99, §5]. If X = X0(D, Np)/W is hyperelliptic, then the hyperelliptic involution ι acts on the normalization of its special fiber, isomorphic to the special f… view at source ↗
read the original abstract

We work towards completely classifying all hyperelliptic Atkin-Lehner quotients of Shimura curves $X_0(D,N)/W$ with level $N$ coprime to $D$ and $W \le W_0(D,N)$, extending, on the one hand, a result of Ogg that provided such a classification for the trivial quotients (the case $W = 1$), and on the other hand, results of Furumoto and Hasegawa that provided such a classification for modular curves (the case $D = 1$). As a byproduct of our methods, building on the works of Guo and Yang, we also obtain models for some quotients of genus at most two, answering some questions of Padurariu and Saia.

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

0 major / 2 minor

Summary. The manuscript works towards classifying all hyperelliptic Atkin-Lehner quotients of Shimura curves X_0(D,N)/W with N coprime to D and W ≤ W_0(D,N). It extends Ogg's classification for the case W=1 and Furumoto-Hasegawa's results for the case D=1. As a byproduct, building on Guo and Yang, it constructs models for some quotients of genus at most two, addressing questions raised by Padurariu and Saia.

Significance. If the extensions and constructions hold, the work provides a natural incremental advance in classifying hyperelliptic quotients beyond the modular curve and trivial-quotient settings, together with explicit low-genus models that are of independent value. The explicit use of Guo-Yang methods to produce these models is a concrete strength.

minor comments (2)
  1. Abstract: the phrasing 'work towards completely classifying' is appropriately cautious, but the introduction should state explicitly which families of (D,N,W) are fully resolved versus those left for future work.
  2. The manuscript should include a short table or list summarizing the new cases treated beyond the Ogg and Furumoto-Hasegawa results, to make the incremental contribution immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, the recognition of the incremental advance over Ogg and Furumoto-Hasegawa, and the recommendation for minor revision. We appreciate the note on the independent value of the low-genus models obtained via Guo-Yang methods.

Circularity Check

0 steps flagged

No significant circularity; derivation extends independent external results

full rationale

The paper's central claim is an incremental classification extending Ogg (for W=1) and Furumoto-Hasegawa (for D=1), with models obtained by building on Guo-Yang methods. No equations, definitions, or load-bearing steps reduce to self-citation, fitted inputs renamed as predictions, or ansatzes smuggled via the authors' own prior work. All cited foundations are from non-overlapping authors and are treated as external. This matches the default expectation of a self-contained, non-circular extension.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.1-grok · 5654 in / 944 out tokens · 23529 ms · 2026-06-25T22:12:30.902589+00:00 · methodology

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