Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

On a non-abelian analogue of a conjecture of Michael Stoll

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that for a rank-0 elliptic curve E over Q, the quadratic Chabauty locus of E minus the origin is, at every good prime p, exactly the Q_p-points of a single finite subscheme defined over Q.

desk verdict Proves a real new case of non-abelian Stoll and makes the rank-zero punctured-elliptic-curve quadratic Chabauty locus explicit; one external application in §5.4 needs a careful hypothesis check. read the letter →

arxiv 2508.19947 v1 pith:HN2FXTJB submitted 2025-08-27 math.NT

L. Alexander Betts This is my paper · ORCID
classification math.NT MSC 11G0514G0514K1211G30
keywords non-abelianChabautyquadraticellipticcurvesManin-MumfordunlikelyintersectionsATvarietiesMordell-Weilrankzerop-adicheights
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's target is a non-abelian analogue of the conjecture that the abelian Chabauty locus is interpolated by a p-independent finite subscheme. It proves the rank-zero quadratic case, including the case where Y is an elliptic curve minus its origin and E/Q has Mordell-Weil rank 0. In that case the quadratic Chabauty locus at every good prime is shown to equal Z(Q_p) for one explicit finite Galois-stable subscheme Z consisting of torsion points; the apparent p-adic dependence collapses into the valuations of a single function H^st. This matters because it turns quadratic Chabauty for these curves from a p-adic analytic construction into an algebraic subset of bounded size, uniformly in p.

What carries the argument

The central object is the split quadratic Albanese variety, the universal G_m^r-torsor over the relevant abelian variety (an AT variety: abelian-by-torus) which realises the maximal quotient of the pro-unipotent fundamental group that is an extension of V_pJ by Q_p(1)^r. From it the paper builds an isogeny system β_n : P → P_n using the identity P_n = P^{⊗(1+n)} ⊗ [-1]^*P^{⊗(1-n)} and the cube formula [n]^*P_n ≅ P^{⊗2n^2}; these are isogenies of AT varieties. Taking the filtered colimit gives a map f_∞ : Y(Z_p) → lim P_n(Q_p), and the isogeny geometric quadratic Chabauty locus is f_∞^{-1}(W) for a finite set W of integral points. The proof's load-bearing finiteness result is a Manin-Mumford

What would settle it

Build the semiabelian scheme B→Y from the once-punctured elliptic curve construction and check whether its section meets the torsion locus in infinitely many points outside the three cases of the relative Manin-Mumford trichotomy; any such intersection would break Theorem D and Theorem A. Alternatively, on a rank-0 curve with a torsion point Q satisfying both criteria of Theorem 6.13, evaluate the actual Coleman functions defining quadratic Chabauty at a good prime p; a contradiction would disprove the claimed equality.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the 'skimmed' quadratic Chabauty locus for a once-punctured rank-0 elliptic curve is a p-independent algebraic object. The paper constructs an isogeny-geometric quadratic Chabauty locus Y(Z_p)^f from the split quadratic Albanese variety P of Y and proves it coincides with the usual non-abelian quadratic Chabauty locus (Theorem 4.1). Writing the finite set W in the rational colimit lim P_n(Q), the isogeny locus is the inverse image of W, and the main finiteness theorem says quadratic torsion packets are finite unless the image lies in a strict AT subvariety (Theorem D). For Y = E minus the origin, the resulting subscheme Z is explicit: Z

Load-bearing premise

The rank-0 punctured-elliptic-curve case depends on a classification of when a family of curves can contain infinitely many torsion points, and the paper assumes that classification is complete for the particular semiabelian family it constructs; if a non-listed example could occur, the main theorem for punctured elliptic curves would not follow.

Editorial extensions

If this is right

  • For every rank-0 elliptic curve E/Q, the quadratic Chabauty locus of E minus the origin has size bounded independently of p, and is explicitly computable from torsion points and local heights.
  • A torsion point lies in the locus exactly when H^st(Q) is rational and its valuations match the Kodaira-type table; this is a simultaneous necessary and sufficient test, refining previous sufficient tests.
  • The isogeny-geometric variant gives the equality Y(Z_p)^f = Z(Q_p) without requiring a rational base point or good reduction at p, so in rank 0 the p-adic method can be replaced by a geometric description.
  • Under the same rank-zero hypothesis, the Selmer Section Conjecture for hyperbolic curves follows once such a finite p-independent subscheme exists.
  • The appendix shows the full depth-2 unipotent quotient is not realisable by a morphism to a smooth variety for projective curves of genus at least 4, so the AT quotient is near the limit of geometric realisability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension the paper does not spell out: because Theorem 6.13 is a finite arithmetic criterion, one can enumerate torsion points up to growing height and search for the subscheme Z purely from H^st and the W^st_ℓ table, without p-adic integration.
  • The structural reading this suggests: Stoll-type p-independence is governed by Manin-Mumford for quadratic torsion packets in AT varieties, so any future counterexample would most plausibly arise from a curve whose image lies in a strict AT subvariety, not from the generic locus.
  • For computational practice, the isogeny formula indicates that H^st can be evaluated through isogenies rather than division polynomials; using cyclic isogenies of small degree might make the criterion feasible on large torsion subgroups, but the paper leaves that implementation open.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper formulates a non-abelian analogue of Stoll's conjecture (Conjecture 1.2) and proves the rank-zero quadratic case. The main theorems are: construction of the (split) quadratic Albanese variety of a smooth curve (Theorems B and B'), an exact comparison between the isogeny geometric quadratic Chabauty locus and the usual quadratic Chabauty locus under rank-zero hypotheses (Theorem C / Theorem 4.1), a Manin-Mumford-type finiteness theorem for quadratic torsion packets in AT varieties (Theorem D), and an explicit description of the finite subscheme cutting out the quadratic Chabauty locus for once-punctured elliptic curves of Mordell-Weil rank zero (Theorem 6.13). The strategy is to replace the p-adic Chabauty condition by a purely geometric condition — membership in the inverse image of a finite set W under the map f∞ — and then to prove finiteness of the relevant fibres via unlikely-intersections results, with the elliptic-curve case ultimately depending on a relative Manin-Mumford theorem.

Significance. If correct, the paper is a substantial advance: for once-punctured elliptic curves of rank 0 it gives a p-independent algebraic description of the quadratic Chabauty locus, uniformly bounded in p, and a simultaneous necessary and sufficient criterion for a torsion point to lie in the locus. The paper introduces a useful category of AT varieties, contains detailed conceptual proofs, and involves no fitted parameters; the computations in §6 are explicit and checkable. The exact equality in Theorem 4.1 addresses the known strictness gap between geometric quadratic Chabauty and quadratic Chabauty. I regard the central claims as very likely correct, but one load-bearing external-input verification is missing (see major comment 1).

major comments (2)
  1. [§5.4, Theorem 5.14] The exceptional case dim(A)=1, r=1 of Theorem D rests entirely on the quoted theorem [BMPZ16, Theorem 2(ii)], but the paper neither gives the full hypotheses of that theorem nor verifies them for the specific semiabelian scheme B = \hat f^*P' over Y and section s. In particular, the paper should state and check any standing assumptions on the base field, on Y (e.g. properness or non-isotriviality), on the extension class q, and on the section s (e.g. non-degeneracy or not being contained in a proper semiabelian subscheme). Without this verification, the trichotomy in Theorem 5.14 is not formally justified, and the finiteness of quadratic torsion packets for the once-punctured elliptic curve case, and hence Theorem 6.13 / Theorem A, does not follow from the cited theorem as written.
  2. [§6, Definition 6.11 and Theorem 6.13] The notation Z(Q) and the domain of the function H^st are inconsistent. The theorem states Z(Q) ⊆ Y(Q)_tors, but the proof and the worked examples require points in Y(Qbar)_tors; Definition 6.11 only defines H^st on Q-rational torsion points. Consequently condition (i), 'H^st(Q) ∈ Q ⊗ Q^×', is not well-formed for arbitrary algebraic torsion points, and condition (ii) is not meaningful unless H^st(Q) is known to be a Galois-invariant element of K ⊗ Q^×. The intended meaning is clear — H^st should be defined on all torsion points and condition (i) should be the assertion that its class lies in the rational subspace Q ⊗ Q^× — but the statement of the main explicit theorem should be corrected.
minor comments (4)
  1. [§1.1, Remark 1.3] The term 'of motivic origin' is deliberately left undefined. This is acceptable as a convention for the conjecture, but the paper should make explicit that Theorem A' and Theorem A are theorems about quotients realized by morphisms, not about all quotients of motivic origin.
  2. [§6, Table 1] In the row for I_m^*, the entry for c=4 is '-(m+4)/8'; due to the line break in the table this could be misread as '-m+4/8'. Please reformat for clarity.
  3. [§6.3, Example 6.16] The sentence '8+3√7 is a principal unit' would be clearer as '8+3√7 is a unit of norm 1 in Q(√7)' — the latter formulation directly explains why its valuations depend only on the rational prime below each place.
  4. [Appendix A, Corollary A.2] The phrase 'not realised by any morphism' relies on the paper's convention that a quotient is realised only when it is isomorphic to the target variety's full pro-unipotent fundamental group. A brief reminder of this convention at the start of the appendix would prevent misunderstanding.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main equality is established by Selmer-scheme comparison and independent Manin–Mumford/relative-Manin–Mumford inputs; self-citations are published infrastructure and are not load-bearing in a reductive sense.

full rationale

The derivation chain is: Theorem C/4.1 (isogeny geometric quadratic Chabauty locus Y(Zp)^f equals the non-abelian Chabauty locus Y(Zp)^{U^p}), Theorem 5.2 (this locus equals Zp-points of a finite subscheme Z defined as f∞^{-1}(W)), with finiteness of quadratic torsion packets supplied by Theorem D. None of these steps reduces to its own input by construction. Y(Zp)^f is defined (Definition 3.14) as the inverse image of the finite set W, while Y(Zp)^{U^p} is defined via Selmer schemes (Definition 4.13); Theorem 4.1 proves their equality by comparing Selmer schemes for U and P (Lemmas 4.16–4.22), including a bijectivity proof for the local Kummer map at p (Lemma 4.18) and a one-point global Selmer scheme (Lemma 4.19). The subscheme Z in Theorem 6.13 is described by explicit residue, discriminant and Kodaira–Tamagawa valuation conditions (conditions (i)–(ii)); the equality Y(Zp)^f = Z(Qp) is then verified, and the converse containment uses finiteness of fibres of fn, not a fitted parameter. The decisive finiteness input, Theorem D in the exceptional case dim(A)=1, r=1, rests on the external relative Manin–Mumford theorem of Bertrand–Masser–Pillay–Zannier [BMPZ16, Theorem 2(ii)], quoted as Theorem 5.14; this is an independent published result with no author overlap, so even if its hypotheses were not checked exhaustively for the constructed semiabelian scheme B (a possible correctness concern, not a circularity), the step does not reduce to the paper's own inputs. The self-citations in §4 ([Bet23a, Theorem 1.1] for local constancy, [Bet23b] for Selmer representability, [Bet24] for local heights, [BKL23] for applications) are to published, refereed work used as technical infrastructure; the load-bearing comparisons are proved in this paper, and no self-citation smuggles in the target equality. The one explicitly flagged gap, footnote 10 to Lemma 4.9, states that the tangential-base-point case of Olsson's crystallinity theorem 'the author has not carefully verified this', but immediately explains that the needed application (V a Gm-torsor over an abelian variety) avoids the issue via the principal-part construction of §2.2.1. Finally, Conjecture 1.2 is a framing device with 'motivic origin' deliberately undefined (Remark 1.3); Theorem A is unconditional and does not assume the conjecture. There are no fitted parameters renamed as predictions and no definitional equivalence between the claimed result and its hypotheses.

Assumptions & free parameters 0 free parameters · 7 assumptions · 2 invented entities

No free parameters are fitted to data. The constants in Theorem 6.13 (division polynomial residues, Delta^{1/12}, Kodaira-Tamagawa sets W^st_l) are computed, not chosen. The axioms are standard theorems plus the paper's own declared convention about motivic quotients.

assumptions (7)
  • ad hoc to paper Quotients realized by morphisms to smooth varieties are declared by fiat to be of motivic origin; Section 1.1 declares this and Remark 1.3 leaves 'motivic origin' deliberately undefined.
    Used only to interpret Theorem A' as a case of Conjecture 1.2. The unconditional geometric theorems do not depend on this definition.
  • standard math Faltings' Tate conjecture for H2 of abelian varieties (Fal83, Theorem (b)), invoked in Proposition 2.15 to identify U^p_AT as the largest Artin-Tate weight -2 quotient.
    Identifies the quotient realized by the quadratic Albanese map.
  • standard math Relative Manin-Mumford theorem of Bertrand-Masser-Pillay-Zannier (BMPZ16, Theorem 2(ii)), applied in Section 5.4 to the semiabelian scheme B over Y.
    Supplies the trichotomy controlling infinite torsion intersections in the exceptional dim(A)=1, r=1 case.
  • standard math Manin-Mumford for abelian and semiabelian varieties (Raynaud), used in Section 5.4 for dim(A)>=2, dim(A)=0, and dim(A)=1 with r>=2.
    Gives finiteness of intersections with torsion in those cases.
  • standard math Hain-Zucker mixed Hodge theory on Malcev Lie algebras (HZ87), used in Proposition 2.14 and Lemma 5.11.
    Controls surjectivity on pro-unipotent fundamental groups via quotients of tori and abelian varieties.
  • standard math Olsson's p-adic Hodge theory for path torsors and local constancy of pro-unipotent Kummer maps (Ols11, Bet23a), used in Section 4.
    Underpins Selmer schemes and the non-abelian Chabauty locus.
  • domain assumption The hypotheses of Theorem A': A has Mordell-Weil rank 0, dim(P)>=2, f induces a surjection on pro-unipotent fundamental groups, and p is a prime of good reduction for (Y,b).
    These are stated hypotheses of the theorem, not hidden assumptions.
invented entities (2)
  • AT varieties
    purpose: Universal targets realizing the quotients U^p_AT and U^p_T; the setting for unlikely intersections.
    Definition 2.2; a new mathematical class justified by the paper's theorems, with no external empirical handle.
  • Quadratic torsion packets
    purpose: Fibres of the map f_infinity to Q tensor P; used to define the finite subscheme Z as a finite union of packets.
    Definition 5.3; a bookkeeping notion introduced by the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On a non-abelian analogue of a conjecture of Michael Stoll." pith.science (2026). https://pith.science/paper/HN2FXTJB

@misc{pith2026250819947,
  author       = {Pith},
  title        = {Pith review of: On a non-abelian analogue of a conjecture of Michael Stoll},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HN2FXTJB}},
  note         = {Machine review of arXiv:2508.19947}
}
read the original abstract

We formulate a non-abelian generalisation of a conjecture of Stoll, which conjecturally describes the structure of the loci cut out by Kim's method of non-abelian Chabauty. We prove the rank 0 quadratic case of this conjecture, which in particular determines the structure of the quadratic Chabauty locus for once-punctured elliptic curves of rank 0. The proof involves using a variant of the geometric quadratic Chabauty method of Edixhoven and Lido to reduce to an unlikely intersections problem, and ultimately to known results about the relative Manin--Mumford Conjecture.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Equationless quadratic Chabauty for non-split Cartan modular curves

    math.NT 2026-08 conditional novelty 8.0 of 10

    The paper gives an equationless geometric quadratic Chabauty method for non-split Cartan modular curves and rederives X_ns^+(13)(Q), which consists of the 7 known CM points.

Reference graph

Works this paper leans on

54 extracted references · 47 canonical work pages · cited by 1 Pith paper

  1. [1]

    Quadratic C habauty for A tkin- L ehner quotients of modular curves of prime level and genus 4, 5, 6

    Nikola Ad z aga, Vishal Arul, Lea Beneish, Mingjie Chen, Shiva Chidambaram, Timo Keller, and Boya Wen. Quadratic C habauty for A tkin- L ehner quotients of modular curves of prime level and genus 4, 5, 6. Acta Arith. , 208(1):15--49, 2023

    work page 2023

  2. [2]

    Rational points on hyperelliptic A tkin- L ehner quotients of modular curves and their coverings

    Nikola Ad z aga, Shiva Chidambaram, Timo Keller, and Oana Padurariu. Rational points on hyperelliptic A tkin- L ehner quotients of modular curves and their coverings. Res. Number Theory , 8(4):Paper No. 87, 24, 2022

    work page 2022

  3. [3]

    Steffen M\"uller

    Vishal Arul and J. Steffen M\"uller. Rational points on X_0^+ ( 125 ) . Expo. Math. , 41(3):709--717, 2023

    work page 2023

  4. [4]

    J. S. Balakrishnan, A. J. Best, F. Bianchi, B. Lawrence, J. S. M\"uller, N. Triantafillou, and J. Vonk. Two recent p -adic approaches towards the (effective) M ordell conjecture. In Arithmetic L -functions and differential geometric methods , volume 338 of Progr. Math. , pages 31--74. Birkh\"auser/Springer, Cham, [2021] 2021

    work page 2021

  5. [5]

    Balakrishnan, Amnon Besser, Francesca Bianchi, and J

    Jennifer S. Balakrishnan, Amnon Besser, Francesca Bianchi, and J. Steffen M\"uller. Explicit quadratic C habauty over number fields. Israel J. Math. , 243(1):185--232, 2021

    work page 2021

  6. [6]

    Balakrishnan, Francesca Bianchi, Victoria Cantoral-Farf\'an, Mirela C iperiani, and Anastassia Etropolski

    Jennifer S. Balakrishnan, Francesca Bianchi, Victoria Cantoral-Farf\'an, Mirela C iperiani, and Anastassia Etropolski. Chabauty- C oleman experiments for genus 3 hyperelliptic curves. In Research directions in number theory--- W omen in N umbers IV , volume 19 of Assoc. Women Math. Ser. , pages 67--90. Springer, Cham, [2019] 2019

    work page 2019

  7. [7]

    Balakrishnan and Netan Dogra

    Jennifer S. Balakrishnan and Netan Dogra. Quadratic C habauty and rational points, I : p -adic heights. Duke Math. J. , 167(11):1981--2038, 2018. With an appendix by J. Steffen M\"uller

    1981

  8. [8]

    Balakrishnan, Ishai Dan-Cohen, Minhyong Kim, and Stefan Wewers

    Jennifer S. Balakrishnan, Ishai Dan-Cohen, Minhyong Kim, and Stefan Wewers. A non-abelian conjecture of T ate- S hafarevich type for hyperbolic curves. Math. Ann. , 372(1-2):369--428, 2018

    work page 2018

Show all 54 references
  1. [9]

    Steffen M\"uller, Jan Tuitman, and Jan Vonk

    Jennifer Balakrishnan, Netan Dogra, J. Steffen M\"uller, Jan Tuitman, and Jan Vonk. Explicit C habauty- K im for the split C artan modular curve of level 13. Ann. of Math. (2) , 189(3):885--944, 2019

    2019

  2. [10]

    Balakrishnan, Netan Dogra, J

    Jennifer S. Balakrishnan, Netan Dogra, J. Steffen M\"uller, Jan Tuitman, and Jan Vonk. Quadratic C habauty for modular curves: algorithms and examples. Compos. Math. , 159(6):1111--1152, 2023

    2023

  3. [11]

    Pink's conjecture on unlikely intersections and families of semi-abelian varieties

    Daniel Bertrand and Bas Edixhoven. Pink's conjecture on unlikely intersections and families of semi-abelian varieties. J. \'Ec. polytech. Math. , 7:711--742, 2020

    2020

  4. [12]

    Alexander Betts

    L. Alexander Betts. Local constancy of pro-unipotent K ummer maps. Proc. Lond. Math. Soc. (3) , 127(3):836--888, 2023

    2023

  5. [13]

    Alexander Betts

    L. Alexander Betts. Weight filtrations on S elmer schemes and the effective C habauty- K im method. Compos. Math. , 159(7):1531--1605, 2023

    2023

  6. [14]

    Alexander Betts

    L. Alexander Betts. The motivic anabelian geometry of local heights on abelian varieties. Mem. Amer. Math. Soc. , 302(1518):v+88, 2024

    2024

  7. [15]

    Heights in D iophantine geometry , volume 4 of New Mathematical Monographs

    Enrico Bombieri and Walter Gubler. Heights in D iophantine geometry , volume 4 of New Mathematical Monographs . Cambridge University Press, Cambridge, 2006

    2006

  8. [16]

    Quadratic C habauty for (bi)elliptic curves and K im's conjecture

    Francesca Bianchi. Quadratic C habauty for (bi)elliptic curves and K im's conjecture. Algebra Number Theory , 14(9):2369--2416, 2020

    2020

  9. [17]

    L -functions and T amagawa numbers of motives

    Spencer Bloch and Kazuya Kato. L -functions and T amagawa numbers of motives. In The G rothendieck F estschrift, V ol.\ I , volume 86 of Progr. Math. , pages 333--400. Birkh\"auser Boston, Boston, MA, 1990

    1990

  10. [18]

    Alexander Betts, Theresa Kumpitsch, and Martin L \"u dtke

    L. Alexander Betts, Theresa Kumpitsch, and Martin L \"u dtke. Chabauty--kim and the section conjecture for locally geometric sections, 2023. arXiv preprint, arXiv:2305.09462v1

    2023 arXiv

  11. [19]

    Alexander Betts and Daniel Litt

    L. Alexander Betts and Daniel Litt. Semisimplicity of the F robenius action on _1 . In p -adic H odge theory, singular varieties, and non-abelian aspects , Simons Symp., pages 17--64. Springer, Cham, [2023] 2023

    2023

  12. [20]

    Bertrand, D

    D. Bertrand, D. Masser, A. Pillay, and U. Zannier. Relative M anin- M umford for semi- A belian surfaces. Proc. Edinb. Math. Soc. (2) , 59(4):837--875, 2016

    2016

  13. [21]

    Rational points on rank 2 genus 2 bielliptic curves in the LMFDB

    Francesca Bianchi and Oana Padurariu. Rational points on rank 2 genus 2 bielliptic curves in the LMFDB . In Lu C a NT : LMFDB , computation, and number theory , volume 796 of Contemp. Math. , pages 215--242. Amer. Math. Soc., [Providence], RI, [2024] 2024

    2024

  14. [22]

    Th\'eorie des groupes de L ie

    Claude Chevalley. Th\'eorie des groupes de L ie. T ome III . T h\'eor\`emes g\'en\'eraux sur les alg\`ebres de L ie , volume No. 1226 of Actualit\'es Scientifiques et Industrielles [Current Scientific and Industrial Topics] . Hermann & Cie, Paris, 1955

    1955

  15. [23]

    Pavel C oupek, David T.-B. G. Lilienfeldt, Luciena X. Xiao, and Zijian Yao. Geometric quadratic C habauty over number fields. Trans. Amer. Math. Soc. , 376(4):2573--2613, 2023

    2023

  16. [24]

    J. E. Cremona, M. Prickett, and Samir Siksek. Height difference bounds for elliptic curves over number fields. J. Number Theory , 116(1):42--68, 2006

    2006

  17. [25]

    P. Deligne. Le groupe fondamental de la droite projective moins trois points. In Galois groups over Q ( B erkeley, CA , 1987) , volume 16 of Math. Sci. Res. Inst. Publ. , pages 79--297. Springer, New York, 1989

    1987

  18. [26]

    Quadratic C habauty for modular curves and modular forms of rank one

    Netan Dogra and Samuel Le Fourn. Quadratic C habauty for modular curves and modular forms of rank one. Math. Ann. , 380(1-2):393--448, 2021

    2021

  19. [27]

    Geometric quadratic C habauty and p -adic heights

    Juanita Duque-Rosero, Sachi Hashimoto, and Pim Spelier. Geometric quadratic C habauty and p -adic heights. Expo. Math. , 41(3):631--674, 2023

    2023

  20. [28]

    Geometric quadratic C habauty

    Bas Edixhoven and Guido Lido. Geometric quadratic C habauty. J. Inst. Math. Jussieu , 22(1):279--333, 2023

    2023

  21. [29]

    atze f\"ur abelsche V ariet\

    G. Faltings. Endlichkeitss\"atze f\"ur abelsche V ariet\"aten \"uber Z ahlk\"orpern. Invent. Math. , 73(3):349--366, 1983

    1983

  22. [30]

    Repr\'esentations p -adiques semi-stables

    Jean-Marc Fontaine. Repr\'esentations p -adiques semi-stables. Number 223, pages 113--184. 1994. With an appendix by Pierre Colmez, P\'eriodes p -adiques (Bures-sur-Yvette, 1988)

    1994

  23. [31]

    Principles of algebraic geometry

    Phillip Griffiths and Joseph Harris. Principles of algebraic geometry . Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978

    1978

  24. [32]

    Chabauty- C oleman computations on rank 1 P icard curves

    Sachi Hashimoto and Travis Morrison. Chabauty- C oleman computations on rank 1 P icard curves. In Arithmetic geometry, number theory, and computation , Simons Symp., pages 485--506. Springer, Cham, [2021] 2021

    2021

  25. [33]

    A geometric linear C habauty comparison theorem

    Sachi Hashimoto and Pim Spelier. A geometric linear C habauty comparison theorem. Acta Arith. , 202(1):67--88, 2022

    2022

  26. [34]

    Hain and Steven Zucker

    Richard M. Hain and Steven Zucker. Unipotent variations of mixed H odge structure. Invent. Math. , 88(1):83--124, 1987

    1987

  27. [35]

    The motivic fundamental group of P^1 \ 0,1, \ and the theorem of S iegel

    Minhyong Kim. The motivic fundamental group of P^1 \ 0,1, \ and the theorem of S iegel. Invent. Math. , 161(3):629--656, 2005

    2005

  28. [36]

    The unipotent A lbanese map and S elmer varieties for curves

    Minhyong Kim. The unipotent A lbanese map and S elmer varieties for curves. Publ. Res. Inst. Math. Sci. , 45(1):89--133, 2009

    2009

  29. [37]

    Tangential localization for S elmer varieties

    Minhyong Kim. Tangential localization for S elmer varieties. Duke Math. J. , 161(2):173--199, 2012

    2012

  30. [38]

    The l -component of the unipotent A lbanese map

    Minhyong Kim and Akio Tamagawa. The l -component of the unipotent A lbanese map. Math. Ann. , 340(1):223--235, 2008

    2008

  31. [39]

    Elliptic curves: D iophantine analysis , volume 231 of Grundlehren der Mathematischen Wissenschaften

    Serge Lang. Elliptic curves: D iophantine analysis , volume 231 of Grundlehren der Mathematischen Wissenschaften . Springer-Verlag, Berlin-New York, 1978

    1978

  32. [40]

    Sur les rev\^etements non ramifi\'es des vari\'et\'es alg\'ebriques

    Serge Lang and Jean-Pierre Serre. Sur les rev\^etements non ramifi\'es des vari\'et\'es alg\'ebriques. Amer. J. Math. , 79:319--330, 1957

    1957

  33. [41]

    J. S. Milne. \'Etale cohomology , volume 33 of Princeton Mathematical Series . Princeton University Press, Princeton, NJ, [2025] 1980

    2025

  34. [42]

    Abelian varieties , volume 5 of Tata Institute of Fundamental Research Studies in Mathematics

    David Mumford. Abelian varieties , volume 5 of Tata Institute of Fundamental Research Studies in Mathematics . Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the...

    2008

  35. [43]

    Martin C. Olsson. Towards non-abelian p -adic H odge theory in the good reduction case. Mem. Amer. Math. Soc. , 210(990):vi+157, 2011

    2011

  36. [44]

    F. Oort. Commutative group schemes , volume 15 of Lecture Notes in Mathematics . Springer-Verlag, Berlin-New York, 1966

    1966

  37. [45]

    M. Raynaud. Courbes sur une vari\'et\'e ab\'elienne et points de torsion. Invent. Math. , 71(1):207--233, 1983

    1983

  38. [46]

    O-minimal geometry of higher A lbanese manifolds, 2025

    Vasily Rogov. O-minimal geometry of higher A lbanese manifolds, 2025. arXiv preprint, arXiv:2505.07632v2

    2025 arXiv

  39. [47]

    Algebraic groups and class fields , volume 117 of Graduate Texts in Mathematics

    Jean-Pierre Serre. Algebraic groups and class fields , volume 117 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1988. Translated from the French

    1988

  40. [48]

    Galois cohomology

    Jean-Pierre Serre. Galois cohomology . Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2002. Translated from the French by Patrick Ion and revised by the author

    2002

  41. [49]

    Silverman

    Joseph H. Silverman. Computing heights on elliptic curves. Math. Comp. , 51(183):339--358, 1988

    1988

  42. [50]

    Silverman

    Joseph H. Silverman. The arithmetic of elliptic curves , volume 106 of Graduate Texts in Mathematics . Springer, Dordrecht, second edition, 2009

    2009

  43. [51]

    On the birational section conjecture with local conditions

    Jakob Stix. On the birational section conjecture with local conditions. Invent. Math. , 199(1):239--265, 2015

    2015

  44. [52]

    Finite descent and rational points

    Michael Stoll. Finite descent and rational points. Unpublished preprint version available at http://www.mathe2.uni-bayreuth.de/stoll/papers/covers8.pdf

  45. [53]

    Finite descent obstructions and rational points on curves

    Michael Stoll. Finite descent obstructions and rational points on curves. Algebra Number Theory , 1(4):349--391, 2007

    2007

  46. [54]

    Galois groups and fundamental groups , volume 117 of Cambridge Studies in Advanced Mathematics

    Tam\'as Szamuely. Galois groups and fundamental groups , volume 117 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2009

    2009

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.