Pith. sign in

REVIEW 2 major objections 6 minor 1 cited by

The Torelli locus and Newton polygons

T0 review · 2 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read For every prime p, this paper constructs supersingular curves of genus δp(p−1)/2 for infinitely many δ, partially answering the existence question for supersingular curves.

desk verdict Solid survey that is worth a referee, but the paper's only new result—the corrected genus formula—has a real bug in the proof when the base-p expansion of δ has consecutive 1s. read the letter →

arxiv 2509.00998 v1 pith:PNRU4QLO submitted 2025-08-31 math.AG math.NT

Rachel Pries This is my paper
classification math.AGmath.NT MSC 11G1014H1014H4014K1011G1811G2011M3814K15
keywords supersingularcurvesArtin–SchreiercoversJacobianisogenyNewtonpolygonTorellilocusp-rankpositivecharacteristicmoduliof
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

For every prime p, this paper produces supersingular curves of genus δp(p−1)/2 whenever the base-p expansion of δ uses only the digits 0 and 1. Supersingular means the Jacobian has the most degenerate possible Frobenius action: every slope of its Newton polygon is 1/2. Since there are infinitely many such δ for each p, the construction yields infinite families of supersingular curves in every characteristic, and it partially answers the longstanding question of whether such curves exist for every genus and prime. The proof works by gluing Artin–Schreier covers into a fiber product and transferring supersingularity from the pieces to the whole Jacobian by isogeny. The same result corrects an earlier published genus formula, which had forgotten to projectivize the parameter space.

What carries the argument

The central mechanism is the fiber product of Artin–Schreier covers. An Artin–Schreier cover is the degree-p cyclic cover of P^1 given by y^p − y = h(x); here h is x f with f an additive polynomial. The paper assembles, for all f in the projectivized F_p-vector space P(L), the covers C_f and their fiber product Y over P^1. The load-bearing identity is the isogeny decomposition Jac(Y) ≃ ⊕_{f∈P(L)} Jac(C_f), cited as [KR89, Theorem B]. It transfers the supersingularity of each component (supplied by an additive-polynomial theorem for Artin–Schreier curves, [vdGvdV92, Theorem 13.7]) to the whole Jacobian. The delicate part is the count of projective classes of f with specified degrees, which co

What would settle it

Take p=3 and δ=4 (base-3 digits 11), so the paper predicts genus 12. Construct the two-dimensional F_3-space L of additive polynomials used in the proof, form the fiber product Y of the four covers y^3−y = x f over P^1, and compute the genus of the smooth model by Riemann–Hurwitz; if the result is not 12, the projectivized count is wrong. Alternatively, compute the zeta function of Y over F_3 and check that every reciprocal root has p-adic valuation 1/2; a single slope different from 1/2, or a p-rank above 0, would refute the claim.

Watch

Extended reading notes

Core claim

The paper's own central claim is Corollary 4.22. Let p be prime and let δ be an integer whose base-p expansion has only 0 and 1 as digits; then, for g = δp(p−1)/2, there exists a smooth supersingular curve of genus g over a finite field of characteristic p. The construction chooses an F_p-vector space L of additive polynomials, builds the degree-p Artin–Schreier covers C_f : y^p − y = x f for each nonzero f in L up to F_p^*-scaling, and forms the fiber product Y over P^1. An isogeny theorem decomposes Jac(Y) as the direct sum of the Jac(C_f); each piece is supersingular because f is additive, and supersingularity is preserved under isogeny. The genus is obtained by counting projective classe

Load-bearing premise

The isogeny decomposition that identifies the Jacobian of the fiber product with the direct sum of the Jacobians of the individual Artin–Schreier covers is the load-bearing premise; if it fails for this particular fiber product, the genus count and the transfer of supersingularity collapse.

Editorial extensions

If this is right

  • For each fixed prime p, infinitely many genera g now have a known supersingular curve, because the digit condition on δ allows infinitely many values.
  • When p=2 the digit condition is vacuous, so the construction recovers the classical fact that supersingular curves exist in every genus in characteristic 2.
  • The result gives a partial positive answer to the open question of existence of supersingular curves for all genera and primes: the known genera form an infinite, sparse set inside every characteristic.
  • The earlier corollary that motivated the construction must be replaced: its genus δp(p−1)^2/2 is corrected to δp(p−1)/2, changing the statement and the resulting count.
  • The curves produced come with an explicit isogeny class: each Jacobian is isogenous to a product of known supersingular Artin–Schreier Jacobians, so the construction is not just an existence proof.

Reading between the lines

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

  • A testable extension is to choose the additive-polynomial space L so that the component curves have Newton polygons other than the supersingular one; if the same isogeny decomposition behaves well, the construction would yield curves with prescribed mixed slopes.
  • The correction suggests that anywhere else in the survey a parameter space carries an F_p^*-scaling action, dimensions should be computed after projectivization; dropping that quotient is a natural source of genus overcounts.
  • The family only reaches genera of the form δp(p−1)/2 with digit-restricted δ; closing the gap to all genera would require genuinely new families of covers, so this result is a lower bound on the flexibility needed rather than the end of the question.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This survey, based on the author's 2024 Arizona Winter School lectures, covers the geometry of the Torelli locus and the arithmetic invariants of Jacobians in positive characteristic. It treats abelian varieties and p-torsion invariants, existence results for curves with prescribed Newton polygons and Ekedahl–Oort types, moduli spaces and their boundaries, cyclic covers, and ℓ-adic monodromy. The manuscript is carefully referenced and explicitly corrects a prior error in [Pri19, Corollary 2.6]. Its main original contribution is Corollary 4.22, which asserts that for every δ whose base-p expansion has only digits 0 and 1, the integer g=δp(p−1)/2 is the genus of a supersingular curve over a finite field of characteristic p. The proof constructs a fiber product Y of Artin–Schreier curves from an F_p-vector space L of additive polynomials, computes its genus by counting projective classes, and transfers supersingularity via Kani–Rosen. I agree with the stress-test concern: the genus computation is incorrect for the substantial subrange where the base-p expansion of δ contains adjacent 1s. The proof works only when all blocks have length 1; the other expository sections are largely sound.

Significance. Strengths: the survey is comprehensive, well-organized, and draws on a wide literature; it gives explicit statements of open problems; it presents fully worked proofs for representative results (e.g., Theorem 5.23, Theorem 6.18 sketch); and it acknowledges earlier mistakes. If Corollary 4.22 were correct, it would provide infinite families of supersingular curves in every characteristic, partially answering Question 4.23. However, the flaw identified below means the corollary in its stated generality is not established by the given proof. The survey's overall value as an exposition survives this defect: the other existence results (Theorem 4.19, Theorem 4.21, Theorem 4.16, Theorem 6.16) are independent of the flawed step, and the manuscript remains a useful reference for the Torelli locus and Newton polygon literature. The paper's emphasis on the author's own recent work is understandable in a survey, but the status of several cited preprints should be clarified.

major comments (2)
  1. [Section 4.6.4, proof of Corollary 4.22] The genus of the fiber product is overcounted when the base-p expansion of δ contains a block of two or more consecutive 1s (r_i≥1). The claim that f with a nonzero component in L_i and no components in L_j for j>i satisfies g_{C_f}=p^{u_i}(p−1)/2 requires deg(f)=p^{u_i}. But L_i is a d_i-dimensional subspace (d_i=r_i+1) of the additive polynomials of degree at most p^{u_i}; for d_i≥2, L_i necessarily contains nonzero elements of degree <p^{u_i}, whose curves have smaller genus. Example p=2, δ=3: the construction forces L_1=span{x,x^2}, so P(L) has classes [x],[x^2],[x^2+x]; the corresponding curves have genera 0,1,1 (y^2+y=x^2 is irreducible but rational), summing to 2, not δp(p−1)/2=3. For p=3, δ=4, the sum is 10, not 12. Thus the proof establishes the result only for δ with no adjacent 1s (all r_i=0).
  2. [Section 4.6.4, isogeny step] The transfer of supersingularity from the individual C_f to the fiber product Y depends on the unstated hypotheses of [KR89, Theorem B]. The author should state the theorem used and verify that Jac(Y) is isogenous to ⊕_{f∈P(L)} Jac(C_f) with no additional isogeny factors, especially when some C_f is rational (e.g., p=2, f=x). This step is load-bearing for the conclusion 'Jac(Y) is supersingular', though secondary to the genus error above.
minor comments (6)
  1. [Theorem 4.21 (Section 4.6.3)] The formula p^h(p−1)/2 is non-integral for h=0 and p=2 (the curve y^2+y=x^2 has genus 0). The statement should require h≥1, or include the p=2 caveat for the degree-1 additive polynomial R(x)=x.
  2. [Section 4.6.4] The phrase 'the vector subspace of F_p[x] of additive polynomials of degree p^{u_i}' is ambiguous: it must mean 'degree at most p^{u_i}', since the space of exact degree p^{u_i} has dimension 1 and cannot contain a subspace of dimension d_i≥2.
  3. [Theorem 9.4 (Section 9.3)] Typo: 'then S has big ℓ-adic monodromy' should read 'then W has big ℓ-adic monodromy'.
  4. [Throughout] Typos: Section 4.3 'V ershiebung' → 'Verschiebung'; Remark 4.13 'field field' → 'field'; Section 2.3.3 'sympletic group' → 'symplectic group'.
  5. [Remark 6.19, Section 8.4.2] The manuscript explicitly suppresses technical details and full statements in these places; this is acceptable for a survey but the reader should be directed to the original papers (e.g., [FvdG04], [AP08], [LMPT22]) for the complete hypotheses.
  6. [Corollary 6.27 and related citations] Several results rely on works marked 'in progress' or arXiv preprints, including [DP], [BPa], [BPb], [Pri25], [Draa], [Drab], and [KYY21]. For a journal version, the status of each citation (published, preprint, in preparation) should be indicated.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: Corollary 4.22 and the survey's central claims rest on independent external theorems; self-citations are not load-bearing.

full rationale

The manuscript is a survey/lecture-note. The one new-looking result, Corollary 4.22, is derived explicitly from two independent external inputs: Theorem 4.21 (quoted from [vdGvdV92, Thm 13.7], also [Bla12] and [BHM+16]) for the supersingularity of Artin-Schreier curves y^p-y=xR(x), and Kani-Rosen [KR89, Theorem B] for the Jacobian isogeny of the fiber product. The genus computation is a direct summation of genera of the C_f; no parameter is fitted to a subset of data and then renamed as a prediction, and no equation is used to define its own conclusion. The author's self-citations ([Pri19], [Pri25], [AP08], [LMPT19a], [LMPT22]) report previously published results or in-progress work and are not used as the sole justification for the main derivation; where they appear, the same statements are typically anchored in external results (e.g., [FvdG04], [KHS20], [Oor91], [vdGvdV92], [KR89]). The review's concern about the proof of Corollary 4.22 (possible overcount when the base-p expansion of δ has adjacent 1s) is a mathematical correctness issue, not a circularity: the proof does not assume the genus it claims to compute. The score 2 reflects the presence of minor self-citations that are not load-bearing; there are no circular steps.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The survey rests entirely on prior literature; no constants are fitted to data. With the paper's own exception of the corrected genus formula, every claim is an imported theorem, and the ledger records the imports the new proof depends on most directly.

assumptions (6)
  • domain assumption [KR89, Theorem B]: the Jacobian of the fiber product Y = ∏_{f∈P(L)} C_f over P^1 is isogenous to ⊕_{f∈P(L)} Jac(C_f)
    Invoked directly in the proof of Corollary 4.22 (Section 4.6.4) as 'By [KR89, Theorem B]'; load-bearing for the genus computation and for supersingularity of Jac(Y).
  • domain assumption [vdGvdV92, Theorem 13.7]: curves y^p − y = xR(x) with additive R(x) of degree p^h are supersingular of genus p^h(p−1)/2
    Used to assert Jac(C_f) is supersingular for each f; also quoted as Theorem 4.21; the genus and supersingularity come from this external theorem.
  • domain assumption Grothendieck-Katz specialization and de Jong-Oort purity of Newton polygons (Theorems 6.2, 6.3)
    Core to the codimension and dimension arguments in Chapters 6 and 8, e.g., the proofs of Theorem 6.16 and Theorem 6.18.
  • domain assumption Small-genus existence results: ordinary curves of all genus (Miller 1972), supersingular genus 2 (Serre), genus 3 (Oort), genus 4 (KHS20)
    Reported as background facts in Chapter 4; the survey does not reprove them.
  • standard math Wild Riemann-Hurwitz and Deuring-Shafarevich formulas for Artin-Schreier covers
    Used for genus and p-rank computations of Artin-Schreier curves in Sections 2.4.4 and 4.6.
  • standard math Dieudonné-Manin classification and symmetric BT1 classification (Oort, Moonen)
    Underlies the definitions of Newton polygon and Ekedahl-Oort type in Chapter 3.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Torelli locus and Newton polygons." pith.science (2026). https://pith.science/paper/PNRU4QLO

@misc{pith2026250900998,
  author       = {Pith},
  title        = {Pith review of: The Torelli locus and Newton polygons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PNRU4QLO}},
  note         = {Machine review of arXiv:2509.00998}
}
read the original abstract

This manuscript is about abelian varieties that are Jacobians of curves. I started writing it for a lecture series at the Arizona Winter School in 2024 on abelian varieties. A longer more descriptive title might be: The Torelli locus in the moduli space of abelian varieties, with applications to Newton polygons of curves in positive characteristic. To elaborate, this manuscript covers two topics: the first is about the geometry of the Torelli locus; the second is about the arithmetic invariants of abelian varieties that occur for Jacobians of smooth curves in positive characteristic.

Discussion (0). Sign in 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. Generic ordinarity for abelian coverings of the projective line

    math.AG 2026-07 conditional novelty 6.0 of 10

    The generic Newton polygon of any prime-to-p abelian cover of P^1 equals the μ-ordinary polygon of the smallest Shimura variety containing its Torelli image.

Reference graph

Works this paper leans on

195 extracted references · 77 canonical work pages · cited by 1 Pith paper

  1. [1]

    \' A lvarez, The p -rank of the reduction mod \, p of J acobians and J acobi sums , Int

    A. \' A lvarez, The p -rank of the reduction mod \, p of J acobians and J acobi sums , Int. J. Number Theory 10 (2014), no. 8, 2097--2114. 3273477

    2014

  2. [2]

    Griffiths, Geometry of algebraic curves

    Enrico Arbarello, Maurizio Cornalba, and Phillip A. Griffiths, Geometry of algebraic curves. V olume II , Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 268, Springer, Heidelberg, 2011, With a contribution by Joseph Daniel Harris. 2807457

    2011

  3. [3]

    Arbarello, M

    E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. V ol. I , Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. 770932

    1985

  4. [4]

    Achter, Darren Glass, and Rachel Pries, Curves of given p -rank with trivial automorphism group , Michigan Math

    Jeffrey D. Achter, Darren Glass, and Rachel Pries, Curves of given p -rank with trivial automorphism group , Michigan Math. J. 56 (2008), no. 3, 583--592. 2490647

    2008

  5. [5]

    Achter and Everett W

    Jeffrey D. Achter and Everett W. Howe, Hasse- W itt and C artier- M anin matrices: a warning and a request , Arithmetic geometry: computation and applications, Contemp. Math., vol. 722, Amer. Math. Soc., [Providence], RI, [2019] 2019, pp. 1--18. 3896846

    2019

  6. [6]

    Avritzer and H

    D. Avritzer and H. Lange, The moduli spaces of hyperelliptic curves and binary forms, Math. Z. 242 (2002), no. 4, 615--632. 1981190

    2002

  7. [7]

    Andreotti and A

    A. Andreotti and A. L. Mayer, On period relations for abelian integrals on algebraic curves, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 189--238. 220740

    1967

  8. [8]

    Noboru Aoki, On the zeta function of some cyclic quotients of F ermat curves , Comment. Math. Univ. St. Pauli 57 (2008), no. 2, 163--185. 2492579

    2008

Show all 195 references
  1. [9]

    Achter and Rachel Pries, Monodromy of the p -rank strata of the moduli space of curves , Int

    Jeffrey D. Achter and Rachel Pries, Monodromy of the p -rank strata of the moduli space of curves , Int. Math. Res. Not. IMRN (2008), no. 15, Art. ID rnn053, 25. 2438069

    2008

  2. [10]

    , The p -rank strata of the moduli space of hyperelliptic curves , Adv. Math. 227 (2011), no. 5, 1846--1872. 2803789

    2011

  3. [11]

    , Generic N ewton polygons for curves of given p -rank , Algebraic curves and finite fields, Radon Ser. Comput. Appl. Math., vol. 16, De Gruyter, Berlin, 2014, pp. 1--21. 3287680

    2014

  4. [12]

    Baker, Cartier points on curves, Internat

    Matthew H. Baker, Cartier points on curves, Internat. Math. Res. Notices (2000), no. 7, 353--370. 1749740 (2001g:11096)

    2000

  5. [13]

    3, 587--641

    Jeremy Booher and Bryden Cais, a -numbers of curves in A rtin- S chreier covers , Algebra Number Theory 14 (2020), no. 3, 587--641. 4113776

    2020

  6. [14]

    Thomas Bouchet, Erik Davis, Steven Groen, Zachary Porat, and Benjamin York, Heuristics for (ir)reducibility of p -rank strata of the moduli space of hyperelliptic curves, https://arxiv.org/abs/2506.06457

    arXiv

  7. [15]

    Arnaud Beauville, Prym varieties: a survey, Theta functions--- B owdoin 1987, P art 1 ( B runswick, ME , 1987), Proc. Sympos. Pure Math., vol. 49, Part 1, Amer. Math. Soc., Providence, RI, 1989, pp. 607--620. 1013156

    1987

  8. [16]

    Women Math

    Irene Bouw, Wei Ho, Beth Malmskog, Renate Scheidler, Padmavathi Srinivasan, and Christelle Vincent, Zeta functions of a class of A rtin- S chreier curves with many automorphisms , Directions in number theory, Assoc. Women Math. Ser., vol. 3, Springer, [Cham], 2016, pp. 87--124...

    2016

  9. [17]

    302, Springer-Verlag, Berlin, 2004

    Christina Birkenhake and Herbert Lange, Complex abelian varieties, second ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. 2062673

    2004

  10. [18]

    Number Theory 132 (2012), no

    R\'egis Blache, Valuation of exponential sums and the generic first slope for A rtin- S chreier curves , J. Number Theory 132 (2012), no. 10, 2336--2352. 2944758

    2012

  11. [19]

    21, Springer-Verlag, Berlin, 1990

    Siegfried Bosch, Werner L \"u tkebohmert, and Michel Raynaud, N\'eron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. 1045822 (91i:14034)

    1990

  12. [20]

    Bouw, The p -rank of ramified covers of curves , Compositio Math

    Irene I. Bouw, The p -rank of ramified covers of curves , Compositio Math. 126 (2001), no. 3, 295--322. 1834740 (2002e:14045)

    2001

  13. [21]

    Jeremy Booher and Rachel Pries, Producing supersingular curves of genus five, https://arxiv.org/abs/2410.20262

  14. [22]

    , Supersingular curves via the S himura-- T aniyama method , https://arxiv.org/abs/2503.04878

    arXiv

  15. [23]

    Pierre Cartier, Une nouvelle op\'eration sur les formes diff\'erentielles, C. R. Acad. Sci. Paris 244 (1957), 426--428. 0084497 (18,870b)

    1957

  16. [24]

    195, American Mathematical Society, Providence, RI, 2014

    Ching-Li Chai, Brian Conrad, and Frans Oort, Complex multiplication and lifting problems, Mathematical Surveys and Monographs, vol. 195, American Mathematical Society, Providence, RI, 2014. 3137398

    2014

  17. [25]

    Kevin Chang, Dusan Dragutinovic, Steven Groen, Yuxin Lin Lin, Natalia Pacheco-Tallaj, and Deepesh Singhal, The p -rank stratification of the moduli space of double covers of a fixed elliptic curve, https://arxiv.org/abs/2502.09540

  18. [26]

    Herbert Clemens and Phillip A

    C. Herbert Clemens and Phillip A. Griffiths, The intermediate J acobian of the cubic threefold , Ann. of Math. (2) 95 (1972), 281--356. 302652

    1972

  19. [27]

    Ching-Li Chai, Monodromy of H ecke-invariant subvarieties , Pure Appl. Math. Q. 1 (2005), no. 2, 291--303. 2194726

    2005

  20. [28]

    Daebeom Choi, Complete subvarieties of M _ g,n and a lifting problem , https://arxiv.org/abs/2304.08568

    arXiv

  21. [29]

    Ching-Li Chai and Frans Oort, Monodromy and irreducibility of leaves, Ann. of Math. (2) 173 (2011), no. 3, 1359--1396. 2800716

    2011

  22. [30]

    , Abelian varieties isogenous to a J acobian , Ann. of Math. (2) 176 (2012), no. 1, 589--635. 2925391

    2012

  23. [31]

    Alberto Collino, A new proof of the R an- M atsusaka criterion for J acobians , Proc. Amer. Math. Soc. 92 (1984), no. 3, 329--331. 759646

    1984

  24. [32]

    Coleman, Torsion points on curves, Galois representations and arithmetic algebraic geometry ( K yoto, 1985/ T okyo, 1986), Adv

    Robert F. Coleman, Torsion points on curves, Galois representations and arithmetic algebraic geometry ( K yoto, 1985/ T okyo, 1986), Adv. Stud. Pure Math., vol. 12, North-Holland, Amsterdam, 1987, pp. 235--247. 948246

    1985

  25. [33]

    176 (2025), no

    Renzo Cavalieri and Rachel Pries, Mass formula for non-ordinary curves in one dimensional families, Manuscripta Math. 176 (2025), no. 1, Paper No. 10, 30. 4852257

    2025

  26. [34]

    Bryden Cais and Douglas Ulmer, p -torsion for unramified A rtin-- S chreier covers of curves , https://arxiv.org/abs/2307.16346

    arXiv

  27. [35]

    11, American Mathematical Society, Providence, RI; Soci\'et\'e Math\'ematique de France, Paris, 2005

    Olivier Debarre, Complex tori and abelian varieties, french ed., SMF/AMS Texts and Monographs, vol. 11, American Mathematical Society, Providence, RI; Soci\'et\'e Math\'ematique de France, Paris, 2005. 2158864

    2005

  28. [36]

    Max Deuring, Die T ypen der M ultiplikatorenringe elliptischer F unktionenk\"orper , Abh. Math. Sem. Univ. Hamburg 14 (1941), no. 1, 197--272. 3069722

    1941

  29. [37]

    Olivier de Gaay Fortman and Stefan Schreieder, Abelian varieties with no power isogenous to a J acobian , https://arxiv.org/pdf/2401.06577

  30. [38]

    Steven Diaz, A bound on the dimensions of complete subvarieties of M _ g , Duke Math. J. 51 (1984), no. 2, 405--408. 747872 (85j:14042)

    1984

  31. [39]

    , Complete subvarieties of the moduli space of smooth curves, Algebraic geometry, B owdoin, 1985 ( B runswick, M aine, 1985), Proc. Sympos. Pure Math., vol. 46, Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 77--81. 927950

    1985

  32. [40]

    Math., vol

    Johan de Jong and Rutger Noot, Jacobians with complex multiplication, Arithmetic algebraic geometry ( T exel, 1989), Progr. Math., vol. 89, Birkh\"auser Boston, Boston, MA, 1991, pp. 177--192. 1085259

    1989

  33. [41]

    A. J. de Jong and F. Oort, Purity of the stratification by N ewton polygons , J. Amer. Math. Soc. 13 (2000), no. 1, 209--241. 1703336

    2000

  34. [42]

    Deligne and D

    P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes \' E tudes Sci. Publ. Math. (1969), no. 36, 75--109. 262240

    1969

  35. [43]

    Deligne and G

    P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes \'Etudes Sci. Publ. Math. (1986), no. 63, 5--89. 849651

    1986

  36. [44]

    Dusan Dragutinovi\'c and Rachel Pries, Newton polygons and a -numbers for curves of p -rank 0 , in progress

  37. [45]

    Dusan Dragutinovi\'c, E kedahl-- O ort types of stable curves , https://arxiv.org/abs/2307.13445

  38. [46]

    , Oort's conjecture and automorphisms of supersingular curves of genus four, arxiv.org/abs/2405.01282

    arXiv

  39. [47]

    , Supersingular curves of genus four in characteristic two, Proc. Amer. Math. Soc. 152 (2024), no. 6, 2333--2347. 4741231

    2024

  40. [48]

    Elkies, Everett W

    Noam D. Elkies, Everett W. Howe, and Christophe Ritzenthaler, Genus bounds for curves with fixed F robenius eigenvalues , Proc. Amer. Math. Soc. 142 (2014), no. 1, 71--84. 3119182

    2014

  41. [49]

    Torsten Ekedahl, On supersingular curves and abelian varieties, Math. Scand. 60 (1987), no. 2, 151--178. 914332

    1987

  42. [50]

    Math., vol

    , Boundary behaviour of H urwitz schemes , The moduli space of curves ( T exel I sland, 1994), Progr. Math., vol. 129, Birkh\"auser Boston, Boston, MA, 1995, pp. 173--198. 1363057

    1994

  43. [51]

    Algebra 327 (2011), 1--12

    Arsen Elkin, The rank of the C artier operator on cyclic covers of the projective line , J. Algebra 327 (2011), 1--12. 2746026

    2011

  44. [52]

    3, 507--532, arXiv:1007.1226

    Arsen Elkin and Rachel Pries, Ekedahl-- O ort strata of hyperelliptic curves in characteristic 2 , Algebra Number Theory 7 (2013), no. 3, 507--532, arXiv:1007.1226. 3095219

    2013 arXiv

  45. [53]

    Torsten Ekedahl and Jean-Pierre Serre, Exemples de courbes alg\' e briques \`a jacobienne compl\`etement d\' e composable , C. R. Acad. Sci. Paris S\' e r. I Math. 317 (1993), no. 5, 509--513. 1239039

    1993

  46. [54]

    Torsten Ekedahl and Gerard van der Geer, Cycle classes of the E - O stratification on the moduli of abelian varieties , Algebra, arithmetic, and geometry: in honor of Y u. I . M anin. V ol. I , Progr. Math., vol. 269, Birkh\"auser Boston Inc., Boston, MA, 2009, pp. 567--636. 2...

    2009

  47. [55]

    Bas Edixhoven, Gerald van der Geer, and Ben Moonen, Abelian varieties, http://van-der-geer.nl/ gerard/AV.pdf

  48. [56]

    Benson Farb, Irrationality of the general smooth quartic 3-fold using intermediate J acobians , Adv. Math. 465 (2025), Paper No. 110160, 6. 4864700

    2025

  49. [57]

    22, Springer-Verlag, Berlin, 1990, With an appendix by David Mumford

    Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990, With an appendix by David Mumford. 1083353

    1990

  50. [58]

    Paola Frediani, Alessandro Ghigi, and Matteo Penegini, Shimura varieties in the T orelli locus via G alois coverings , Int. Math. Res. Not. IMRN (2015), no. 20, 10595--10623. 3455876

    2015

  51. [59]

    13, Springer-Verlag, Berlin, 1988, Translated from the German by Betty S

    Eberhard Freitag and Reinhardt Kiehl, \'etale cohomology and the W eil conjecture , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 13, Springer-Verlag, Berlin, 1988, Translated from the German by Betty S. Waterhouse an...

    1988

  52. [60]

    William Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math. (2) 90 (1969), 542--575. 0260752

    1969

  53. [61]

    Reine Angew

    Carel Faber and Gerard van der Geer, Complete subvarieties of moduli spaces and the P rym map , J. Reine Angew. Math. 573 (2004), 117--137. 2084584

    2004

  54. [62]

    Gabber, On space filling curves and A lbanese varieties , Geom

    O. Gabber, On space filling curves and A lbanese varieties , Geom. Funct. Anal. 11 (2001), no. 6, 1192--1200. 1878318

    2001

  55. [63]

    Harvey, On complete curves in moduli space

    Gabino Gonz\' a lez D\' ez and William J. Harvey, On complete curves in moduli space. I , II , Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 3, 461--466, 467--472. 1120481

    1991

  56. [64]

    Samuel Grushevsky, Gabriele Mondello, Riccardo Salvati Manni, and Jakob Tsimerman, Compact subvarieties of the moduli space of complex abelian varieties, https://arxiv.org/abs/2404.06009

  57. [65]

    Goren, Lectures on H ilbert modular varieties and modular forms , CRM Monograph Series, vol

    E. Goren, Lectures on H ilbert modular varieties and modular forms , CRM Monograph Series, vol. 14, American Mathematical Society, Providence, RI, 2002, With M.-H. Nicole. 2003c:11038

    2002

  58. [66]

    117 (2005), no

    Darren Glass and Rachel Pries, Hyperelliptic curves with prescribed p -torsion , Manuscripta Math. 117 (2005), no. 3, 299--317. 2154252

    2005

  59. [67]

    Gross and David E

    Benedict H. Gross and David E. Rohrlich, Some results on the M ordell- W eil group of the J acobian of the F ermat curve , Invent. Math. 44 (1978), no. 3, 201--224. 0491708

    1978

  60. [68]

    Samuel Grushevsky, The S chottky problem , Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, pp. 129--164. 2931868

    2012

  61. [69]

    R. C. Gunning, Some curves in abelian varieties, Invent. Math. 66 (1982), no. 3, 377--389. 662597

    1982

  62. [70]

    Chris Hall, Big symplectic or orthogonal monodromy modulo l , Duke Math. J. 141 (2008), no. 1, 179--203. 2372151

    2008

  63. [71]

    Hansen, Deligne- L usztig varieties and group codes , Coding theory and algebraic geometry ( L uminy, 1991), Lecture Notes in Math., vol

    Johan P. Hansen, Deligne- L usztig varieties and group codes , Coding theory and algebraic geometry ( L uminy, 1991), Lecture Notes in Math., vol. 1518, Springer, Berlin, 1992, pp. 63--81. 1186416 (94e:94024)

    1991

  64. [72]

    Algebraic Geom

    Shushi Harashita, Ekedahl- O ort strata and the first N ewton slope strata , J. Algebraic Geom. 16 (2007), no. 1, 171--199. 2257323

    2007

  65. [73]

    David Harvey, Kedlaya's algorithm in larger characteristic, Int. Math. Res. Not. IMRN (2007), no. 22, Art. ID rnm095, 29. 2376210

    2007

  66. [74]

    Shushi Harashita, Generic N ewton polygons of E kedahl- O ort strata: O ort's conjecture , Ann. Inst. Fourier (Grenoble) 60 (2010), no. 5, 1787--1830. 2766230

    2010

  67. [75]

    B90, Res

    , Supersingular abelian varieties and curves, and their moduli spaces, with a remark on the dimension of the moduli of supersingular curves of genus 4, Theory and A pplications of S upersingular C urves and S upersingular A belian V arieties, RIMS K\^ o ky\^ u roku Bessatsu, v...

    2022

  68. [76]

    orper vom P rimzahlgrade p \

    Helmut Hasse, Existenz separabler zyklischer unverzweigter E rweiterungsk\"orper vom P rimzahlgrade p \"uber elliptischen F unktionenk\"orpern der C harakteristik p , J. Reine Angew. Math. 172 (1935), 77--85. 1581440

    1935

  69. [77]

    187, Springer-Verlag, New York, 1998

    Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. 1631825

    1998

  70. [78]

    Taira Honda, On the J acobian variety of the algebraic curve y^ 2 =1-x^ l \ over a field of characteristic p>0 , Osaka J. Math. 3 (1966), 189--194. 0225777

    1966

  71. [79]

    Jeroen Hanselman, Andreas Pieper, and Sam Schiavone, Equations of genus 4 curves from their theta constants, https://arxiv.org/pdf/2402.03160

    arXiv

  72. [80]

    Sutherland, Computing H asse- W itt matrices of hyperelliptic curves in average polynomial time , LMS J

    David Harvey and Andrew V. Sutherland, Computing H asse- W itt matrices of hyperelliptic curves in average polynomial time , LMS J. Comput. Math. 17 (2014), 257--273. 3240808

    2014

  73. [81]

    Math., vol

    , Computing H asse- W itt matrices of hyperelliptic curves in average polynomial time, II , Frobenius distributions: L ang- T rotter and S ato- T ate conjectures, Contemp. Math., vol. 663, Amer. Math. Soc., Providence, RI, 2016, pp. 127--147. 3502941

    2016

  74. [82]

    14, Springer, Cham, 2018, With an appendix by Olivier Ta\" bi, pp

    Klaus Hulek and Orsola Tommasi, The topology of A _g and its compactifications , Geometry of moduli, Abel Symp., vol. 14, Springer, Cham, 2018, With an appendix by Olivier Ta\" bi, pp. 135--193. 3968046

    2018

  75. [83]

    111, Springer-Verlag, New York, 2004, With appendices by Otto Forster, Ruth Lawrence and Stefan Theisen

    Dale Husem\"oller, Elliptic curves, second ed., Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 2004, With appendices by Otto Forster, Ruth Lawrence and Stefan Theisen. 2024529

    2004

  76. [84]

    Tomoyoshi Ibukiyama, Principal polarizations of supersingular abelian surfaces, J. Math. Soc. Japan 72 (2020), no. 4, 1161--1180. 4165927

    2020

  77. [85]

    Jun-ichi Igusa, Class number of a definite quaternion with prime discriminant, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 312--314. 0098728

    1958

  78. [86]

    57 (1986), no

    Tomoyoshi Ibukiyama, Toshiyuki Katsura, and Frans Oort, Supersingular curves of genus two and class numbers, Compositio Math. 57 (1986), no. 2, 127--152. 827350

    1986

  79. [87]

    Valentijn Karemaker, Geometry and arithmetic of moduli spaces of abelian varieties in positive characteristic, Abelian varieties, Lectures from the Arizona Winter School 2024, AMS Mathematical Surveys and Monographs

    2024

  80. [88]

    Katz, Slope filtration of F -crystals , Journ\'ees de G \'eom\'etrie A lg\'ebrique de R ennes ( R ennes, 1978), V ol

    Nicholas M. Katz, Slope filtration of F -crystals , Journ\'ees de G \'eom\'etrie A lg\'ebrique de R ennes ( R ennes, 1978), V ol. I , Ast\'erisque, vol. 63, Soc. Math. France, Paris, 1979, pp. 113--163. 563463

    1978

  81. [89]

    Kedlaya, Counting points on hyperelliptic curves using M onsky- W ashnitzer cohomology , J

    Kiran S. Kedlaya, Counting points on hyperelliptic curves using M onsky- W ashnitzer cohomology , J. Ramanujan Math. Soc. 16 (2001), no. 4, 323--338. 1877805

    2001

  82. [90]

    Sci., vol

    , Computing zeta functions via p -adic cohomology , Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 3076, Springer, Berlin, 2004, pp. 1--17. 2137340

    2004

  83. [91]

    Howe, Algorithms to enumerate superspecial H owe curves of genus 4 , A NTS XIV --- P roceedings of the F ourteenth A lgorithmic N umber T heory S ymposium, Open Book Ser., vol

    Momonari Kudo, Shushi Harashita, and Everett W. Howe, Algorithms to enumerate superspecial H owe curves of genus 4 , A NTS XIV --- P roceedings of the F ourteenth A lgorithmic N umber T heory S ymposium, Open Book Ser., vol. 4, Math. Sci. Publ., Berkeley, CA, 2020, pp. 301--31...

    2020

  84. [92]

    Number Theory 6 (2020), no

    Momonari Kudo, Shushi Harashita, and Hayato Senda, The existence of supersingular curves of genus 4 in arbitrary characteristic, Res. Number Theory 6 (2020), no. 4, Paper No. 44, 17. 4170348

    2020

  85. [93]

    Kleiman, The P icard scheme , Fundamental algebraic geometry, Math

    Steven L. Kleiman, The P icard scheme , Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 235--321. 2223410

    2005

  86. [94]

    Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol

    Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. 772569

    1985

  87. [95]

    Knudsen, The projectivity of the moduli space of stable curves

    Finn F. Knudsen, The projectivity of the moduli space of stable curves. II . T he stacks M_ g,n , Math. Scand. 52 (1983), no. 2, 161--199. 702953

    1983

  88. [96]

    31 (1975), no

    Neal Koblitz, p -adic variation of the zeta-function over families of varieties defined over finite fields , Compositio Math. 31 (1975), no. 2, 119--218. 414557

    1975

  89. [97]

    Kottwitz, Isocrystals with additional structure, Compositio Math

    Robert E. Kottwitz, Isocrystals with additional structure, Compositio Math. 56 (1985), no. 2, 201--220. 809866

    1985

  90. [98]

    II , Compositio Math

    , Isocrystals with additional structure. II , Compositio Math. 109 (1997), no. 3, 255--339. 1485921

    1997

  91. [99]

    Kowalski, The large sieve, monodromy and zeta functions of curves, J

    E. Kowalski, The large sieve, monodromy and zeta functions of curves, J. Reine Angew. Math. 601 (2006), 29--69. 2289204

    2006

  92. [100]

    Pure Appl

    Valentijn Karemaker and Rachel Pries, Fully maximal and fully minimal abelian varieties, J. Pure Appl. Algebra 223 (2019), no. 7, 3031--3056. 3912957

    2019

  93. [101]

    Kani and M

    E. Kani and M. Rosen, Idempotent relations and factors of J acobians , Math. Ann. 284 (1989), no. 2, 307--327

    1989

  94. [102]

    Bereich Bonn, September 1975, 86 pp

    Hanspeter Kraft, Kommutative algebraische p - G ruppen (mit A nwendungen auf p -divisible G ruppen und abelsche V ariet\"aten) , Sonderforsch. Bereich Bonn, September 1975, 86 pp

    1975

  95. [103]

    Krichever, Integrable linear equations and the R iemann- S chottky problem , Algebraic geometry and number theory, Progr

    I. Krichever, Integrable linear equations and the R iemann- S chottky problem , Algebraic geometry and number theory, Progr. Math., vol. 253, Birkh\" a user Boston, Boston, MA, 2006, pp. 497--514. 2263198

    2006

  96. [104]

    Igor Krichever, Characterizing J acobians via trisecants of the K ummer variety , Ann. of Math. (2) 172 (2010), no. 1, 485--516. 2680424

    2010

  97. [105]

    Katz and Peter Sarnak, Random matrices, F robenius eigenvalues, and monodromy , American Mathematical Society Colloquium Publications, vol

    Nicholas M. Katz and Peter Sarnak, Random matrices, F robenius eigenvalues, and monodromy , American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. 1659828

    1999

  98. [106]

    Sean Keel and Lorenzo Sadun, Oort's conjecture for A_g C , J. Amer. Math. Soc. 16 (2003), no. 4, 887--900. 1992828

    2003

  99. [107]

    Kr\"amer and R

    T. Kr\"amer and R. Weissauer, On the T annaka group attached to the theta divisor of a generic principally polarized abelian variety , Math. Z. 281 (2015), no. 3-4, 723--745. 3421638

    2015

  100. [108]

    Valentijn Karemaker and Chia-Fu Yu, Supersingular E kedahl-- O ort strata and O ort's conjecture , arxiv.org/abs/2406.19748

  101. [109]

    Valentijn Karemaker, Fuetaro Yobuko, and Chia-Fu Yu, Mass formula and O ort's conjecture for supersingular abelian threefolds , Adv. Math. 386 (2021), Paper No. 107812, 52. 4267516

    2021

  102. [110]

    Aaron Landesman, Exercises on the moduli spaces and the T orelli map , https://swc-math.github.io/aws/2024/2024LandesmanProblems.pdf

    2024

  103. [111]

    Herbert Lange, Abelian varieties over the complex numbers---a graduate course, Grundlehren Text Editions, Springer, Cham, [2023] 2023. 4573077

    2023

  104. [112]

    Number Theory 5 (2019), no

    Wanlin Li, Elena Mantovan, Rachel Pries, and Yunqing Tang, Newton polygons arising from special families of cyclic covers of the projective line, Res. Number Theory 5 (2019), no. 1, Paper No. 12, 31. 3897613

    2019

  105. [113]

    Women Math

    , Newton polygons of cyclic covers of the projective line branched at three points, Research directions in number theory--- W omen in N umbers IV , Assoc. Women Math. Ser., vol. 19, Springer, Cham, [2019] 2019, pp. 115--132. 4069381

    2019

  106. [114]

    , Newton polygon stratification of the T orelli locus in unitary S himura varieties , Int. Math. Res. Not. IMRN (2022), no. 9, 6464--6511. 4411461

    2022

  107. [115]

    Yuxin Lin, Elena Mantovan, and Deepesh Singhal, Non- -ordinary smooth cyclic covers of P ^1 , https://arxiv.org/abs/2406.09632

    arXiv

  108. [116]

    , Abelian covers of P ^1 of p -ordinary E kedahl- O ort type , Int. Math. Res. Not. IMRN (2024), no. 23, 14369--14392. 4838305

    2024

  109. [117]

    1680, Springer-Verlag, Berlin, 1998

    Ke-Zheng Li and Frans Oort, Moduli of supersingular abelian varieties, Lecture Notes in Mathematics, vol. 1680, Springer-Verlag, Berlin, 1998. 1611305 (99e:14052)

    1998

  110. [118]

    1680, Springer-Verlag, Berlin, 1998

    , Moduli of supersingular abelian varieties, Lecture Notes in Mathematics, vol. 1680, Springer-Verlag, Berlin, 1998

    1998

  111. [119]

    Eduard Looijenga, On the tautological ring of M _g , Invent. Math. 121 (1995), no. 2, 411--419. 1346214 (96g:14021)

    1995

  112. [120]

    Ju. I. Manin, The H asse- W itt matrix of an algebraic curve , Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 153--172. 124324

    1961

  113. [121]

    Nauk 18 (1963), no

    , Theory of commutative formal groups over fields of finite characteristic, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 3--90. 0157972 (28 \#1200)

    1963

  114. [122]

    Teruhisa Matsusaka, On a generating curve of an A belian variety , Natur. Sci. Rep. Ochanomizu Univ. 3 (1952), 1--4. 49604

    1952

  115. [123]

    , On a characterization of a J acobian variety , Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 32 (1959), 1--19. 108497

    1959

  116. [124]

    Mumford, J

    D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. 1304906

    1994

  117. [125]

    James Milne, Abelian varieties, https://www.jmilne.org/math/CourseNotes/AV.pdf

  118. [126]

    Leonhard Miller, Curves with invertible H asse- W itt-matrix , Math. Ann. 197 (1972), 123--127. 314849

    1972

  119. [127]

    J. S. Milne, Abelian varieties, Arithmetic geometry ( S torrs, C onn., 1984), Springer, New York, 1986, pp. 103--150. 861974

    1984

  120. [128]

    167--212

    , Jacobian varieties, Arithmetic geometry ( S torrs, C onn., 1984), Springer, New York, 1986, pp. 167--212. 861976

    1984

  121. [129]

    5, American Mathematical Society, Providence, RI, 1995

    Rick Miranda, Algebraic curves and R iemann surfaces , Graduate Studies in Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1995. 1326604

    1995

  122. [130]

    Ben Moonen and Frans Oort, The T orelli locus and special subvarieties , Handbook of moduli. V ol. II , Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 549--594. 3184184

    2013

  123. [131]

    Math., vol

    Ben Moonen, Group schemes with additional structures and W eyl group cosets , Moduli of abelian varieties ( T exel I sland, 1999), Progr. Math., vol. 195, Birkh\"auser, Basel, 2001, pp. 255--298. 1827024 (2002c:14074)

    1999

  124. [132]

    , Special subvarieties arising from families of cyclic covers of the projective line, Doc. Math. 15 (2010), 793--819. 2735989 (2012a:14071)

    2010

  125. [133]

    E kedahl- O ort types of curves , J

    , Computing discrete invariants of varieties in positive characteristic: I . E kedahl- O ort types of curves , J. Pure Appl. Algebra 226 (2022), no. 11, Paper No. 107100, 19. 4412228

    2022

  126. [134]

    David Mumford, Picard groups of moduli problems, Arithmetical A lgebraic G eometry ( P roc. C onf. P urdue U niv., 1963), Harper & Row, New York, 1965, pp. 33--81. 201443

    1963

  127. [135]

    , Curves and their J acobians , University of Michigan Press, Ann Arbor, MI, 1975. 419430

    1975

  128. [136]

    5, Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008, With appendices by C

    , Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, 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 second (1974) edit...

    2008

  129. [137]

    Ben Moonen and Torsten Wedhorn, Discrete invariants of varieties in positive characteristic, Int. Math. Res. Not. (2004), no. 72, 3855--3903. 2104263

    2004

  130. [138]

    David Masser and Umberto Zannier, Abelian varieties isogenous to no J acobian , Ann. of Math. (2) 191 (2020), no. 2, 635--674. 4076633

    2020

  131. [139]

    Peter Norman and Frans Oort, Moduli of abelian varieties, Ann. of Math. (2) 112 (1980), no. 3, 413--439. 595202

    1980

  132. [140]

    Tadao Oda, The first de R ham cohomology group and D ieudonn\'e modules , Ann. Sci. \'Ecole Norm. Sup. (4) 2 (1969), 63--135. 0241435 (39 \#2775)

    1969

  133. [141]

    Frans Oort, Subvarieties of moduli spaces, Invent. Math. 24 (1974), 95--119. 0424813 (54 \#12771)

    1974

  134. [142]

    Math., vol

    , Hyperelliptic supersingular curves, Arithmetic algebraic geometry ( T exel, 1989), Progr. Math., vol. 89, Birkh\"auser Boston, Boston, MA, 1991, pp. 247--284. 1085262 (92c:14043)

    1989

  135. [143]

    , Newton polygons and formal groups: conjectures by M anin and G rothendieck , Ann. of Math. (2) 152 (2000), no. 1, 183--206. 1792294

    2000

  136. [144]

    Math., vol

    , Newton polygon strata in the moduli space of abelian varieties, Moduli of abelian varieties ( T exel I sland, 1999), Progr. Math., vol. 195, Birkh\"auser, Basel, 2001, pp. 417--440. 1827028

    1999

  137. [145]

    Math., vol

    , A stratification of a moduli space of abelian varieties, Moduli of abelian varieties (Texel Island, 1999), Progr. Math., vol. 195, Birkh\"auser, Basel, 2001, pp. 345--416. 2002b:14055

    1999

  138. [146]

    , Abelian varieties isogenous to a J acobian; in problems from the W orkshop on A utomorphisms of C urves , Rend. Sem. Mat. Univ. Padova 113 (2005), 129--177. 2168985

    2005

  139. [147]

    , Special subvarieties in the T orelli locus , Open problems in arithmetic algebraic geometry, Adv. Lect. Math. (ALM), vol. 46, Int. Press, Somerville, MA, [2019] 2019, pp. 71--93. 3971182

    2019

  140. [148]

    Ekin Ozman and Rachel Pries, Ordinary and almost ordinary P rym varieties , Asian J. Math. 23 (2019), no. 3, 455--477. 3983755

    2019

  141. [149]

    Ekin Ozman, Rachel Pries, and Colin Weir, The boundary of the p -rank 0 stratum of the moduli space of cyclic covers of the projective line , Nagoya Math. J. 248 (2022), 865--887. 4508269

    2022

  142. [150]

    Andreas Pieper, Theta nullvalues of supersingular abelian varieties, https://arxiv.org/pdf/2208.12492

    arXiv

  143. [151]

    Number Theory 8 (2022), no

    , Constructing all genus 2 curves with supersingular J acobian , Res. Number Theory 8 (2022), no. 2, Paper No. 32, 26. 4423681

    2022

  144. [152]

    Rojas, Completely decomposable J acobian varieties in new genera , Exp

    Jennifer Paulhus and Anita M. Rojas, Completely decomposable J acobian varieties in new genera , Exp. Math. 26 (2017), no. 4, 430--445. 3684576

    2017

  145. [153]

    Math., vol

    Rachel Pries, A short guide to p -torsion of abelian varieties in characteristic p , Computational arithmetic geometry, Contemp. Math., vol. 463, Amer. Math. Soc., Providence, RI, 2008, math.NT/0609658, pp. 121--129. MR2459994 (2009m:11085)

    2008

  146. [154]

    , The p -torsion of curves with large p -rank , Int. J. Number Theory 5 (2009), no. 6, 1103--1116. MR2569747

    2009

  147. [155]

    , Current results on N ewton polygons of curves , Open problems in arithmetic algebraic geometry, Adv. Lect. Math. (ALM), vol. 46, Int. Press, Somerville, MA, [2019] 2019, pp. 179--207. 3971184

    2019

  148. [156]

    , Some cases of O ort's conjecture about N ewton polygons of curves , Nagoya Math. J. 257 (2025), 93--103. 4897411

    2025

  149. [157]

    Jennifer Paulhus and Drew Sutherland, Completely decomposable modular J acobians , https://arxiv.org/html/2502.16007v1

    arXiv

  150. [158]

    Rachel Pries and Douglas Ulmer, On BT_1 group schemes and F ermat curves , New York J. Math. 27 (2021), 705--739. 4250272

    2021

  151. [159]

    , Every BT_1 group scheme appears in a J acobian , Proc. Amer. Math. Soc. 150 (2022), no. 2, 525--537. 4356165

    2022

  152. [160]

    Rachel Pries and Colin Weir, The E kedahl- O ort type of J acobians of H ermitian curves , Asian J. Math. 19 (2015), no. 5, 845--869. 3431681

    2015

  153. [161]

    Rachel Pries and Hui June Zhu, The p -rank stratification of A rtin- S chreier curves , Ann. Inst. Fourier (Grenoble) 62 (2012), no. 2, 707--726. 2985514

    2012

  154. [162]

    Ziv Ran, On subvarieties of abelian varieties, Invent. Math. 62 (1981), no. 3, 459--479. 604839

    1981

  155. [163]

    Algebra 236 (2001), no

    Riccardo Re, The rank of the C artier operator and linear systems on curves , J. Algebra 236 (2001), no. 1, 80--92. 1808346

    2001

  156. [164]

    1975, Springer-Verlag, Berlin, 2009

    Jan Christian Rohde, Cyclic coverings, C alabi- Y au manifolds and complex multiplication , Lecture Notes in Mathematics, vol. 1975, Springer-Verlag, Berlin, 2009. 2510071

    1975

  157. [165]

    Rapoport and M

    M. Rapoport and M. Richartz, On the classification and specialization of F -isocrystals with additional structure , Compositio Math. 103 (1996), no. 2, 153--181. 1411570

    1996

  158. [166]

    Ren\' e Schoof, Elliptic curves over finite fields and the computation of square roots mod p , Math. Comp. 44 (1985), no. 170, 483--494. 777280

    1985

  159. [167]

    H. P. F. Swinnerton-Dyer, Analytic theory of abelian varieties, London Mathematical Society Lecture Note Series, vol. No. 14, Cambridge University Press, London-New York, 1974. 366934

    1974

  160. [168]

    Serre, Corps locaux, Hermann, 1968

    J.-P. Serre, Corps locaux, Hermann, 1968

    1968

  161. [169]

    Bordeaux I, Talence, 1983, pp

    Jean-Pierre Serre, Nombres de points des courbes alg\' e briques sur F q , Seminar on number theory, 1982--1983 ( T alence, 1982/1983), Univ. Bordeaux I, Talence, 1983, pp. Exp. No. 22, 8. 750323

    1982

  162. [170]

    Goro Shimura, On purely transcendental fields automorphic functions of several variable, Osaka Math. J. 1 (1964), no. 1, 1--14. 176113

    1964

  163. [171]

    Takahiro Shiota, Characterization of J acobian varieties in terms of soliton equations , Invent. Math. 83 (1986), no. 2, 333--382. 818357

    1986

  164. [172]

    Silverman, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol

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

    2009

  165. [173]

    254, Springer-Verlag, Berlin, 2009

    Henning Stichtenoth, Algebraic function fields and codes, second ed., Graduate Texts in Mathematics, vol. 254, Springer-Verlag, Berlin, 2009. 2464941 (2010d:14034)

    2009

  166. [174]

    Marco Streng, Explicit supersingular cyclic curves, https://arxiv.org/pdf/2501.14902

    arXiv

  167. [175]

    16 (1975), no

    Dor \'e Subrao, The p -rank of A rtin- S chreier curves , Manuscripta Math. 16 (1975), no. 2, 169--193. 0376693

    1975

  168. [176]

    Reine Angew

    Karl-Otto St\" o hr and Jos\' e Felipe Voloch, A formula for the C artier operator on plane algebraic curves , J. Reine Angew. Math. 377 (1987), 49--64. 887399

    1987

  169. [177]

    John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134--144. 0206004 (34 \#5829)

    1966

  170. [178]

    Jacob Tsimerman, The existence of an abelian variety over Q isogenous to no J acobian , Ann. of Math. (2) 176 (2012), no. 1, 637--650. 2925392

    2012

  171. [179]

    , The A ndr\'e- O ort conjecture for A_g , Ann. of Math. (2) 187 (2018), no. 2, 379--390. 3744855

    2018

  172. [180]

    E33, Friedr

    Gerard van der Geer, Cycles on the moduli space of abelian varieties, Moduli of curves and abelian varieties, Aspects Math., vol. E33, Friedr. Vieweg, Braunschweig, 1999, pp. 65--89. 1722539

    1999

  173. [181]

    E33, Friedr

    Gerard van der Geer and Frans Oort, Moduli of abelian varieties: a short introduction and survey, Moduli of curves and abelian varieties, Aspects Math., vol. E33, Friedr. Vieweg, Braunschweig, 1999, pp. 1--21. 1722536

    1999

  174. [182]

    I , Compositio Math

    Gerard van der Geer and Marcel van der Vlugt, Reed- M uller codes and supersingular curves. I , Compositio Math. 84 (1992), no. 3, 333--367. 1189892 (93k:14038)

    1992

  175. [183]

    Reine Angew

    , On the existence of supersingular curves of given genus, J. Reine Angew. Math. 458 (1995), 53--61. 1310953 (95k:11084)

    1995

  176. [184]

    Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613--670. MR1005008 (90k:14004)

    1989

  177. [185]

    Algebra 16 (1988), no

    Jos\'e Felipe Voloch, A note on algebraic curves in characteristic 2 , Comm. Algebra 16 (1988), no. 4, 869--875. 932639

    1988

  178. [186]

    Eva Viehmann and Torsten Wedhorn, Ekedahl- O ort and N ewton strata for S himura varieties of PEL type , Math. Ann. 356 (2013), no. 4, 1493--1550. 3072810

    2013

  179. [187]

    Ind., no

    Andr \'e Weil, Sur les courbes alg\'ebriques et les vari\'et\'es qui s'en d\'eduisent, Actualit\'es Sci. Ind., no. 1041, Hermann et Cie., Paris, 1948

    1948

  180. [188]

    Ind., no

    , Vari\'et\'es ab\'eliennes et courbes alg\'ebriques, Actualit\'es Sci. Ind., no. 1064, Hermann & Cie., Paris, 1948

    1948

  181. [189]

    G. E. Welters, A criterion for J acobi varieties , Ann. of Math. (2) 120 (1984), no. 3, 497--504. 769160

    1984

  182. [190]

    Stefan Wewers, Construction of H urwitz spaces , Dissertation, 1998

    1998

  183. [191]

    Algebra 52 (1978), no

    Noriko Yui, On the J acobian varieties of hyperelliptic curves over fields of characteristic p>2 , J. Algebra 52 (1978), no. 2, 378--410. 491717

    1978

  184. [192]

    Algebra 65 (1980), no

    , On the J acobian variety of the F ermat curve , J. Algebra 65 (1980), no. 1, 1--35. 578793

    1980

  185. [193]

    Algebra 322 (2009), no

    Chia-Fu Yu and Jeng-Daw Yu, Mass formula for supersingular abelian surfaces, J. Algebra 322 (2009), no. 10, 3733--3743. 2568360

    2009

  186. [194]

    C. G. Zaal, A complete surface in M_6 in characteristic >2 , Compositio Math. 119 (1999), no. 2, 209--212. 1723129

    1999

  187. [195]

    Zijian Zhou, Ekedahl- O ort strata on the moduli space of curves of genus four , Rocky Mountain J. Math. 50 (2020), no. 2, 747--761. 4104409

    2020

Pith tools

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