Pith. sign in

REVIEW 1 major objections 5 minor 32 references

\'Etale cohomology of rank one $\ell$-adic local systems in positive characteristic

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · 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 Frobenius-stable closed subsets of the parameter space of rank-one $\ell$-adic local systems are finite unions of torsion-translated formal Lie subgroups, and derives Hard Lefschetz, quasi-linear jumping loci, and…

desk verdict A genuinely new structural theorem about Frobenius-invariant closed subsets of ℓ-adic character spaces, with real applications to Hard Lefschetz and jumping loci; the main soft spot is an explicitly unproved specialization step in Theorem 5.4. read the letter →

arxiv 1908.08291 v4 pith:3S44JDWG submitted 2019-08-22 math.AG math.NT

Hélène Esnault , Moritz Kerz This is my paper · ORCID
classification math.AGmath.NT MSC 14G1714G22
keywords étalecohomologyrankoneℓ-adiclocalsystemsHardLefschetztheoremjumpinglocigenericvanishingformalLiegroupsanalysispositivecharacteristic
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

This paper proves a structure theorem for Zariski-closed, Frobenius-invariant subsets of the spaces that parametrize rank-one $\ell$-adic local systems on varieties in positive characteristic: any such subset must be a finite union of torsion-translated formal Lie subgroups, a property the authors call quasi-linearity. From this single statement they derive three results previously available only over the complex numbers or for torsion local systems: Hard Lefschetz for rank-one $\ell$-adic local systems twisted by arithmetic semi-simple perverse sheaves, a quasi-linearity theorem for cohomological jumping loci, and lower bounds on the codimension of generic vanishing loci. The point of the paper is that non-archimedean tools, such as Tate algebras, $\ell$-adic logarithms, and formal Lie groups, can replace the complex-analytic and weight arguments that carried the characteristic-zero proofs.

What carries the argument

The central objects are multiplicative formal Lie groups: for a free $\mathbb{Z}_\ell$-module $\pi$, the space $G(\mathbb{Q}_\ell) = \operatorname{Hom}_{\mathrm{cont}}(\pi, \mathbb{Q}_\ell^\times)$ is the $\mathbb{Q}_\ell$-points of $\operatorname{Spf}(\mathbb{Z}_\ell[[\pi]])$ and is identified with the maximal spectrum of the Jacobson ring $R = \mathbb{Z}_\ell[[\pi]] \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell$. This is the parameter space for rank-one local systems, since each continuous character of the abelian fundamental group gives one. The proof is carried by two $\ell$-adic analytic propositions: in a Tate algebra $E\langle T_1,\ldots,T_b\rangle$, if a diagonal automorphism $\sigma$ has eigenvalues whose only root relations are trivial, then any $\sigma$-invariant ideal not contained in the maximal ideal is the whole ring, and any radical $\sigma$-invariant ideal is homogeneous; after transfer through the $\ell$-adic exponential map this makes the closed set conical around a torsion point, and a formal-linearity theorem converts conicality into quasi-linearity. A second mechanism is the generalized Fourier--Mellin transform $R\Gamma(X, F \otimes L_R)$, valued in finitely generated $R$-modules, whose flatness and vanishing are controlled by a Galois tower and give the generic-vanishing estimates.

What would settle it

Find a smooth projective variety over an algebraically closed field of positive characteristic, an arithmetic semi-simple perverse sheaf, and a rank-one $\ell$-adic local system for which some cup-product map $\cup\eta^i$ is not an isomorphism; that would refute Theorem 5.4. More directly, exhibit a Zariski-closed subset $S$ of a multiplicative formal Lie group that is invariant under a semi-simple automorphism with all complex eigenvalues of equal absolute value different from $1$, yet is not a finite union of torsion-translated formal Lie subgroups; that would refute the Main Theorem.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.4: let $\pi$ be a finitely generated free $\mathbb{Z}_\ell$-module and let $G(\mathbb{Q}_\ell) = \operatorname{Hom}_{\mathrm{cont}}(\pi, \mathbb{Q}_\ell^\times)$ be the character group of the associated multiplicative formal Lie group, equipped with its Zariski topology. If $\sigma$ is a $\mathbb{Z}_\ell$-linear automorphism of $\pi$ acting semi-simply on $\pi \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell$ with all complex eigenvalues of the same absolute value, different from $1$, and $S \subset G(\mathbb{Q}_\ell)$ is Zariski closed with $\sigma(S) = S$, then $S = \bigcup_{r\in I} s_r H_r(\mathbb{Q}_\ell)$, where the $s_r$ are torsion points and each $H_r$ is a formal Lie subgroup. The proof reduces to an $\ell$-adic analytic statement about ideals in Tate algebras: a radical $\sigma$-invariant ideal in coordinates where $\sigma$ is diagonal is homogeneous, and after multiplication by $\ell^n$ and a logarithm chart the invariant closed set is conical, so a formal-linearity theorem can finish the argument. The applications follow by identifying Hard-Lefschetz failure loci and jumping loci as such $\sigma$-invariant closed sets, with the Frobenius action in the role of $\sigma$; because torsion points are dense in a quasi-linear set, Deligne's torsion Hard Lefschetz rules out any nonempty failure locus.

Load-bearing premise

The argument depends on a reduction that moves the problem from general algebraically closed fields to the algebraic closure of a finite field, and this reduction is quoted from an existing observation rather than fully proved in this paper; if that transfer fails in positive characteristic, the central proof no longer applies.

Editorial extensions

If this is right

  • Hard Lefschetz holds in positive characteristic for rank-one $\ell$-adic local systems: for smooth projective $X$ and any arithmetic semi-simple perverse sheaf $F$, the cup-product map $\cup\eta^i: H^{-i}(X,F\otimes L) \to H^i(X,F\otimes L)$ is an isomorphism for all $i$.
  • Cohomological jumping loci $\Sigma^i(F,j)$ of arithmetic sheaves are quasi-linear when the abelian fundamental-group quotient has pure nonzero weight; this is the first structure theorem of its kind in positive characteristic.
  • Generic vanishing holds when a polarization becomes divisible in the Galois tower: $\operatorname{codim}(\Sigma^i(F,0)) \geq i$, and for abelian varieties the bound improves to $\operatorname{codim}(\Sigma^i(F,0)) \geq 2i$.
  • Because torsion points are dense in every quasi-linear set, any Frobenius-stable bad locus that contains no torsion local system must be empty; this is exactly how Deligne's known torsion case is leveraged.

Reading between the lines

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

  • Editorial inference: the Main Theorem should apply to any Zariski-closed subset of a multiplicative formal Lie group carrying a semi-simple automorphism of equal, nontrivial absolute eigenvalues, so the technique may transfer to other deformation spaces where Frobenius acts with a pure weight.
  • Editorial inference: a mixed-weight version of the quoted formal-linearity result would likely remove the purity hypothesis and settle the paper's Remark 6.3; the structure of the proof points directly at this missing input.
  • Editorial inference: for non-arithmetic sheaves the expected statement keeps the finite-union-of-subgroups form but drops the torsion condition on the translates; separating these two phenomena could be tested on finite-field examples where Frobenius invariance is unavailable.
  • Editorial inference: the Fourier--Mellin vanishing of Proposition 4.6 gives a cohomological route to generic vanishing that does not depend on complex Hodge or D-module theory, and the paper notes this could also supply a direct proof in characteristic zero.
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

1 major / 5 minor

Summary. The paper proves a structure theorem for Zariski closed subsets of the character group of a free Z_ell-module that are invariant under a semisimple linear automorphism all of whose eigenvalues have the same complex absolute value, different from 1: any such subset is a finite union of torsion-translated formal Lie subgroups (Theorem 3.4, restated as Theorem 1.7). The proof is carried out through ell-adic Tate algebras and a result of de Jong. The paper then derives three applications in positive characteristic: Hard Lefschetz for rank one Q_ell-local systems twisted by arithmetic semisimple perverse sheaves (Theorem 5.4), quasi-linearity of cohomological jumping loci for arithmetic sheaves (Theorem 6.2), and codimension bounds for generic vanishing (Theorem 6.8 and Corollaries 6.9, 6.10). A generalized Fourier-Mellin transform and a divisibility lemma (Proposition 4.6) are introduced to obtain the vanishing estimates.

Significance. The main theorem is a clean and non-obvious statement about Frobenius-stable loci in deformation spaces of rank one ell-adic local systems, and the analytic core (Propositions 2.1 and 2.5 and Theorem 3.4) is proved with complete arguments. The applications are substantial: the Hard Lefschetz theorem for rank one local systems with arithmetic twists and the quasi-linearity of ell-adic jumping loci are new in positive characteristic, and the generic vanishing bounds strengthen earlier partial results. The paper is carefully organized, and the main theorem is not used to prove itself; the applications are genuinely derived from it. The only serious caveat is a specialization step that is invoked rather than proved, which affects the stated generality of several applications.

major comments (1)
  1. [Section 5.2 (and the proof of Theorem 6.2)] The reduction from an arbitrary algebraically closed field F of characteristic p to F = \bar F_p is load-bearing for Theorem 5.4 and is also invoked at the start of the proof of Theorem 6.2. The text cites [BBD82, Lem. 6.1.9] together with [Dri01, Rmk. 1.7], but [BBD82, Lem. 6.1.9] is formulated for passage from C to positive characteristic, and [Dri01, Rmk. 1.7] is only a remark rather than a stated theorem with hypotheses and proof. Since the Frobenius automorphism sigma to which Theorem 3.4 is applied is produced only after this reduction, the application of the Main Theorem depends exactly on this transfer. Please either prove the positive-characteristic spreading/specialization principle, or state the precise form of [Dri01, Rmk. 1.7] being used and verify that it applies to the simultaneous descent of X, F, L, and eta, including preservation of arithmeticity and semi-simplicity. Until this is supplied, Theorem 5.4 and the results depending on it (Theorem 6.8, Corollaries 6.9 and 6.10) are proved only over \bar F_p, not over every algebraically closed field of characteristic p.
minor comments (5)
  1. [Section 5.2] The notation 'F = F_p' is misleading because F is assumed algebraically closed; the authors mean the algebraic closure of F_p. Please clarify.
  2. [Section 3.3, proof of Theorem 3.4] The assertion that there exists n > 0 with [\ell^n](S) \cap G(\rho/\ell^w) non-empty deserves a short justification; it follows from the fact that for any s in S, the characters [\ell^n](s) converge to 1 in the ell-adic topology.
  3. [Section 5.2] After Theorem 3.4 gives S = \bigcup_r s_r H_r(Q_ell), the statement 'by Lemma 3.1, the torsion points are dense in S' is not immediate from the lemma as stated, since the lemma concerns G(Q_ell); it follows by applying Lemma 3.1 to each formal Lie subgroup H_r appearing in the quasi-linear decomposition.
  4. [Section 4.2, Proposition 4.6] The isomorphism of pro-rings in equation (3) is justified via equations (4) and (5) with the phrase 'one easily sees'; since this is a key technical step, expanding the verification of the two pro-system isomorphisms would improve readability.
  5. [Section 6.2, proof of Corollary 6.9] The phrase 'for cohomological dimension reasons that, i.e.' contains a typo; it should read 'for cohomological dimension reasons, i.e.'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Main Theorem is proved from Tate-algebra arguments and de Jong's formal linearity result, and the applications are derived from it rather than used as inputs.

full rationale

The paper's central claim (Theorem 3.4 / Introduction Theorem 1.7) is proved self-containedly: after a logarithmic change of coordinates, a sigma-invariant closed subset gives a homogeneous radical ideal in a Tate algebra (Propositions 2.1 and 2.5), and de Jong's formal linearity result [deJ00, Prop. 1.2(1)] converts the resulting conical structure into quasi-linearity. Nothing in that proof uses the Hard Lefschetz, jumping-locus, or generic-vanishing theorems; the applications are consequences, not inputs. The only self-citation to prior work by the authors is [EK20, Prop. 2.2], invoked in Lemma 3.2 for the fact that a certain completed tensor product is a reduced Jacobson ring; this is a background algebraic fact, not a reformulation of the target conclusion, and the density statement of Lemma 3.2 is then proved from it rather than assumed. The Hard Lefschetz proof reduces to F_p via [BBD82, Lem. 6.1.9] and [Dri01, Rmk. 1.7]; that cited transfer is external and not reproduced, which is a correctness risk about the stated generality, but it is not a circular step because the Main Theorem is not being presupposed. The torsion case of Hard Lefschetz (Deligne), the arithmetic perverse case (BBD82 plus Lafforgue), Tate's semisimplicity theorem, and Drinfeld's strategy are all external benchmarks, not fitted inputs or renaming of the conclusions. No equation in the paper reduces to its own input, and no fitted parameter is relabeled as a prediction. Accordingly the circularity score is 0.

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

The Main Theorem's proof is self-contained apart from standard rigid analysis and de Jong's formal linearity theorem. The applications add hypotheses: F arithmetic and π pure of non-zero weight. No free parameters are fitted and no entities are invented; the Fourier-Mellin transform is a variant of existing constructions. The proof does not use the target theorems as inputs, so circularity burden is low.

assumptions (6)
  • domain assumption The spreading/specialization lemma [BBD82, Lem. 6.1.9] extends from complex numbers to positive characteristic as asserted in [Dri01, Rmk. 1.7], allowing reduction to F = F_p in the proof of Theorem 5.4.
    Invoked in Section 5.2, first paragraph; the paper does not prove this transfer in detail and notes the original lemma is written only for passage from C to positive characteristic.
  • standard math de Jong's formal linearity theorem [deJ00, Prop. 1.2(1)] characterizes closed subsets of formal Lie groups stable under [ℓ] as quasi-linear.
    Used as a black box at the end of the proof of Theorem 3.4 to conclude that [ℓ]-stability of [ℓ^n](S) implies S is a finite union of torsion-translated formal Lie subgroups.
  • standard math Frobenius acts semi-simply on the Tate module of an abelian variety (Tate [Tat66, Thm. 2]) and with pure weight -1 on the ℓ-adic completion of the abelianized fundamental group of a smooth projective variety (Deligne [Del80, Thm. 1]).
    These results supply the eigenvalue hypotheses (semi-simplicity, all |ι(α_i)| equal and different from 1) needed to apply Theorem 3.4 in Sections 5.2 and 6.2.
  • standard math Deligne's Hard Lefschetz for torsion rank-one local systems [Del80, Thm. 4.1.1] and the Hard Lefschetz theorem for arithmetic semi-simple perverse sheaves (Theorem 5.3, assembled from [BBD82, Thm. 6.2.10] and Lafforgue [Laf02]) hold.
    Used in Section 5.2 to rule out torsion points in the bad locus; these are external benchmarks, not re-derived in the paper.
  • standard math Ekedahl's adic formalism provides finiteness, base change, limit, and duality properties for m-adic étale cohomology [Eke90, Thm. 6.3, 7.2].
    This underlies the Fourier-Mellin transform in Section 4.1 and the constructibility argument in Corollary 4.3.
  • domain assumption The arithmeticity hypothesis (Definition 5.1): the sheaf F is fixed up to quasi-isomorphism by the Galois group of a finitely generated field, so a geometric Frobenius stabilizes F.
    This is a hypothesis of Theorems 5.4, 6.2, and 6.8, not a derived fact; the paper conjectures in Remarks 6.3 and 6.11 that it can be dropped.

how reviews work

0 comments
Cite this review

Pith. "Pith review of \'Etale cohomology of rank one $\ell$-adic local systems in positive characteristic." pith.science (2026). https://pith.science/paper/3S44JDWG

@misc{pith2026190808291,
  author       = {Pith},
  title        = {Pith review of: \'Etale cohomology of rank one $\ell$-adic local systems in positive characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3S44JDWG}},
  note         = {Machine review of arXiv:1908.08291}
}
abstract

We show that in positive characteristic special loci of deformation spaces of rank one $\ell$-adic local systems are quasilinear. From this we deduce the Hard Lefschetz theorem for rank one $\ell$-adic local systems and a generic vanishing theorem. Last version: a few typos corrected

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    : Vanishing theorems for perverse sheaves on abelian varieties, revisited , Selecta Math

    Bhatt, B., Schnell, C., Scholze, P. : Vanishing theorems for perverse sheaves on abelian varieties, revisited , Selecta Math. 24 (2018), no. 1, 63--84

    work page 2018

  2. [2]

    : Faisceaux pervers , Analysis and topology on singular spaces, I (Luminy, 1981), Ast\'erisque 100 (1982), 5--171

    Beilinson, A., Bernstein, J., Deligne, P. : Faisceaux pervers , Analysis and topology on singular spaces, I (Luminy, 1981), Ast\'erisque 100 (1982), 5--171

    work page 1982

  3. [3]

    : Cohomology jump loci of quasi-projective varieties , Ann

    Budur, N., Wang, B. : Cohomology jump loci of quasi-projective varieties , Ann. Sci. \'Ecole Norm. Sup. 48 (2005), no. 1, 227--236

    work page 2005

  4. [4]

    : Local Fields , London Mathematical Society Student Texts 3 (1986), Cambridge University Press, xiv + 360 pp

    Cassels, J. : Local Fields , London Mathematical Society Student Texts 3 (1986), Cambridge University Press, xiv + 360 pp

    work page 1986

  5. [5]

    : A result on formal linearity , J

    de Jong, J. : A result on formal linearity , J. of Algebra 225 (2000), 936--942

    work page 2000

  6. [6]

    : La conjecture de Weil II , Publ

    Deligne, P. : La conjecture de Weil II , Publ. math. I.H.\'E.S. 42 (1980), 137--252

    work page 1980

  7. [7]

    : On a conjecture of Kashiwara , Math

    Drinfeld, V. : On a conjecture of Kashiwara , Math. Res. Lett. 8 (2001) no 5-6, 713--728

    work page 2001

  8. [8]

    On a conjecture of Deligne , Mosc

    Drinfeld, V. On a conjecture of Deligne , Mosc. Math. J. 12 (2012), no. 3, 515--542, 668

    work page 2012

Show all 32 references
  1. [9]

    : On the adic formalism , The Grothendieck Festschrift, Vol

    Ekedahl, T. : On the adic formalism , The Grothendieck Festschrift, Vol. II, 197--218, Progr. Math. 87 , Birkh\"auser Boston, Boston, MA, 1990

    1990

  2. [10]

    : Cohomological dimension in pro-p-towers , http://page.mi.fu-berlin.de/esnault/preprints/helene/130_scholze.pdf, to appear in Int

    Esnault, H. : Cohomological dimension in pro-p-towers , http://page.mi.fu-berlin.de/esnault/preprints/helene/130_scholze.pdf, to appear in Int. Math. Res. Not

  3. [11]

    : Arithmetic subspaces of moduli spaces of rank one local systems , preprint 2019, 19 pages, http://page.mi.fu-berlin.de/esnault/preprints/helene/133_esn_kerz.pdf

    Esnault, H., Kerz, M. : Arithmetic subspaces of moduli spaces of rank one local systems , preprint 2019, 19 pages, http://page.mi.fu-berlin.de/esnault/preprints/helene/133_esn_kerz.pdf

    2019

  4. [12]

    : Rigid analytic geometry and its applications , Progress in Mathematics 218 (2004), 296 p., Birkh\"auser Verlag

    Fresnel, J., van der Put, M. : Rigid analytic geometry and its applications , Progress in Mathematics 218 (2004), 296 p., Birkh\"auser Verlag

    2004

  5. [13]

    : Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent

    Green, M., Lazarsfeld, R. : Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. math. 90 (1987), 389--407

    1987

  6. [14]

    : Higher obstructions to deforming cohomology groups of line bundles , J

    Green, M., Lazarsfeld, R. : Higher obstructions to deforming cohomology groups of line bundles , J. Amer. Math. Soc. 4 (1991), no. 1, 87--103

    1991

  7. [15]

    : Faisceaux pervers -adiques sur un tore , Duke

    Gabber, O., Loeser, F. : Faisceaux pervers -adiques sur un tore , Duke. Math. J. 83 (3) (1996), 1--106

    1996

  8. [16]

    : Weights in arithmetic geometry , Jpn

    Jannsen, U. : Weights in arithmetic geometry , Jpn. J. Math. 5 (2010), no. 1, 73--102

    2010

  9. [17]

    : Vanishing theorems for constructible sheaves on abelian varieties , J

    Kr\"amer, T., Weissauer, R. : Vanishing theorems for constructible sheaves on abelian varieties , J. Alg. Geom. 24 (3) (2015), 531--568

    2015

  10. [18]

    : Chtoucas de Drinfeld et correspondance de Langlands , Invent

    Lafforgue, L. : Chtoucas de Drinfeld et correspondance de Langlands , Invent. math. 147 (2002), no. 1, 1--241

    2002

  11. [19]

    : Faisceaux pervers, transformation de Mellin et d\'eterminants , M\'emoires de la Soc

    Loeser, F. : Faisceaux pervers, transformation de Mellin et d\'eterminants , M\'emoires de la Soc. Math. de France 66 (1996), 105 pp

    1996

  12. [20]

    : Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D -modules, Part 2, Memoirs of the Am

    Mochizuki, T. : Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D -modules, Part 2, Memoirs of the Am. Math. Soc. Vol. 185 (2007), 564 pp

    2007

  13. [21]

    : Abelian varieties , Tata Institute of Fundamental Research Studies in Mathematics 5 (1970), 242 pp

    Mumford, D. : Abelian varieties , Tata Institute of Fundamental Research Studies in Mathematics 5 (1970), 242 pp

    1970

  14. [22]

    : A conjecture of Beauville and Catanese revisited , Math

    Pink, R., R\"ossler, D. : A conjecture of Beauville and Catanese revisited , Math. Ann. 330 (2004) 2, 293--308

    2004

  15. [23]

    : Lieu des p\^oles d'un syst\`eme holonome d'\'equations aux diff\'erences finies , Bull

    Sabbah, C. : Lieu des p\^oles d'un syst\`eme holonome d'\'equations aux diff\'erences finies , Bull. Soc. Math. France 120 (1992), no. 3, 371--396

    1992

  16. [24]

    : Polarizable twistor -modules , Ast\'erisque 300 (2005), vi+ 208 pp

    Sabbah, C. : Polarizable twistor -modules , Ast\'erisque 300 (2005), vi+ 208 pp

    2005

  17. [25]

    : Holonomic -modules on abelian varieties , Publ

    Schnell, C. : Holonomic -modules on abelian varieties , Publ. math. I.H.\'E.S. 121 (1) (2015), 1--55

    2015

  18. [26]

    10, 1--22

    Serre, J.-P.: Morphismes universels et vari\'et\'e d'Albanese , S\'eminaire Claude Chevalley 4 (1958-1959), exp. 10, 1--22

    1958

  19. [27]

    : Higgs bundles and local systems , Publ

    Simpson, C. : Higgs bundles and local systems , Publ. math. I.H.\'E.S. 75 (1992), 5--95

    1992

  20. [28]

    : Subspaces of moduli spaces of rank one local systems , Annales de l'\'E

    Simpson, C. : Subspaces of moduli spaces of rank one local systems , Annales de l'\'E. N. S. 4\`eme s\'erie 26 (3) (1993), 361--401

    1993

  21. [29]

    : Endomorphisms of abelian varieties over finite fields , Invent

    Tate, J. : Endomorphisms of abelian varieties over finite fields , Invent. math. 2 (1966), 134--144

    1966

  22. [30]

    : Vanishing theorems for constructible sheaves on abelian varieties over finite fields , Math

    Weissauer, R. : Vanishing theorems for constructible sheaves on abelian varieties over finite fields , Math. Ann. 365 (1-2) (2016), 559--578

    2016

  23. [31]

    : \'El\'ements de G\'eom\'etrie Alg\'ebrique, III, \'Etude cohomologique des faisceaux coh\'erents , Publ

    Grothendieck, A., Dieudonn\'e, J. : \'El\'ements de G\'eom\'etrie Alg\'ebrique, III, \'Etude cohomologique des faisceaux coh\'erents , Publ. Math. I.H.\'E.S. 11 (1961) and 17 (1963)

    1961

  24. [32]

    : S\'eminaire de G\'eom\'etrie Alg\'ebrique du Bois Marie - 1962--64 - Sch\'emas en groupes , Lecture Notes in Mathematics 151 , 152 and 153 (1970)

    Grothendieck, A., Demazure, M. : S\'eminaire de G\'eom\'etrie Alg\'ebrique du Bois Marie - 1962--64 - Sch\'emas en groupes , Lecture Notes in Mathematics 151 , 152 and 153 (1970)

    1970

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.