arxiv: 2605.04696 · v1 · submitted 2026-05-06 · 🧮 math.NT · math.AG

Recognition: unknown

mathbb{G}_m-cohomology of p-adic Stein spaces

Damien Junger, Sally Gilles

Pith reviewed 2026-05-08 15:37 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords p-adic cohomologyétale G_m-cohomologyStein spacesprincipal unitsp-adic Hodge theoryKummer sequencesDrinfeld upper half-space

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The pith

The étale G_m-cohomology of p-adic rigid analytic Stein spaces is computed by filtering through principal units.

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

The paper computes the étale G_m-cohomology for certain p-adic rigid analytic Stein spaces. The key step is to filter G_m by its subgroup of principal units. This splits the cohomology into the part for the units, which is found using p-adic Hodge theory after switching to the pro-étale site and applying comparison theorems, and the part for the quotient, which is found using Kummer exact sequences. The resulting explicit formula applies in particular to the Drinfeld upper half-space.

Core claim

We compute the étale G_m-cohomology of some p-adic rigid analytic Stein spaces. The computation is done by considering the filtration induced by the subgroup of principal units U=1+ m O^+ of G_m. We then determine the U-cohomology via methods from p-adic Hodge theory (passage to the pro-étale site, comparison theorems with p-adic cohomologies), while the G_m/U-cohomology is obtained using Kummer exact sequences. In particular, our formula applies to the case of Drinfeld upper-half space.

What carries the argument

The filtration induced by the principal units subgroup U of G_m, which permits separate treatment of U-cohomology by p-adic Hodge theory and G_m/U-cohomology by Kummer sequences.

If this is right

The formula gives the cohomology groups for the Drinfeld upper half-space.
U-cohomology is accessible through pro-étale site passage and p-adic cohomology comparisons.
G_m/U-cohomology follows from Kummer exact sequences.
The method covers a range of p-adic rigid analytic Stein spaces.

Where Pith is reading between the lines

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

This provides a method that could be tested on additional Stein spaces to verify the general applicability.
The separation of the multiplicative group into units and quotient parts may connect to computations in other p-adic cohomology theories.

Load-bearing premise

The filtration by the principal units subgroup U allows the U-cohomology to be determined via p-adic Hodge theory methods and the G_m/U-cohomology via Kummer exact sequences.

What would settle it

A mismatch between the predicted cohomology groups from the formula and a direct calculation of the étale G_m-cohomology groups on the Drinfeld upper half-space.

read the original abstract

We compute the \'etale $\mathbb{G}_m$-cohomology of some $p$-adic rigid analytic Stein spaces. The computation is done by considering the filtration induced by the subgroup of principal units $U=1+ \mathfrak{m} \mathcal{O}^+$ of $\mathbb{G}_m$. We then determine the $U$-cohomology via methods from $p$-adic Hodge theory (passage to the pro-\'etale site, comparison theorems with $p$-adic cohomologies), while the $\mathbb{G}_m/U$-cohomology is obtained using Kummer exact sequences. In particular, our formula applies to the case of Drinfeld upper-half space.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

Paper gives explicit Gm-cohomology formula on p-adic Stein spaces including Drinfeld half-space by splitting via principal units, but the key comparison step for non-proper spaces needs checking.

read the letter

The main point is that this paper works out an explicit formula for the étale Gm-cohomology of some p-adic rigid analytic Stein spaces. They filter Gm by the subgroup of principal units U, compute the U-cohomology with pro-étale comparisons from p-adic Hodge theory, and handle the Gm/U part with Kummer sequences. The formula is then applied to the Drinfeld upper half-space as the main example. This is a concrete calculation rather than a general theory, and it fills in a spot where explicit answers have been missing. The approach sticks to standard tools, so the overall structure is direct and the derivations look like they follow from known results without new inventions. That makes the computation useful for anyone who needs numbers on these spaces. The soft spot is the extension of the p-adic Hodge comparisons to non-proper Stein spaces. Most statements of those comparisons assume properness or some form of compactness to control the cohomology groups and ensure the identifications hold without extra terms. The Drinfeld space is smooth but open, so the paper has to show why the pro-étale site still gives the right U-cohomology here. If the justification is only a citation to the proper case, that step could be thin. The rest of the argument, including the Kummer sequences, appears routine once the U part is settled. This is for specialists in p-adic Hodge theory and rigid analytic geometry who already know the Drinfeld space and want a usable cohomology formula. A reader working on periods or Galois representations attached to these spaces would get direct value from the result. It is not foundational, but the engagement with the literature is honest and the claim is specific enough that it deserves a serious referee. A referee can verify the comparison step on the Stein case and confirm the exact sequences produce the stated formula. I would send it out for peer review.

Referee Report

1 major / 1 minor

Summary. The paper computes the étale G_m-cohomology of certain p-adic rigid analytic Stein spaces. It proceeds by filtering G_m via the subgroup U of principal units, determining the U-cohomology through p-adic Hodge theory (pro-étale site and comparison theorems with p-adic cohomologies) and the G_m/U-cohomology via Kummer exact sequences. The resulting formula is applied in particular to the Drinfeld upper half space.

Significance. If the comparison theorems extend to the non-proper Stein setting, the explicit formulas would give concrete descriptions of G_m-cohomology for key examples in p-adic rigid geometry, including the Drinfeld upper half space. This would strengthen the toolkit for relating étale cohomology to de Rham and crystalline data in non-proper cases.

major comments (1)

[U-cohomology via pro-étale site and p-adic Hodge comparisons] The central step identifying H^*(X, U) via pro-étale comparisons with p-adic cohomologies (abstract and the section on U-cohomology) is load-bearing but lacks explicit justification for non-proper Stein spaces. Standard pro-étale-to-de Rham or crystalline comparisons are typically stated for proper smooth rigid spaces over O_C; the manuscript does not cite a reduction, compact-support argument, or theorem covering the Stein case (e.g., Drinfeld upper half space).

minor comments (1)

[Abstract] The abstract refers to 'some p-adic rigid analytic Stein spaces' without an explicit list; adding a sentence naming the spaces treated (beyond the Drinfeld example) would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the paper's significance and for identifying the need for clearer justification of the key comparison step. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [U-cohomology via pro-étale site and p-adic Hodge comparisons] The central step identifying H^*(X, U) via pro-étale comparisons with p-adic cohomologies (abstract and the section on U-cohomology) is load-bearing but lacks explicit justification for non-proper Stein spaces. Standard pro-étale-to-de Rham or crystalline comparisons are typically stated for proper smooth rigid spaces over O_C; the manuscript does not cite a reduction, compact-support argument, or theorem covering the Stein case (e.g., Drinfeld upper half space).

Authors: We agree that the current manuscript does not provide an explicit reduction or citation justifying the pro-étale comparisons for non-proper Stein spaces, even though the pro-étale site and the underlying p-adic Hodge comparison theorems are formulated in a way that permits passage to direct limits over affinoid subdomains. Stein spaces are exhausted by increasing unions of affinoids, and the relevant cohomology groups commute with these direct limits in the pro-étale topology. In the revised version we will add a short subsection (or appendix) that spells out this reduction, citing the relevant statements from the literature on pro-étale cohomology that allow the extension to the Stein case, together with a brief verification that the Drinfeld upper half space satisfies the necessary hypotheses. This will make the load-bearing step fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; computation chains to external p-adic Hodge and Kummer methods

full rationale

The derivation splits Gm-cohomology via the U-filtration, then invokes standard pro-étale comparisons and Kummer sequences for the pieces. These are presented as established external tools rather than results derived inside the paper. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided abstract or summary. The approach is therefore self-contained against external benchmarks, yielding only a minor score for routine citation of prior theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the full set of assumptions cannot be audited. The approach rests on standard background results in étale cohomology and p-adic Hodge theory.

axioms (2)

standard math Standard comparison theorems between étale cohomology and p-adic cohomologies on the pro-étale site
Invoked to determine U-cohomology
standard math Kummer exact sequences for the quotient Gm/U
Used to obtain Gm/U-cohomology

pith-pipeline@v0.9.0 · 5410 in / 1249 out tokens · 77732 ms · 2026-05-08T15:37:19.484962+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

153 extracted references · 64 canonical work pages

[1]

Revisiting the de

Ardakov, Konstantin , TITLE =. Ast\'erisque , FJOURNAL =. 2021 , PAGES =. doi:10.24033/ast , URL =

work page doi:10.24033/ast 2021
[2]

Represent

Ardakov, Konstantin , TITLE =. Represent. Theory , FJOURNAL =. 2023 , PAGES =. doi:10.1090/ert/642 , URL =

work page doi:10.1090/ert/642 2023
[3]

, TITLE =

Ardakov, Konstantin and Wadsley, Simon J. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2019 , PAGES =. doi:10.1515/crelle-2016-0016 , URL =

work page doi:10.1515/crelle-2016-0016 2019
[4]

Ardakov, Konstantin and Wadsley, Simon , TITLE =. J. Algebraic Geom. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/jag/709 , URL =

work page doi:10.1090/jag/709 2018
[5]

Ardakov, Konstantin and Bode, Andreas and Wadsley, Simon , TITLE =. Compos. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1112/s0010437x21007521 , URL =

work page doi:10.1112/s0010437x21007521 2021
[6]

Abe, Tomoyuki , TITLE =. Compos. Math. , FJOURNAL =. 2010 , NUMBER =. doi:10.1112/S0010437X09004485 , URL =

work page doi:10.1112/s0010437x09004485 2010
[7]

Abe, Tomoyuki , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 2015 , NUMBER =. doi:10.24033/bsmf.2680 , URL =

work page doi:10.24033/bsmf.2680 2015
[8]

Abe, Tomoyuki , TITLE =. Rend. Semin. Mat. Univ. Padova , FJOURNAL =. 2014 , PAGES =. doi:10.4171/RSMUP/131-7 , URL =

work page doi:10.4171/rsmup/131-7 2014
[9]

Abe, Tomoyuki and Marmora, Adriano , TITLE =. J. Inst. Math. Jussieu , FJOURNAL =. 2015 , NUMBER =. doi:10.1017/S1474748014000024 , URL =

work page doi:10.1017/s1474748014000024 2015
[10]

, TITLE =

Garnier, L. , TITLE =. J. Algebra , FJOURNAL =. 1998 , NUMBER =. doi:10.1006/jabr.1997.7373 , URL =

work page doi:10.1006/jabr.1997.7373 1998
[11]

Bartenwerfer, Wolfgang , TITLE =. Nederl. Akad. Wetensch. Indag. Math. , FJOURNAL =. 1982 , NUMBER =

1982
[12]

, Isbn =

Berkovich, Vladimir G. , Isbn =. Spectral theory and analytic geometry over non-
[13]

, Fjournal =

Berkovich, Vladimir G. , Fjournal =. \'. Inst. Hautes \'. 1993 , Bdsk-Url-1 =

1993
[14]

Berkovich, Vladimir G. , Doi =. Vanishing cycles for formal schemes , Url =. Invent. Math. , Mrclass =. 1994 , Bdsk-Url-1 =

1994
[15]

Berkovich, Vladimir G. , Doi =. On the comparison theorem for \'. Israel J. Math. , Mrclass =. 1995 , Bdsk-Url-1 =

1995
[16]

, Fjournal =

Berkovich, Vladimir G. , Fjournal =. The automorphism group of the. C. R. Acad. Sci. Paris S\'
[17]

Berkovich, Vladimir G. , Doi =. Vanishing cycles for formal schemes. Invent. Math. , Mrclass =. 1996 , Bdsk-Url-1 =

1996
[18]

, TITLE =

Berkovich, Vladimir G. , TITLE =. Invent. Math. , FJOURNAL =. 1999 , NUMBER =. doi:10.1007/s002220050323 , URL =

work page doi:10.1007/s002220050323 1999
[19]

1996 , PAGES =

Huber, Roland , TITLE =. 1996 , PAGES =. doi:10.1007/978-3-663-09991-8 , URL =

work page doi:10.1007/978-3-663-09991-8 1996
[20]

Berthelot, Pierre , TITLE =. Ann. Sci. \'Ecole Norm. Sup. (4) , FJOURNAL =. 1996 , NUMBER =

1996
[21]

Berthelot, Pierre , TITLE =. M\'em. Soc. Math. Fr. (N.S.) , FJOURNAL =. 2000 , PAGES =

2000
[22]

Ast\'erisque , FJOURNAL =

Berthelot, Pierre , TITLE =. Ast\'erisque , FJOURNAL =. 2002 , PAGES =

2002
[23]

Journ\'ees de

Berthelot, Pierre and Messing, William , TITLE =. Journ\'ees de. 1979 , MRCLASS =

1979
[24]

Th\'eorie de

Berthelot, Pierre and Breen, Lawrence and Messing, William , TITLE =. 1982 , PAGES =. doi:10.1007/BFb0093025 , URL =

work page doi:10.1007/bfb0093025 1982
[25]

Berthelot, Pierre and Messing, William , TITLE =. The. 1990 , ISBN =

1990
[26]

Prépublication IRMAR , Volume =

Berthelot, Pierre , title =. Prépublication IRMAR , Volume =
[27]

Ast\'erisque , FJOURNAL =

Berthelot, Pierre , TITLE =. Ast\'erisque , FJOURNAL =. 1984 , PAGES =

1984
[28]

Berthelot, Pierre , TITLE =. Invent. Math. , FJOURNAL =. 1997 , NUMBER =. doi:10.1007/s002220050143 , URL =

work page doi:10.1007/s002220050143 1997
[29]

Dualit\'

Berthelot, Pierre , Doi =. Dualit\'. C. R. Acad. Sci. Paris S\'. 1997 , Bdsk-Url-1 =

1997
[30]

Study group on ultrametric analysis, 9th year: 1981/82,

Berthelot, Pierre , TITLE =. Study group on ultrametric analysis, 9th year: 1981/82,. 1983 , ISBN =

1981
[31]

Colmez, Pierre , TITLE =. Ast\'. 2010 , PAGES =

2010
[32]

On the cohomology of the affine space , BOOKTITLE =

Colmez, Pierre and Nizio. On the cohomology of the affine space , BOOKTITLE =. [2020] 2020 , ISBN =. doi:10.1007/978-3-030-43844-9\_2 , URL =

work page doi:10.1007/978-3-030-43844-9 2020
[33]

Colmez, Pierre and Nizio. On. M\"unster J. Math. , FJOURNAL =. 2020 , NUMBER =

2020
[34]

On the cohomology of

Colmez, Pierre and Nizio. On the cohomology of. J. Algebraic Geom. , FJOURNAL =. 2025 , NUMBER =

2025
[35]

Breuil, Christophe , TITLE =. Ann. Sci. \'. 2004 , NUMBER =. doi:10.1016/j.ansens.2004.02.001 , URL =

work page doi:10.1016/j.ansens.2004.02.001 2004
[36]

The image of

Pask\=. The image of. Publ. Math. Inst. Hautes \'. 2013 , PAGES =. doi:10.1007/s10240-013-0049-y , URL =

work page doi:10.1007/s10240-013-0049-y 2013
[37]

Scholze, Peter , TITLE =. Publ. Math. Inst. Hautes \'. 2012 , PAGES =. doi:10.1007/s10240-012-0042-x , URL =

work page doi:10.1007/s10240-012-0042-x 2012
[38]

Scholze, Peter , TITLE =. Invent. Math. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00222-012-0420-5 , URL =

work page doi:10.1007/s00222-012-0420-5 2013
[39]

Forum Math

Scholze, Peter , TITLE =. Forum Math. Pi , FJOURNAL =. 2013 , PAGES =. doi:10.1017/fmp.2013.1 , URL =

work page doi:10.1017/fmp.2013.1 2013
[40]

Current developments in mathematics 2012 , PAGES =

Scholze, Peter , TITLE =. Current developments in mathematics 2012 , PAGES =. 2013 , ISBN =

2012
[41]

Scholze, Peter , TITLE =. Ann. Sci. \'. 2018 , NUMBER =. doi:10.24033/asens.2367 , URL =

work page doi:10.24033/asens.2367 2018
[42]

Bhatt, Bhargav and Morrow, Matthew and Scholze, Peter , TITLE =. Publ. Math. Inst. Hautes \'. 2018 , PAGES =. doi:10.1007/s10240-019-00102-z , URL =

work page doi:10.1007/s10240-019-00102-z 2018
[43]

Bhatt, Bhargav and Morrow, Matthew and Scholze, Peter , TITLE =. Publ. Math. Inst. Hautes \'. 2019 , PAGES =. doi:10.1007/s10240-019-00106-9 , URL =

work page doi:10.1007/s10240-019-00106-9 2019
[44]

Scholze, Peter and Weinstein, Jared , TITLE =. Camb. J. Math. , FJOURNAL =. 2013 , NUMBER =. doi:10.4310/CJM.2013.v1.n2.a1 , URL =

work page doi:10.4310/cjm.2013.v1.n2.a1 2013
[45]

Ertl, Veronika and Gilles, Sally and Nizio. On the. Int. Math. Res. Not. IMRN , FJOURNAL =. 2024 , NUMBER =. doi:10.1093/imrn/rnae185 , URL =

work page doi:10.1093/imrn/rnae185 2024
[46]

Forum Math

Heuer, Ben , TITLE =. Forum Math. Sigma , FJOURNAL =. 2022 , PAGES =. doi:10.1017/fms.2022.72 , URL =

work page doi:10.1017/fms.2022.72 2022
[47]

Raynaud, Michel , Fjournal =. Sch\'. Bull. Soc. Math. France , Mrclass =. 1974 , Bdsk-Url-1 =

1974
[49]

Cohomologie

Colmez, Pierre and Dospinescu, Gabriel and Nizio , Wies. Cohomologie. J. Amer. Math. Soc. , Mrclass =. 2020 , Bdsk-Url-1 =. doi:10.1090/jams/935 , Fjournal =

work page doi:10.1090/jams/935 2020
[50]

Cohomology of

Colmez, Pierre and Dospinescu, Gabriel and Nizio , Wies. Cohomology of. Invent. Math. , Mrclass =. 2020 , Bdsk-Url-1 =. doi:10.1007/s00222-019-00923-z , Fjournal =

work page doi:10.1007/s00222-019-00923-z 2020
[51]

Integral

Colmez, Pierre and Dospinescu, Gabriel and Nizio , Wies. Integral. Duke Math. J. , FJOURNAL =. 2021 , NUMBER =. doi:10.1215/00127094-2020-0084 , URL =

work page doi:10.1215/00127094-2020-0084 2021
[52]

Colmez, Pierre and Dospinescu, Gabriel and Hauseux, Julien and Nizio , Wies. Math. Ann. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00208-020-02139-6 , URL =

work page doi:10.1007/s00208-020-02139-6 2021
[53]

Dospinescu, Gabriel and Le Bras, Arthur-C\'. Rev\^. Ann. of Math. (2) , FJOURNAL =. 2017 , NUMBER =. doi:10.4007/annals.2017.186.2.1 , URL =

work page doi:10.4007/annals.2017.186.2.1 2017
[54]

Duke Math

Le Bras, Arthur-C\'esar , TITLE =. Duke Math. J. , FJOURNAL =. 2018 , NUMBER =. doi:10.1215/00127094-2018-0034 , URL =

work page doi:10.1215/00127094-2018-0034 2018
[55]

Orlik, Sascha , TITLE =. Invent. Math. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s00222-005-0452-1 , URL =

work page doi:10.1007/s00222-005-0452-1 2005
[56]

Orlik, Peter and Solomon, Louis , TITLE =. Invent. Math. , FJOURNAL =. 1980 , NUMBER =. doi:10.1007/BF01392549 , URL =

work page doi:10.1007/bf01392549 1980
[57]

Bosco, Guido , Note =
[58]

Lubin, Jonathan and Tate, John , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1965 , PAGES =. doi:10.2307/1970622 , URL =

work page doi:10.2307/1970622 1965
[59]

Lubin, Jonathan and Tate, John , TITLE =. Bull. Soc. Math. France , FJOURNAL =. 1966 , PAGES =

1966
[60]

Saito, Takeshi , TITLE =. Invent. Math. , FJOURNAL =. 1993 , NUMBER =. doi:10.1007/BF01244312 , URL =

work page doi:10.1007/bf01244312 1993
[61]

, Mrclass =

Froohlich, A. , Mrclass =. Formal groups , Year =
[62]

Supercuspidal representations in the cohomology of

Harris, Michael , Doi =. Supercuspidal representations in the cohomology of. Invent. Math. , Mrclass =. 1997 , Bdsk-Url-1 =

1997
[63]

Boyer, P. , Doi =. Mauvaise r\'. Invent. Math. , Mrclass =. 1999 , Bdsk-Url-1 =

1999
[64]

Uniformisation des vari\'

Hausberger, Thomas , Fjournal =. Uniformisation des vari\'. Ann. Inst. Fourier (Grenoble) , Mrclass =. 2005 , Bdsk-Url-1 =

2005
[65]

, Fjournal =

Van der Put, M. , Fjournal =. Cohomology on affinoid spaces , Url =. Compositio Math. , Mrclass =. 1982 , Bdsk-Url-1 =

1982
[66]

Groupe de travail d'analyse ultram\'etrique , note =

Van der Put, Marius , title =. Groupe de travail d'analyse ultram\'etrique , note =. 1981-1982 , zbl =

1981
[67]

Logarithmic differential forms on

Iovita, Adrian and Spiess, Michael , Doi =. Logarithmic differential forms on. Duke Math. J. , Mrclass =. 2001 , Bdsk-Url-1 =

2001
[68]

and Stuhler, U

Schneider, P. and Stuhler, U. , Doi =. The cohomology of. Invent. Math. , Mrclass =. 1991 , Bdsk-Url-1 =

1991
[69]

and Stuhler, U

Schneider, P. and Stuhler, U. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1993 , PAGES =. doi:10.1515/crll.1993.436.19 , URL =

work page doi:10.1515/crll.1993.436.19 1993
[70]

Schneider, Peter and Stuhler, Ulrich , TITLE =. Inst. Hautes \'Etudes Sci. Publ. Math. , FJOURNAL =. 1997 , PAGES =

1997
[71]

Residues on buildings and de

de Shalit, Ehud , Doi =. Residues on buildings and de. Duke Math. J. , Mrclass =. 2001 , Bdsk-Url-1 =

2001
[72]

On the cohomology of

Alon, Gil and de Shalit, Ehud , Date-Modified =. On the cohomology of. Israel J. Math. , Mrclass =. 2002 , Bdsk-Url-1 =. doi:10.1007/BF02773150 , Fjournal =

work page doi:10.1007/bf02773150 2002
[73]

On non-abelian

Yoshida, Teruyoshi , Booktitle =. On non-abelian
[74]

and Lusztig, G

Deligne, P. and Lusztig, G. , Doi =. Representations of reductive groups over finite fields , Url =. Ann. of Math. (2) , Mrclass =. 1976 , Bdsk-Url-1 =

1976
[75]

Rigid analytic spaces with overconvergent structure sheaf , Url =

Grosse-Kl\". Rigid analytic spaces with overconvergent structure sheaf , Url =. J. Reine Angew. Math. , Mrclass =. 2000 , Bdsk-Url-1 =. doi:10.1515/crll.2000.018 , Fjournal =

work page doi:10.1515/crll.2000.018 2000
[76]

Finiteness of de

Grosse-Kl\". Finiteness of de. Duke Math. J. , Mrclass =. 2002 , Bdsk-Url-1 =. doi:10.1215/S0012-7094-02-11312-X , Fjournal =

work page doi:10.1215/s0012-7094-02-11312-x 2002
[77]

Grosse-Kl\". De. Math. Z. , Mrclass =. 2004 , Bdsk-Url-1 =. doi:10.1007/s00209-003-0544-9 , Fjournal =

work page doi:10.1007/s00209-003-0544-9 2004
[78]

Sheaves of bounded

Grosse-Kl\". Sheaves of bounded. Ann. Sci. \'. 2007 , Bdsk-Url-1 =. doi:10.1016/j.ansens.2007.04.001 , Fjournal =

work page doi:10.1016/j.ansens.2007.04.001 2007
[79]

On the crystalline cohomology of

Grosse-Kl\". On the crystalline cohomology of. Finite Fields Appl. , Mrclass =. 2007 , Bdsk-Url-1 =. doi:10.1016/j.ffa.2006.06.001 , Fjournal =

work page doi:10.1016/j.ffa.2006.06.001 2007
[80]

Integral structures in automorphic line bundles on the

Grosse-Kl\". Integral structures in automorphic line bundles on the. Math. Ann. , FJOURNAL =. 2004 , NUMBER =. doi:10.1007/s00208-004-0512-7 , URL =

work page doi:10.1007/s00208-004-0512-7 2004
[81]

L'espace sym\'

Wang, Haoran , Doi =. L'espace sym\'. Math. Z. , Mrclass =. 2014 , Bdsk-Url-1 =

2014

Showing first 80 references.