
@incollection{BB19,
  author = {Barthel, Tobias and Beaudry, Agn{\`e}s},
  title = {Chromatic structures in stable homotopy theory},
  booktitle = {Handbook of Homotopy Theory},
  editor = {Miller, Haynes},
  publisher = {CRC Press},
  address = {Boca Raton, FL},
  year = {2020},
  pages = {163--220},
  note = {Citations use arXiv:1901.09004v2, 29 April 2019},
  url = {https://arxiv.org/abs/1901.09004v2}
}

@incollection{Hovey93,
  author = {Hovey, Mark},
  title = {{Bousfield} localization functors and {Hopkins}' chromatic splitting conjecture},
  booktitle = {The {\v C}ech Centennial},
  series = {Contemporary Mathematics},
  volume = {181},
  publisher = {American Mathematical Society},
  address = {Providence, RI},
  year = {1995},
  pages = {225--250},
  note = {Citations use the author manuscript dated July 1993},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/chromatic-splitting.pdf}
}

@unpublished{BMR26,
  author = {Barthel, Tobias and Mann, Lucas and Ray, Rin and Schlank, Tomer M.
            and Senger, Andrew and Weinstein, Jared and Zhou, Xinyu},
  title = {On the chromatic splitting conjecture in coheight 1},
  year = {2026},
  note = {Preprint dated 26 August 2026; citations use that version},
  url = {https://math.bu.edu/people/jsweinst/ChromaticSplitting.pdf}
}

@article{ALB25,
  author = {Ansch{\"u}tz, Johannes and Le Bras, Arthur-C{\'e}sar},
  title = {A {Fourier} transform for {Banach--Colmez} spaces},
  journal = {Journal of the European Mathematical Society},
  volume = {27},
  number = {9},
  year = {2025},
  pages = {3651--3712},
  doi = {10.4171/JEMS/1480},
  url = {https://ems.press/journals/jems/articles/14297884},
  note = {Citations use the published numbering}
}

@article{FS21,
  author = {Fargues, Laurent and Scholze, Peter},
  title = {Geometrization of the local {Langlands} correspondence},
  journal = {Ast{\'e}risque},
  volume = {466},
  year = {2026},
  note = {Numbered citations use the accepted version arXiv:2102.13459v4,
          27 November 2024},
  url = {https://arxiv.org/abs/2102.13459v4}
}

@article{ScholzeDiamonds17,
  author = {Scholze, Peter},
  title = {{\'E}tale cohomology of diamonds},
  journal = {Ast{\'e}risque},
  volume = {466},
  year = {2026},
  note = {Numbered citations use arXiv:1709.07343v4, 14 April 2026},
  url = {https://arxiv.org/abs/1709.07343v4}
}

@article{SW13,
  author = {Scholze, Peter and Weinstein, Jared},
  title = {Moduli of {$p$}-divisible groups},
  journal = {Cambridge Journal of Mathematics},
  volume = {1},
  number = {2},
  year = {2013},
  pages = {145--237},
  url = {https://archive.intlpress.com/site/pub/files/_fulltext/journals/cjm/2013/0001/0002/CJM-2013-0001-0002-a001.pdf}
}

@article{LT66,
  author = {Lubin, Jonathan and Tate, John},
  title = {Formal moduli for one-parameter formal {Lie} groups},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume = {94},
  year = {1966},
  pages = {49--59},
  doi = {10.24033/bsmf.1633},
  url = {https://www.numdam.org/articles/10.24033/bsmf.1633/},
  note = {Citations use the original published numbering in the Numdam scan}
}

@article{Lazard55,
  author = {Lazard, Michel},
  title = {Sur les groupes de {Lie} formels {\`a} un param{\`e}tre},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume = {83},
  year = {1955},
  pages = {251--274},
  doi = {10.24033/bsmf.1462},
  url = {https://www.numdam.org/articles/10.24033/bsmf.1462/},
  note = {Citations use the original published numbering in the Numdam scan}
}

@article{Lau10,
  author = {Lau, Eike},
  title = {{Tate} modules of universal {$p$}-divisible groups},
  journal = {Compositio Mathematica},
  volume = {146},
  number = {1},
  year = {2010},
  pages = {220--232},
  doi = {10.1112/S0010437X09004242},
  url = {https://doi.org/10.1112/S0010437X09004242},
  note = {Citations use the published numbering}
}

@article{OortZink02,
  author = {Oort, Frans and Zink, Thomas},
  title = {Families of {$p$}-divisible groups with constant {Newton} polygon},
  journal = {Documenta Mathematica},
  volume = {7},
  year = {2002},
  pages = {183--201},
  doi = {10.4171/DM/123},
  url = {https://ems.press/content/serial-article-files/25903}
}

@article{DH04,
  author = {Devinatz, Ethan S. and Hopkins, Michael J.},
  title = {Homotopy fixed point spectra for closed subgroups of the {Morava} stabilizer groups},
  journal = {Topology},
  volume = {43},
  number = {1},
  year = {2004},
  pages = {1--47},
  doi = {10.1016/S0040-9383(03)00029-6},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/dh04.pdf},
  note = {Citations use the published version}
}

@article{Mathew16,
  author = {Mathew, Akhil},
  title = {The {Galois} group of a stable homotopy theory},
  journal = {Advances in Mathematics},
  volume = {291},
  year = {2016},
  pages = {403--541},
  doi = {10.1016/j.aim.2015.12.017},
  url = {https://arxiv.org/abs/1404.2156v2},
  note = {Numbered citations use arXiv:1404.2156v2}
}

@article{HS99,
  author = {Hovey, Mark and Strickland, Neil P.},
  title = {{Morava} {$K$}-theories and localisation},
  journal = {Memoirs of the American Mathematical Society},
  volume = {139},
  number = {666},
  year = {1999},
  pages = {viii+100},
  doi = {10.1090/memo/0666},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/kn.pdf},
  note = {Citations use the author manuscript hosted by Ravenel}
}

@unpublished{Hovey04,
  author = {Hovey, Mark},
  title = {Some spectral sequences in {Morava} {$E$}-theory},
  year = {2004},
  month = {1 March},
  note = {Preprint},
  url = {https://drewkh.folk.ntnu.no/hovey_ss.pdf}
}

@article{BGHS22,
  author = {Beaudry, Agn{\`e}s and Goerss, Paul G. and Hopkins, Michael J.
            and Stojanoska, Vesna},
  title = {Dualizing spheres for compact {$p$}-adic analytic groups and duality in chromatic homotopy},
  journal = {Inventiones Mathematicae},
  volume = {229},
  number = {3},
  year = {2022},
  pages = {1301--1434},
  doi = {10.1007/s00222-022-01120-1},
  url = {https://arxiv.org/abs/2010.09518v2}
}

@article{AW06,
  author = {Ardakov, Konstantin and Wadsley, Simon},
  title = {Characteristic elements for {$p$}-torsion {Iwasawa} modules},
  journal = {Journal of Algebraic Geometry},
  volume = {15},
  number = {2},
  year = {2006},
  pages = {339--377},
  doi = {10.1090/S1056-3911-05-00415-7},
  url = {https://people.maths.ox.ac.uk/ardakov/eulerchars.pdf},
  note = {Citations use the Oxford author manuscript}
}

@incollection{Tate67,
  author = {Tate, John T.},
  title = {{$p$}-Divisible Groups},
  booktitle = {Proceedings of a Conference on Local Fields},
  editor = {Springer, T. A.},
  publisher = {Springer},
  address = {Berlin, Heidelberg},
  year = {1967},
  pages = {158--183},
  doi = {10.1007/978-3-642-87942-5_12},
  url = {https://www.math.purdue.edu/~tongliu/teaching/598/p-divisible.pdf},
  note = {Citations use the original proceedings pagination}
}

@misc{CompanionFiltered,
  author = {{OpenAI}},
  title = {{Filtered chromatic splitting at generic primes}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf}{OAI:Filtered-chromatic-splitting-at-generic-primes-September-25-2026}},
  year = {2026}
}

@misc{CompanionRational,
  author = {{OpenAI}},
  title = {{A rational obstruction to strong chromatic splitting at height three}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026/paper.pdf}{OAI:A-rational-obstruction-to-strong-chromatic-splitting-at-height-three-September-25-2026}},
  year = {2026}
}

@article{Beaudry2017,
  author = {Beaudry, Agn{\`e}s},
  title = {The chromatic splitting conjecture at {$n=p=2$}},
  journal = {Geometry \& Topology},
  volume = {21},
  number = {6},
  year = {2017},
  pages = {3213--3230},
  doi = {10.2140/gt.2017.21.3213},
  url = {https://msp.org/gt/2017/21-6/gt-v21-n6-p02-s.pdf},
  note = {Statement locators refer to the published version}
}

@article{BGH2022,
  author = {Beaudry, Agn{\`e}s and Goerss, Paul G. and Henn, Hans-Werner},
  title = {Chromatic splitting for the {$K(2)$}-local sphere at {$p=2$}},
  journal = {Geometry \& Topology},
  volume = {26},
  number = {1},
  year = {2022},
  pages = {377--476},
  doi = {10.2140/gt.2022.26.377},
  eprint = {1712.08182v4},
  archivePrefix = {arXiv},
  primaryClass = {math.AT},
  url = {https://arxiv.org/abs/1712.08182v4},
  note = {Statement locators refer to arXiv:1712.08182v4, submitted 6 October 2020}
}

@article{Quillen69,
  author = {Quillen, Daniel},
  title = {On the formal group laws of unoriented and complex cobordism theory},
  journal = {Bulletin of the American Mathematical Society},
  volume = {75},
  year = {1969},
  pages = {1293--1298},
  doi = {10.1090/S0002-9904-1969-12401-8},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/QuillenBP.pdf}
}

@article{Powell12,
  author = {Powell, Geoffrey},
  title = {On the double transfer and the {$f$}-invariant},
  journal = {Glasgow Mathematical Journal},
  volume = {54},
  number = {3},
  year = {2012},
  pages = {547--577},
  doi = {10.1017/S0017089512000158},
  url = {https://arxiv.org/abs/1005.0918v2},
  note = {Numbered citations use arXiv:1005.0918v2}
}

@article{Ravenel84,
  author = {Ravenel, Douglas C.},
  title = {Localization with respect to certain periodic homology theories},
  journal = {American Journal of Mathematics},
  volume = {106},
  number = {2},
  year = {1984},
  pages = {351--414},
  doi = {10.2307/2374308},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/mypapers/loc.pdf}
}

@article{Scholze12,
  author = {Scholze, Peter},
  title = {Perfectoid spaces},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume = {116},
  year = {2012},
  pages = {245--313},
  doi = {10.1007/s10240-012-0042-x},
  url = {https://arxiv.org/abs/1111.4914v1},
  note = {Numbered citations use arXiv:1111.4914v1}
}

@article{Cartier58,
  author = {Cartier, Pierre},
  title = {Questions de rationalit{\'e} des diviseurs en g{\'e}om{\'e}trie alg{\'e}brique},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume = {86},
  year = {1958},
  pages = {177--251},
  doi = {10.24033/bsmf.1503},
  url = {https://www.numdam.org/articles/10.24033/bsmf.1503/}
}

@incollection{Serre58Topology,
  author = {Serre, Jean-Pierre},
  title = {Sur la topologie des vari{\'e}t{\'e}s alg{\'e}briques en caract{\'e}ristique {$p$}},
  booktitle = {Symposium internacional de topolog{\'i}a algebraica},
  publisher = {Universidad Nacional Aut{\'o}noma de M{\'e}xico and UNESCO},
  address = {Mexico City},
  year = {1958},
  pages = {24--53},
  url = {https://www.math.uchicago.edu/~emerton/prismatic/Serre-mexico.pdf},
  note = {Citations use the original proceedings pagination}
}

@article{DH95,
  author = {Devinatz, Ethan S. and Hopkins, Michael J.},
  title = {The action of the {Morava} stabilizer group on the {Lubin--Tate} moduli space of lifts},
  journal = {American Journal of Mathematics},
  volume = {117},
  number = {3},
  year = {1995},
  pages = {669--710},
  doi = {10.2307/2375086},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/dh95.pdf},
  note = {Citations use the published June 1995 version}
}

@incollection{GH04,
  author = {Goerss, Paul G. and Hopkins, Michael J.},
  title = {Moduli spaces of commutative ring spectra},
  booktitle = {Structured Ring Spectra},
  editor = {Baker, Andrew and Richter, Birgit},
  series = {London Mathematical Society Lecture Note Series},
  volume = {315},
  publisher = {Cambridge University Press},
  address = {Cambridge},
  year = {2004},
  pages = {151--200},
  doi = {10.1017/CBO9780511529955.009},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/sum.pdf},
  note = {Numbered citations and manuscript page locators use the undated authors' manuscript version at the URL}
}

@incollection{Illusie85,
  author = {Illusie, Luc},
  title = {D\'eformations de groupes de {Barsotti--Tate}},
  booktitle = {S\'eminaire sur les pinceaux arithm\'etiques : la conjecture de {Mordell}},
  series = {Ast\'erisque},
  number = {127},
  publisher = {Soci\'et\'e Math\'ematique de France},
  year = {1985},
  pages = {151--198},
  url = {https://www.numdam.org/item/AST_1985__127__151_0/},
  note = {Citations use the original published numbering in the Numdam scan}
}

@book{Messing72,
  author = {Messing, William},
  title = {The Crystals Associated to {Barsotti--Tate} Groups: With Applications to Abelian Schemes},
  series = {Lecture Notes in Mathematics},
  volume = {264},
  publisher = {Springer-Verlag},
  address = {Berlin, Heidelberg},
  year = {1972},
  doi = {10.1007/BFb0058301},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/messing72.pdf},
  note = {Chapter and page locators use the original book pagination}
}

@incollection{Gross79,
  author = {Gross, Benedict H.},
  title = {Ramification in {$p$}-adic {Lie} extensions},
  booktitle = {Journ\'ees de g\'eom\'etrie alg\'ebrique de Rennes (1978), III},
  series = {Ast\'erisque},
  number = {65},
  publisher = {Soci\'et\'e Math\'ematique de France},
  year = {1979},
  pages = {81--102},
  url = {https://www.numdam.org/item/AST_1979__65__81_0/},
  note = {Citations use the original published numbering}
}

@article{Torii03,
  author = {Torii, Takeshi},
  title = {On degeneration of one-dimensional formal group laws and applications to stable homotopy theory},
  journal = {American Journal of Mathematics},
  volume = {125},
  number = {5},
  year = {2003},
  pages = {1037--1077},
  doi = {10.1353/ajm.2003.0036},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/torii1.pdf},
  note = {Citations use the published numbering}
}

@article{Torii11,
  author = {Torii, Takeshi},
  title = {{$K(n)$}-localization of the {$K(n+1)$}-local {$E_{n+1}$}-{Adams} spectral sequences},
  journal = {Pacific Journal of Mathematics},
  volume = {250},
  number = {2},
  year = {2011},
  pages = {439--471},
  doi = {10.2140/pjm.2011.250.439},
  url = {https://msp.org/pjm/2011/250-2/pjm-v250-n2-p10-s.pdf},
  note = {Citations use the published numbering}
}

@article{Heyer23,
  author = {Heyer, Claudius},
  title = {The left adjoint of derived parabolic induction},
  journal = {Mathematische Zeitschrift},
  volume = {305},
  year = {2023},
  doi = {10.1007/s00209-023-03385-5},
  url = {https://link.springer.com/article/10.1007/s00209-023-03385-5},
  note = {Article 46, 60 pages. Citations use the published numbering; Corollary 5.3.4 is on pages 57--58}
}

@misc{HeyerGeo24,
  author = {Heyer, Claudius},
  title = {The Geometrical Lemma for Smooth Representations in Natural Characteristic},
  year = {2024},
  eprint = {2303.14721v2},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2303.14721v2},
  note = {Preprint version 2, 18 January 2024. Numbered citations use the versioned PDF, not experimental HTML}
}

@article{Sinha09,
  author = {Sinha, Dev P.},
  title = {The topology of spaces of knots: cosimplicial models},
  journal = {American Journal of Mathematics},
  volume = {131},
  number = {4},
  pages = {945--980},
  year = {2009},
  eprint = {math/0202287v7},
  archivePrefix = {arXiv},
  primaryClass = {math.AT},
  url = {https://arxiv.org/abs/math/0202287v7},
  note = {The cited definition and theorem numbers are those of the finalized arXiv version, 16 March 2009}
}

@book{Lurie09,
  author = {Lurie, Jacob},
  title = {Higher Topos Theory},
  series = {Annals of Mathematics Studies},
  number = {170},
  publisher = {Princeton University Press},
  address = {Princeton and Oxford},
  year = {2009},
  url = {https://people.math.harvard.edu/~lurie/papers/HTT.pdf}
}

@article{ShimomuraYabe95,
  author = {Shimomura, Katsumi and Yabe, Atsuko},
  title = {The homotopy groups {$\pi_*(L_2S^0)$}},
  journal = {Topology},
  volume = {34},
  number = {2},
  year = {1995},
  pages = {261--289},
  doi = {10.1016/0040-9383(94)00032-G},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/shimomura-yabe.pdf}
}

@article{Behrens12,
  author = {Behrens, Mark},
  title = {The homotopy groups of {$S_{E(2)}$} at {$p\geq5$} revisited},
  journal = {Advances in Mathematics},
  volume = {230},
  number = {2},
  year = {2012},
  pages = {458--492},
  doi = {10.1016/j.aim.2012.02.023},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/Behrens-revisit.pdf}
}

@article{GHM14,
  author = {Goerss, Paul G. and Henn, Hans-Werner and Mahowald, Mark},
  title = {The rational homotopy of the {$K(2)$}-local sphere and the chromatic splitting conjecture for the prime {$3$} and level {$2$}},
  journal = {Documenta Mathematica},
  volume = {19},
  year = {2014},
  pages = {1271--1290},
  url = {https://ems.press/content/serial-article-files/26260},
  note = {Citations use the published Theorem~5.11}
}
