Mathematical  Culture and Thought

Mathematical Culture and Thought

Cohomology ‎T‎heories and Changing the Weight of Galios Representations

Document Type : Original

Authors
School of Mathematics‎, ‎IPM‎, ‎Iran
Abstract
In this paper, we first introduce various cohomology theories and their relationship with Galois group representations. Then, we present a method that naturally associates certain Galois group representations with Hodge-Tate weights greater than one to representations with lower weights, and under certain conditions, to representations with weights zero and one. More precisely, we prove the existence of a fully faithful functor from the category of graded sparse regular representations with good prismatic reduction of the Galois group to the category of graded representations with good prismatic reduction and weights zero and one.
Keywords
Subjects

[۱] بهزادی‌پور، حسین؛ پیردایه، جلال؛ رستگار، آرش، نظریۀ میدان‌های رده‌ای روی میدان تک عضوی، فرهنگ و اندیشه ریاضی، ۴۴ (۱۴۰۴)، شماره ۲، ۲۱۹-۲۳۴.
[۲] کاشانی، سید محمدباقر، تقارن آینه‌ای، فرهنگ و اندیشه ریاضی، ۴۴ (۱۴۰۴)، شماره ۱، ۱۷۷-۲۱۰.
[3]    André, Y., La conjecture du facteur direct, Publ. Math. IHES, 127 (2018), 71-93.
[4]    Berthelot, P., Ogus, A., Notes on Crystalline Cohomology, vol. 21, Princeton University Press Princeton, 2015.
[5]    Bhatt, B., Scholze, P., Prismatic F -crystals and crystalline Galois representations (2021), available at arXiv:2106.14735.
[6]    Bhatt, B., Scholze, P., Prisms and prismatic cohomology, Ann. of Math., 196 (2022), 1135-1275.
[7]    Brinon, O., Conrad, B., Preparatory Notes on p-adic Hodge Theory, Clay Mathematical Institute Summer School, Honolulu, Hawaii, 2009.
[8]    Bültel, O., PEL moduli spaces without C-valued points (2008), available at arXiv:0808.4091.
[9]    Faltings, G., p-adic Hodge theory, J. Amer. Math. Soc., 1 (1988), 255-299.
[10]    Hahn, J., Raksit, A., Wilson, D., A motivic filtration on the topological cyclic homology of commutative ring spectra (2022), available at arXiv:2206.11208.
111 Hedayatzadeh, S., 211002110, A., Deformations of prismatic higher ((,/لم )-displays over quasi-syntomic rings (2024), available at arXiv:2402.12879.
[12]    Hedayatzadeh, S., Partofard, A., Embeddings of some exceptional Shimura varieties into Siegel variety (work in progress).
[13]    Katz, N., Slope filtration of F-crystals, Astérisque, 63 (1979), 979.
[14]    Kisin, M., Crystalline representations and F -crystals, in Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday, G. David, ed., Birkhäuser Boston, Boston, MA, 2006, 459-496.
[15]    Milne, J. S. , Motives—Grothendieck’s dream, in Open Problems and Surveys of Contemporary Mathematics, vol. 6, 2013, 325-342.
[16]    Scholze, P., Weinstein, J., Berkeley Lectures on p-adic Geometry, Princeton University Press, Princeton, 2020.
[17]    Voevodsky, V., Triangulated categories of motives over a field, in Cycles, Transfers, and Motivic Homology Theories, Princeton University Press, Princeton, 2000.
[18]    Zink, T., The display of a formal p-divisible group, Astérisque, 278 (2002), 127-248.

  • Receive Date 15 March 2025
  • Revise Date 10 October 2025
  • Accept Date 11 October 2025
  • Publish Date 22 May 2026