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題名 投射有限群表現之形變理論
Deformation Theory of Representations of Profinite Groups
作者 周惠雯
Chou, Hui Wen
貢獻者 余屹正
Yu, Yih Jeng
周惠雯
Chou, Hui Wen
關鍵詞 投射有限群
表現
形變
泛形變
泛形變環
扎里斯基切空間
Profinite groups
Representations
Deformations
Universal deformations
Universal deformation rings
Zariski tangent space
Group cohomology
日期 2012
上傳時間 1-Feb-2013 16:53:18 (UTC+8)
摘要 在本碩士論文中, 我們闡述了投射有限群表現, 以及其形變理論。 我們亦特別研究這些表示在 GL_1 和 GL_2 之形變, 並且給了可表示化 的判定準則。 最後, 我們解釋相對應的泛形變環之扎里斯基切空間與 群餘調之關連, 並計算了 GL_1 的泛形變表現。
In this master thesis, we give an exposition of the deformation theory of representations for GL_1 and GL_2, respectively, of certain profinite groups. We give rigidity conditions of the fixed representation and verify several conditions for the representability. Finally, we interpret the Zariski tangent spaces of respective universal deformation rings as certain group cohomology and calculate the universal deformation for GL_1.
參考文獻 [1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
[2] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939.
[3] J. W. S. Cassels and A. Fro ̈hlich (eds.), Algebraic Number Theory (2nd edition), London, United Kingdom, London Mathematical Society, 2010, Reprint of the 1967 original. MR 911121 (88h:11073)
[4] L. Clozel, M. Harris, and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. Inst. Hautes E ́tudes Sci. (2008), no. 108, 1–181, With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vigne ́ras.
[5] B. Conrad, F. Diamond, and R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 3, 521–567.
[6] B. de Smit and H. W. Lenstra, Jr. , Explicit construction of universal deformation rings, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, 1997, pp. 313–326.
[7] F. Diamond, On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), no. 1, 137–166.
[8] J.-M. Fontaine and B. Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78.
[9] A. Grothendieck, Technique de descente et th ́eor"emes d’existence en g ́eom ́etrie alg ́ebrique. II. Le th ́eor"eme d’existence en th ́eorie formelle des modules, Se ́minaire Bourbaki, vol. 5, Socie ́te ́ Mathe ́matique de France, 1995, pp. 369–390.
[10] A. Grothendieck and J. Dieudonne ́, El ́ements de G ́eom ́etrie Alg ́ebrique, Publ. Math. IHES,4 (1960), 8 (1961), 11 (1961), 17 (1963), 20 (1964), 24 (1965), 28 (1966), 32 (1967).
[11] K. Haberlan, Galois Cohomology of Algebraic Number Fields, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, With two appendices by Helmut Koch and Thomas Zink.
[12] M. Harris, N.Shepherd-Barron, and R. Taylor, A family of Calabi-Yau varieties and potential automorphy, Ann. of Math. (2) 171 (2010), no. 2, 779–813.
[13] H. Hida, Galois representations into GL_2(Z_p[X]) attached to ordinary cusp forms, Invent. Math. 85 (1986), no. 3, 545–613.
[14] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504.
[15] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586.
[16] M. Kisin, The Fontaine-Mazur conjecture for GL_2, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690.
[17] M. Kisin, Lecture Notes on Deformations of Galois Representations, Clay Mathematics Institute 2009 Summer School on Galois Representations (University of Hawaii at Manoa, Honolulu, Hawaii), June 15 - July 10 2009.
[18] M. Kisin, Moduli of finite flat group schemes and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180.
[19] S. Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, 1998.
[20] B. Mazur, Deforming of Galois Representations, Galois Groups over Q (Y. Ihara, K. Ribet, and J.-P. Serre, eds.), Mathematical Sciences Research Institute Publications, no. 16, Springer-Verlag, 1987, pp. 385–437.
[21] B. Mazur, An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer-Verlag, New York, 1997, pp. 243–311.
[22] J. Neukirch, Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Springer-Verlag, 1999.
[23] R. Ramakrishna, On a variation of Mazur’s deformation functor, Compositio Math. 87 (1993), 269–286.
[24] M. Schlessinger, Functors of Artin rings, Trans. A.M.S. 130 (1968), 208–222.
[25] J.-P. Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, 1979.
[26] J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, 1996.
[27] J.-P. Serre, Galois cohomology, Springer Monographs in Mathematics, Springer-Verlag, 1997.
[28] R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II, Publ. Math. Inst. Hautes E ́tudes Sci. (208), no. 108, 183–239.
[29] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), no. 3, 553–572.
[30] J. Tilouine, Deformations of Galois representations and Hecke algebras, Published for The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, 1996.
[31] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. of Math. 141(1995), no.3, 443–551.
描述 碩士
國立政治大學
應用數學研究所
99751014
101
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0099751014
資料類型 thesis
dc.contributor.advisor 余屹正zh_TW
dc.contributor.advisor Yu, Yih Jengen_US
dc.contributor.author (Authors) 周惠雯zh_TW
dc.contributor.author (Authors) Chou, Hui Wenen_US
dc.creator (作者) 周惠雯zh_TW
dc.creator (作者) Chou, Hui Wenen_US
dc.date (日期) 2012en_US
dc.date.accessioned 1-Feb-2013 16:53:18 (UTC+8)-
dc.date.available 1-Feb-2013 16:53:18 (UTC+8)-
dc.date.issued (上傳時間) 1-Feb-2013 16:53:18 (UTC+8)-
dc.identifier (Other Identifiers) G0099751014en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/56880-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 99751014zh_TW
dc.description (描述) 101zh_TW
dc.description.abstract (摘要) 在本碩士論文中, 我們闡述了投射有限群表現, 以及其形變理論。 我們亦特別研究這些表示在 GL_1 和 GL_2 之形變, 並且給了可表示化 的判定準則。 最後, 我們解釋相對應的泛形變環之扎里斯基切空間與 群餘調之關連, 並計算了 GL_1 的泛形變表現。zh_TW
dc.description.abstract (摘要) In this master thesis, we give an exposition of the deformation theory of representations for GL_1 and GL_2, respectively, of certain profinite groups. We give rigidity conditions of the fixed representation and verify several conditions for the representability. Finally, we interpret the Zariski tangent spaces of respective universal deformation rings as certain group cohomology and calculate the universal deformation for GL_1.en_US
dc.description.tableofcontents 謝辭 v
Abstract vi
摘要 vii
Notations viii
Contents xi
1 Introduction 1
2 Profinite Groups and their Representations 6
2.1 Projective limits 6
2.2 Profinite groups 7
2.3 Representations of profinite groups 9
2.4 The p-finiteness condition 12
3 Deformation Theory 15
3.1 The ring of Witt vectors 15
3.2 The deformation functor 17
3.3 Pro-representability 20
3.4 Schlessinger’s criteria 23
3.5 The Zariski tangent space and its cohomological interpretation 25
4 The Existence of the Universal Deformation 31
4.1 Verification of condition (H1) 31
4.2 Verification of condition (H2) 33
4.3 Verification of condition (H3) 33
4.4 Verification of condition (H4) 34
4.5 The main theorem 36
4.6 Absolutely irreducible representations 36
4.7 Example: the case GL_1 38
A Categories and Functors 40
A.1 Categories 40
A.2 Functors 42
A.3 Representability 43
B Cohomology for profinite groups 45
B.1 G-modules 45
B.2 Cohomology for profinite groups 46
Bibliography 50
Index 53		zh_TW
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0099751014en_US
dc.subject (關鍵詞) 投射有限群zh_TW
dc.subject (關鍵詞) 表現zh_TW
dc.subject (關鍵詞) 形變zh_TW
dc.subject (關鍵詞) 泛形變zh_TW
dc.subject (關鍵詞) 泛形變環zh_TW
dc.subject (關鍵詞) 扎里斯基切空間zh_TW
dc.subject (關鍵詞) Profinite groupsen_US
dc.subject (關鍵詞) Representationsen_US
dc.subject (關鍵詞) Deformationsen_US
dc.subject (關鍵詞) Universal deformationsen_US
dc.subject (關鍵詞) Universal deformation ringsen_US
dc.subject (關鍵詞) Zariski tangent spaceen_US
dc.subject (關鍵詞) Group cohomologyen_US
dc.title (題名) 投射有限群表現之形變理論zh_TW
dc.title (題名) Deformation Theory of Representations of Profinite Groupsen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
[2] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939.
[3] J. W. S. Cassels and A. Fro ̈hlich (eds.), Algebraic Number Theory (2nd edition), London, United Kingdom, London Mathematical Society, 2010, Reprint of the 1967 original. MR 911121 (88h:11073)
[4] L. Clozel, M. Harris, and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. Inst. Hautes E ́tudes Sci. (2008), no. 108, 1–181, With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vigne ́ras.
[5] B. Conrad, F. Diamond, and R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 3, 521–567.
[6] B. de Smit and H. W. Lenstra, Jr. , Explicit construction of universal deformation rings, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, 1997, pp. 313–326.
[7] F. Diamond, On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), no. 1, 137–166.
[8] J.-M. Fontaine and B. Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78.
[9] A. Grothendieck, Technique de descente et th ́eor"emes d’existence en g ́eom ́etrie alg ́ebrique. II. Le th ́eor"eme d’existence en th ́eorie formelle des modules, Se ́minaire Bourbaki, vol. 5, Socie ́te ́ Mathe ́matique de France, 1995, pp. 369–390.
[10] A. Grothendieck and J. Dieudonne ́, El ́ements de G ́eom ́etrie Alg ́ebrique, Publ. Math. IHES,4 (1960), 8 (1961), 11 (1961), 17 (1963), 20 (1964), 24 (1965), 28 (1966), 32 (1967).
[11] K. Haberlan, Galois Cohomology of Algebraic Number Fields, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, With two appendices by Helmut Koch and Thomas Zink.
[12] M. Harris, N.Shepherd-Barron, and R. Taylor, A family of Calabi-Yau varieties and potential automorphy, Ann. of Math. (2) 171 (2010), no. 2, 779–813.
[13] H. Hida, Galois representations into GL_2(Z_p[X]) attached to ordinary cusp forms, Invent. Math. 85 (1986), no. 3, 545–613.
[14] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504.
[15] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586.
[16] M. Kisin, The Fontaine-Mazur conjecture for GL_2, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690.
[17] M. Kisin, Lecture Notes on Deformations of Galois Representations, Clay Mathematics Institute 2009 Summer School on Galois Representations (University of Hawaii at Manoa, Honolulu, Hawaii), June 15 - July 10 2009.
[18] M. Kisin, Moduli of finite flat group schemes and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180.
[19] S. Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, 1998.
[20] B. Mazur, Deforming of Galois Representations, Galois Groups over Q (Y. Ihara, K. Ribet, and J.-P. Serre, eds.), Mathematical Sciences Research Institute Publications, no. 16, Springer-Verlag, 1987, pp. 385–437.
[21] B. Mazur, An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer-Verlag, New York, 1997, pp. 243–311.
[22] J. Neukirch, Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Springer-Verlag, 1999.
[23] R. Ramakrishna, On a variation of Mazur’s deformation functor, Compositio Math. 87 (1993), 269–286.
[24] M. Schlessinger, Functors of Artin rings, Trans. A.M.S. 130 (1968), 208–222.
[25] J.-P. Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, 1979.
[26] J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, 1996.
[27] J.-P. Serre, Galois cohomology, Springer Monographs in Mathematics, Springer-Verlag, 1997.
[28] R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II, Publ. Math. Inst. Hautes E ́tudes Sci. (208), no. 108, 183–239.
[29] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), no. 3, 553–572.
[30] J. Tilouine, Deformations of Galois representations and Hecke algebras, Published for The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, 1996.
[31] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. of Math. 141(1995), no.3, 443–551.		zh_TW