AG-Seminar (Kings) - Revision history

Kuh45866 at 21:50, 3 December 2024

2024-12-03T21:50:09Z

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	Winter Semester 24/25		Winter Semester 24/25

	'''Oberseminar - Explicit reciprocity laws and applications.'''		'''Oberseminar - Explicit reciprocity laws and applications.'''

Kuh45866 at 21:41, 3 December 2024

2024-12-03T21:41:58Z

← Older revision		Revision as of 23:41, 3 December 2024
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	\|'''PD-differential operators.'''		\|'''PD-differential operators.'''
	Cover P. Berthelot and A. Ogus <ref name="BO78"></ref>, §4. Only sketch the proof of Theorem 4.8. For motivation you may need to have a look at §2 as well.		Cover P. Berthelot and A. Ogus <ref name="BO78"></ref>, §4. Only sketch the proof of Theorem 4.8. For motivation you may need to have a look at §2 as well.
	\|~~tba~~		\|Han-Ung Kufner
	\|-		\|-
	\|7		\|7
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	\|Zhenghang Du		\|Zhenghang Du
	\|-		\|-
	\|colspan = 5 \| '''III. ~~Crystals associated to p-divisible groups (optional)'''~~		\|colspan = 5 \| ''' III. Kato’s explicit reciprocity law and applications'''
	\|-
	\|colspan = 5\| In the following two talks certain crystals are attached to a p-divisible group. This is completely independent of part IV, so one can also postpone this. However, these constructions play an important role for Tsuji’s generalization of Kato’s explicit ~~reciprocity law given in T. Tsuji<ref name="Tsu04">T. Tsuji. Explicit~~ reciprocity law and ~~formal moduli for Lubin-Tate formal groups, J. Reine Angew. Math.~~
	~~569, 103-173 (2004).</ref>.~~
	\|-		\|-
	\|9		\|9
	\|11.12.2024		\|11.12.2024
	\|'''~~The crystals E(G~~<sub>0</sub>~~), E(G<sub>0</sub>)~~ and ~~D(G<sub>0</sub>)~~ '''		\|'''B<sub>dR</sub> and the dual exponential map.'''
	Explain ~~the definition of the crystals E(G<sub>0</sub>), E(G<sub>0</sub>) and D(G<sub>0</sub>) following W~~. ~~Messing~~ <ref name="~~Mes72~~">W. ~~Messing~~. ~~The crystals associated~~ to ~~Barsotti~~-~~Tate groups: with applications to Abelian schemes~~,		Explain K. Kato<ref name="Kat93">K. Kato. Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR, Colliot
	~~Lecture Notes~~ in ~~Mathematics. 264~~. Berlin~~-Heidelberg-New York~~: Springer-Verlag. ~~190 p~~. (~~1972~~).</ref>, IV, 2.~~0 – 2.5. Explain~~ the ~~necessary definitions and results about exponentials from Chapter III~~. ~~The key result which will be proven later is Theorem 2~~.2.		Thélène, Jean-Louis et al., Arithmetic algebraic geometry. Lectures given at the 2nd session of the Centro
	\|~~tba~~		Internazionale Matematico Estivo (C.I.M.E.), held in Trento, Italy, June 24- July 2, 1991. Berlin: Springer
			Verlag. Lect. Notes Math. 1553, 50-163 (1993).</ref>, Chapter II, §1.1 where a construction of Fontaine’s period ring B<sub>dR</sub> is given and the formalism of de Rham representations is explained. Proceed to discuss §1.2 on dual exponential maps.
			\|Christoph Fronhöfer
	\|-		\|-
	\|10		\|10
	\|18.12.2024		\|18.12.2024
	\|'''~~Proof of W. Messing <ref name="Mes72"></ref>, IV, Theorem 2.2~~. '''		\|'''The explicit reciprocity law I.'''
	~~Explain the proof~~ of ~~W. Messing <ref name="Mes72"></ref>, IV, Theorem 2.2 given in loc. cit., 2.6~~ and ~~2.7. For this you may also need~~ to ~~explain some results about~~ the ~~formal completion E(G) of the universal vector extension of a p-divisible group following W~~. ~~Messing~~ <ref name="~~Mes72~~"></ref>, ~~IV, 1~~.~~16 –~~ 1.22.		The aim of this and the next talk is to prove the explicit reciprocity law K. Kato<ref name="Kat93"></ref>, Theorem 2.1.7.
	\|~~tba~~		\|Han-Ung Kufner
	\|-
	~~\|colspan = 5 \| ''' IV. Kato’s explicit reciprocity law and applications'''~~
	\|-		\|-
	\|11		\|11
	\|08.01.2025		\|08.01.2025
	\|'''~~B<sub>dR</sub> and the dual exponential map~~.'''		\|'''The explicit reciprocity law II.'''
	~~Explain K. Kato<ref name="Kat93">K. Kato. Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR, Colliot~~		Continuation of Talk 10.
	~~Thélène, Jean-Louis et al., Arithmetic algebraic geometry. Lectures given at the 2nd session of the Centro~~		\|Han-Ung Kufner
	~~Internazionale Matematico Estivo (C.I.M.E.), held in Trento, Italy, June 24- July 2, 1991. Berlin: Springer~~
	~~Verlag. Lect. Notes Math. 1553, 50-163 (1993).</ref>, Chapter II, §1.1 where a construction~~ of ~~Fontaine’s period ring B<sub>dR</sub> is given and the formalism of de Rham representations is explained. Proceed to discuss §1.2 on dual exponential maps~~.
	\|~~Christoph Fronhöfer~~
	\|-		\|-
	\|12		\|12
	\|15.01.2025		\|15.01.2025
	\|'''~~The explicit reciprocity law~~ I.'''		\|'''Relation to special L-values I.'''
	~~The aim of this and the next talk is to prove~~ the explicit reciprocity law K. Kato<ref name="Kat93"></ref>, Theorem 2.~~1.7~~.		By using the explicit reciprocity law, explain and prove K. Kato<ref name="Kat93"></ref>, Theorem 1.2.6.
	\|~~tba~~		\|Guido Kings
	\|-		\|-
	\|13		\|13
	\|22.01.2025		\|22.01.2025
	\|'''~~The explicit reciprocity law~~ II.'''		\|'''Relation to special L-values II.'''
	Continuation of Talk 12.		Continuation of Talk 12.
	\|~~tba~~		\|Guido Kings
			\|-
			\|colspan = 5 \| '''IV. Crystals associated to p-divisible groups (optional)'''
			\|-
			\|colspan = 5\| In the following two talks certain crystals are attached to a p-divisible group. This is completely independent of part IV, so one can also postpone this. However, these constructions play an important role for Tsuji’s generalization of Kato’s explicit reciprocity law given in T. Tsuji<ref name="Tsu04">T. Tsuji. Explicit reciprocity law and formal moduli for Lubin-Tate formal groups, J. Reine Angew. Math.
			569, 103-173 (2004).</ref>.
	\|-		\|-
	\|14		\|14
	\|29.01.~~2025~~		\|11.12.2024
	\|'''~~Relation to special L-values I.~~'''		\|'''The crystals E(G<sub>0</sub>), E(G<sub>0</sub>) and D(G<sub>0</sub>) '''
	~~By using~~ the ~~explicit reciprocity law~~, ~~explain~~ and ~~prove K~~. ~~Kato~~<ref name="~~Kat93~~"></ref>, Theorem 1.2.6.		Explain the definition of the crystals E(G<sub>0</sub>), E(G<sub>0</sub>) and D(G<sub>0</sub>) following W. Messing <ref name="Mes72">W. Messing. The crystals associated to Barsotti-Tate groups: with applications to Abelian schemes,
			Lecture Notes in Mathematics. 264. Berlin-Heidelberg-New York: Springer-Verlag. 190 p. (1972).</ref>, IV, 2.0 – 2.5. Explain the necessary definitions and results about exponentials from Chapter III. The key result which will be proven later is Theorem 2.2.
	\|tba		\|tba
	\|-		\|-
	\|15		\|15
	\|05.02.~~2025~~		\|18.12.2024
	\|'''~~Relation to special L-values II~~.'''		\|'''Proof of W. Messing <ref name="Mes72"></ref>, IV, Theorem 2.2. '''
	~~Continuation~~ of ~~Talk 14~~.		Explain the proof of W. Messing <ref name="Mes72"></ref>, IV, Theorem 2.2 given in loc. cit., 2.6 and 2.7. For this you may also need to explain some results about the formal completion E(G) of the universal vector extension of a p-divisible group following W. Messing <ref name="Mes72"></ref>, IV, 1.16 – 1.22.
	\|tba		\|tba
	\|-		\|-

Kuh45866 at 15:42, 21 October 2024

2024-10-21T15:42:03Z

← Older revision		Revision as of 17:42, 21 October 2024
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	\|'''Crystals.'''		\|'''Crystals.'''
	P. Berthelot and A. Ogus <ref name="BO78"></ref>, §6. Define crystals in 6.1. State Proposition 6.2 and if time permits sketch the proof. Discuss 6.3 – 6.5. Prove Theorem 6.6. Explain the linearization construction 6.9. Finally prove Proposition 6.10 and explain Remark 6.10.1.		P. Berthelot and A. Ogus <ref name="BO78"></ref>, §6. Define crystals in 6.1. State Proposition 6.2 and if time permits sketch the proof. Discuss 6.3 – 6.5. Prove Theorem 6.6. Explain the linearization construction 6.9. Finally prove Proposition 6.10 and explain Remark 6.10.1.
	\|~~tba~~		\|Guillermo Gamarra Segovia
	\|-		\|-
	\|8		\|8

Brv60445 at 09:28, 14 October 2024

2024-10-14T09:28:01Z

← Older revision		Revision as of 11:28, 14 October 2024
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	\|11.12.2024		\|11.12.2024
	\|'''The crystals E(G<sub>0</sub>), E(G<sub>0</sub>) and D(G<sub>0</sub>) '''		\|'''The crystals E(G<sub>0</sub>), E(G<sub>0</sub>) and D(G<sub>0</sub>) '''
	Explain the definition of the crystals E(G<sub>0</sub>), E(G<sub>0</sub>) and D(G<sub>0</sub>) following [Mes72], IV, 2.0 – 2.5. Explain the necessary definitions and results about exponentials from Chapter III. The key result which will be proven later is Theorem 2.2.		Explain the definition of the crystals E(G<sub>0</sub>), E(G<sub>0</sub>) and D(G<sub>0</sub>) following W. Messing <ref name="Mes72">W. Messing. The crystals associated to Barsotti-Tate groups: with applications to Abelian schemes,
			Lecture Notes in Mathematics. 264. Berlin-Heidelberg-New York: Springer-Verlag. 190 p. (1972).</ref>, IV, 2.0 – 2.5. Explain the necessary definitions and results about exponentials from Chapter III. The key result which will be proven later is Theorem 2.2.
	\|tba		\|tba
	\|-		\|-
	\|10		\|10
	\|18.12.2024		\|18.12.2024
	\|'''Proof of W. Messing <ref name="Mes72">~~W. Messing. The crystals associated to Barsotti-Tate groups: with applications to Abelian schemes,~~		\|'''Proof of W. Messing <ref name="Mes72"></ref>, IV, Theorem 2.2. '''
	~~Lecture Notes in Mathematics. 264. Berlin-Heidelberg-New York: Springer-Verlag. 190 p. (1972).~~</ref>, IV, Theorem 2.2. '''
	Explain the proof of W. Messing <ref name="Mes72"></ref>, IV, Theorem 2.2 given in loc. cit., 2.6 and 2.7. For this you may also need to explain some results about the formal completion E(G) of the universal vector extension of a p-divisible group following W. Messing <ref name="Mes72"></ref>, IV, 1.16 – 1.22.		Explain the proof of W. Messing <ref name="Mes72"></ref>, IV, Theorem 2.2 given in loc. cit., 2.6 and 2.7. For this you may also need to explain some results about the formal completion E(G) of the universal vector extension of a p-divisible group following W. Messing <ref name="Mes72"></ref>, IV, 1.16 – 1.22.
	\|tba		\|tba

Brv60445 at 09:26, 14 October 2024

2024-10-14T09:26:49Z

← Older revision		Revision as of 11:26, 14 October 2024
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	\|06.11.2024		\|06.11.2024
	\|'''Divided power structures.'''		\|'''Divided power structures.'''
	Discuss some basic algebraic preliminaries on divided powers, following P. Berthelot and A. Ogus <ref name"BO78"> P. Berthelot, A. Ogus. Notes on crystalline cohomology, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, vi+243 pp (1978).</ref>, §3. More precisely cover 3.1 – 3.5, explain Theorem 3.9 about the divided power algebra without proof, explain extensions and compatible PD-structures 3.14 – 3.17. Discuss PD-envelopes in 3.19. Only sketch the construction if time permits. Cover 3.21 – 3.22, Definitions 3.24 and 3.29, Finally, briefly discuss how to globalize to schemes.		Discuss some basic algebraic preliminaries on divided powers, following P. Berthelot and A. Ogus <ref name="BO78"> P. Berthelot, A. Ogus. Notes on crystalline cohomology, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, vi+243 pp (1978).</ref>, §3. More precisely cover 3.1 – 3.5, explain Theorem 3.9 about the divided power algebra without proof, explain extensions and compatible PD-structures 3.14 – 3.17. Discuss PD-envelopes in 3.19. Only sketch the construction if time permits. Cover 3.21 – 3.22, Definitions 3.24 and 3.29, Finally, briefly discuss how to globalize to schemes.
	\|Lukas Prader		\|Lukas Prader
	\|-		\|-

Brv60445 at 09:26, 14 October 2024

2024-10-14T09:26:27Z

← Older revision		Revision as of 11:26, 14 October 2024
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	\|06.11.2024		\|06.11.2024
	\|'''Divided power structures.'''		\|'''Divided power structures.'''
	Discuss some basic algebraic preliminaries on divided powers, following P. Berthelot and A. Ogus<ref name"BO78">P. Berthelot, A. Ogus. Notes on crystalline cohomology, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, vi+243 pp (1978).</ref>, §3. More precisely cover 3.1 – 3.5, explain Theorem 3.9 about the divided power algebra without proof, explain extensions and compatible PD-structures 3.14 – 3.17. Discuss PD-envelopes in 3.19. Only sketch the construction if time permits. Cover 3.21 – 3.22, Definitions 3.24 and 3.29, Finally, briefly discuss how to globalize to schemes.		Discuss some basic algebraic preliminaries on divided powers, following P. Berthelot and A. Ogus <ref name"BO78"> P. Berthelot, A. Ogus. Notes on crystalline cohomology, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, vi+243 pp (1978).</ref>, §3. More precisely cover 3.1 – 3.5, explain Theorem 3.9 about the divided power algebra without proof, explain extensions and compatible PD-structures 3.14 – 3.17. Discuss PD-envelopes in 3.19. Only sketch the construction if time permits. Cover 3.21 – 3.22, Definitions 3.24 and 3.29, Finally, briefly discuss how to globalize to schemes.
	\|Lukas Prader		\|Lukas Prader
	\|-		\|-

Brv60445 at 09:26, 14 October 2024

2024-10-14T09:26:11Z

← Older revision		Revision as of 11:26, 14 October 2024
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	\|06.11.2024		\|06.11.2024
	\|'''Divided power structures.'''		\|'''Divided power structures.'''
	Discuss some basic algebraic preliminaries on divided powers, following P. Berthelot and A. Ogus<ref name"BO78">P. Berthelot, A. Ogus. Notes on crystalline cohomology, Princeton University Press, Princeton, NJ;		Discuss some basic algebraic preliminaries on divided powers, following P. Berthelot and A. Ogus<ref name"BO78">P. Berthelot, A. Ogus. Notes on crystalline cohomology, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, vi+243 pp (1978).</ref>, §3. More precisely cover 3.1 – 3.5, explain Theorem 3.9 about the divided power algebra without proof, explain extensions and compatible PD-structures 3.14 – 3.17. Discuss PD-envelopes in 3.19. Only sketch the construction if time permits. Cover 3.21 – 3.22, Definitions 3.24 and 3.29, Finally, briefly discuss how to globalize to schemes.
	University of Tokyo Press, Tokyo, vi+243 pp (1978).</ref>, §3. More precisely cover 3.1 – 3.5, explain Theorem 3.9 about the divided power algebra without proof, explain extensions and compatible PD-structures 3.14 – 3.17. Discuss PD-envelopes in 3.19. Only sketch the construction if time permits. Cover 3.21 – 3.22, Definitions 3.24 and 3.29, Finally, briefly discuss how to globalize to schemes.
	\|Lukas Prader		\|Lukas Prader
	\|-		\|-

Brv60445 at 09:25, 14 October 2024

2024-10-14T09:25:00Z

← Older revision		Revision as of 11:25, 14 October 2024
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	\|colspan = 5\| '''I. Hodge-Tate decomposition for p-divisible groups'''		\|colspan = 5\| '''I. Hodge-Tate decomposition for p-divisible groups'''
	\|-		\|-
	\| 1		\|1
	\|16.10.2024		\|16.10.2024
	\|'''p-divisible groups.'''		\|'''p-divisible groups.'''

Brv60445 at 09:24, 14 October 2024

2024-10-14T09:24:53Z

← Older revision		Revision as of 11:24, 14 October 2024
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	\|colspan = 5\| '''I. Hodge-Tate decomposition for p-divisible groups'''		\|colspan = 5\| '''I. Hodge-Tate decomposition for p-divisible groups'''
	\|-		\|-
	\|1		\| 1
	\|16.10.2024		\|16.10.2024
	\|'''p-divisible groups.'''		\|'''p-divisible groups.'''

Brv60445 at 09:24, 14 October 2024

2024-10-14T09:24:20Z

← Older revision		Revision as of 11:24, 14 October 2024
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	\|-		\|-
	\|colspan = 5\| In the following two talks certain crystals are attached to a p-divisible group. This is completely independent of part IV, so one can also postpone this. However, these constructions play an important role for Tsuji’s generalization of Kato’s explicit reciprocity law given in T. Tsuji<ref name="Tsu04">T. Tsuji. Explicit reciprocity law and formal moduli for Lubin-Tate formal groups, J. Reine Angew. Math.		\|colspan = 5\| In the following two talks certain crystals are attached to a p-divisible group. This is completely independent of part IV, so one can also postpone this. However, these constructions play an important role for Tsuji’s generalization of Kato’s explicit reciprocity law given in T. Tsuji<ref name="Tsu04">T. Tsuji. Explicit reciprocity law and formal moduli for Lubin-Tate formal groups, J. Reine Angew. Math.
	569, 103-173 (2004).</ref.		569, 103-173 (2004).</ref>.
	\|-		\|-
	\|9		\|9