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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 [BO78], §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
|-
|-
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|13.11.2024
|13.11.2024
|'''Crystalline cohomology.'''
|'''Crystalline cohomology.'''
Introduce the crystalline site and topos following [BO78], §5 until Example 5.2. Briefly sketch the construction of g∗<sub>crys</sub> and g<sub>crys,∗</sub> given in 5.6 – 5.8 and state Proposition 5.9. Define global sections and crystalline cohomology in Definition 5.15 and the discussion following it. Define the morphisms of topoi u<sub>X/S</sub> and i<sub>X/S</sub> relating the crystalline and Zariski topos. Show Propositions 5.25 – 5.27. Minimize the amount of the categorical nonsense before that.
Introduce the crystalline site and topos following P. Berthelot and A. Ogus <ref name"BO78"></ref>, §5 until Example 5.2. Briefly sketch the construction of g∗<sub>crys</sub> and g<sub>crys,∗</sub> given in 5.6 – 5.8 and state Proposition 5.9. Define global sections and crystalline cohomology in Definition 5.15 and the discussion following it. Define the morphisms of topoi u<sub>X/S</sub> and i<sub>X/S</sub> relating the crystalline and Zariski topos. Show Propositions 5.25 – 5.27. Minimize the amount of the categorical nonsense before that.
|Chiara Sabadin
|Chiara Sabadin
|-
|-
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|20.11.2024
|20.11.2024
|'''PD-differential operators.'''  
|'''PD-differential operators.'''  
Cover [BO78], §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
|tba
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|-
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|27.11.2024
|27.11.2024
|'''Crystals.'''
|'''Crystals.'''
[BO78], §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
|tba
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|-
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|04.12.2024
|04.12.2024
|'''Cohomology of crystals and de Rham cohomology.'''  
|'''Cohomology of crystals and de Rham cohomology.'''  
Explain [BO78], Lemma 6.11 and Theorem 6.12 (the crystalline Poincaré Lemma). Discuss its filtered version Theorem 6.13 and only briefly explain how to obtain Theorem 6.14.2 from it. Finally prove Theorem 7.2. If time permits state Corollaries 7.3 and 7.4.
Explain P. Berthelot and A. Ogus <ref name"BO78"></ref>, Lemma 6.11 and Theorem 6.12 (the crystalline Poincaré Lemma). Discuss its filtered version Theorem 6.13 and only briefly explain how to obtain Theorem 6.14.2 from it. Finally prove Theorem 7.2. If time permits state Corollaries 7.3 and 7.4.
|Zhenghang Du
|Zhenghang Du
|-
|-
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|10
|10
|18.12.2024
|18.12.2024
|'''Proof of [Mes72], IV, Theorem 2.2. (optional)'''  
|'''Proof of W. Messing <ref name="Mes72">W. Messing. The crystals associated to Barsotti-Tate groups: with applications to Abelian schemes,
Explain the proof of [Mes72], 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 [Mes72], IV, 1.16 – 1.22.
Lecture Notes in Mathematics. 264. Berlin-Heidelberg-New York: Springer-Verlag. 190 p. (1972).</ref>, IV, Theorem 2.2. (optional)'''  
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
|-
|-

Revision as of 11:10, 14 October 2024

Winter  Semester 24/25

Oberseminar - Explicit reciprocity laws and applications.

Time and place: Wednesday 10-12, SFB Lecture Hall (M311). [ Explicit reciprocity laws and applications, programme]


No Date Title / Abstract Speaker
1 16.10.2024 p-divisible groups.

Briefly discuss the (co)tangent space and differentials for affine group schemes following G. Tamme[1], 1.1. Define p-divisible groups following loc. cit., 1.3. Discuss section 2.2. Finally explain the relation between p-divisible groups and formal groups by explaining M. Morrow[2], Proposition 4.12. For time reasons, you may focus on how to obtain a p-divisible group from a formal group.

Christoph Fronhöfer
2 23.10.2024 The universal vector extension.

Cover G. Tamme[1], 1.2 on vector groups, section 1.4 on the universal vector extension of a p-divisible group and finally section 2.3 on the module ωE.

Julio de Mello Bezerra
3 30.10.2024 The period pairing and Hodge-Tate decomposition.

Introduce the ring A1inf following G. Tamme[1], 2.1. Define the period pairing following section 3.1. Finally deduce the Hodge-Tate decomposition in section 3.2.

Julio de Mello Bezerra
4 06.11.2024 Divided power structures.

Discuss some basic algebraic preliminaries on divided powers, following P. Berthelot and A. Ogus[3], §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
5 13.11.2024 Crystalline cohomology.

Introduce the crystalline site and topos following P. Berthelot and A. Ogus Cite error: Invalid <ref> tag; refs with no name must have content, §5 until Example 5.2. Briefly sketch the construction of g∗crys and gcrys,∗ given in 5.6 – 5.8 and state Proposition 5.9. Define global sections and crystalline cohomology in Definition 5.15 and the discussion following it. Define the morphisms of topoi uX/S and iX/S relating the crystalline and Zariski topos. Show Propositions 5.25 – 5.27. Minimize the amount of the categorical nonsense before that.

Chiara Sabadin
6 20.11.2024 PD-differential operators.

Cover P. Berthelot and A. Ogus Cite error: Invalid <ref> tag; refs with no name must have content, §4. Only sketch the proof of Theorem 4.8. For motivation you may need to have a look at §2 as well.

tba
7 27.11.2024 Crystals.

P. Berthelot and A. Ogus Cite error: Invalid <ref> tag; refs with no name must have content, §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
8 04.12.2024 Cohomology of crystals and de Rham cohomology.

Explain P. Berthelot and A. Ogus Cite error: Invalid <ref> tag; refs with no name must have content, Lemma 6.11 and Theorem 6.12 (the crystalline Poincaré Lemma). Discuss its filtered version Theorem 6.13 and only briefly explain how to obtain Theorem 6.14.2 from it. Finally prove Theorem 7.2. If time permits state Corollaries 7.3 and 7.4.

Zhenghang Du
9 11.12.2024 The crystals E(G0), E(G0) and D(G0) (optional)

Explain the definition of the crystals E(G0), E(G0) and D(G0) 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.

tba
10 18.12.2024 Proof of W. Messing [4], IV, Theorem 2.2. (optional)

Explain the proof of W. Messing [4], 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 [4], IV, 1.16 – 1.22.

tba
11 08.01.2025 BdR and the dual exponential map.

Explain [Kat93], Chapter II, §1.1 where a construction of Fontaine’s period ring BdR is given and the formalism of de Rham representations is explained. Proceed to discuss §1.2 on dual exponential maps.

Christoph Fronhöfer
12 15.01.2025 The explicit reciprocity law I.

The aim of this and the next talk is to prove the explicit reciprocity law [Kat93], Theorem 2.1.7.

tba
13 22.01.2025 The explicit reciprocity law II.

Continuation of Talk 12.

tba
14 29.01.2025 Relation to special L-values I.

By using the explicit reciprocity law, explain and prove [Kat93], Theorem 1.2.6.

tba
15 05.02.2025 Relation to special L-values II.

Continuation of Talk 14.

tba 


Winter Semester 23/24

Oberseminar - Euler systems

Time and place: Thursday 10-12, SFB Lecture Hall. Programme


No Date Title / Abstract Speaker
1 19.10.2023 Euler systems and main results. Zhenghang Du
2 26.10.2023 Euler systems and main results. Zhenghang Du
3 02.11.2023 Example: cyclotomic units. Chiara Sabadin
4 09.11.2023 Example: elliptic curves with CM I. Christoph Fronhöfer
5 16.11.2023 Example: elliptic curves with CM II. TBA
6 23.11.2023 The derivative construction. Julio de Mello Bezerra
7 30.11.2023 Local properties of derivative classes. Guido Kings
8 07.12.2023 Bounding the order of the Selmer group. Lukas Prader
9 14.12.2023 Twisting of Euler systems. Han-Ung Kufner
10 21.12.2023 Iwasawa theory I. Bence Forrás
11 11.01.2024 Iwasawa theory II. TBA
12 18.01.2024
13 25.01.2024
14 01.02.2024 



Summer Semester 23

Oberseminar - Modular Galois representations

Time and place: Thursday 10-12, SFB Lecture Hall.

No Date Title / Abstract Speaker
1 20.04.2023 Group representations and semi-simple algebras Chiara Sabadin
2 27.04.2023 Group representations and semi-simple algebras Chiara Sabadin
3 04.05.2023 Representations and pseudo-representations with coefficients in Artin rings Guillermo Gamarra-Segovia
4 11.05.2023 Representations and pseudo-representations with coefficients in Artin rings Guillermo Gamarra-Segovia
5 18.05.2023 holiday
6 25.05.2023 Deformation of group representations Zhenghang Du
7 01.06.2023 Deformation of group representations Zhenghang Du
8 08.06.2023 holiday
9 15.06.2023 The q-expansion principle and p-adic Hecke algebras Julio de Mello Bezerra
10 22.06.2023 The q-expansion principle and p-adic Hecke algebras Julio de Mello Bezerra
11 29.06.2023 no meeting due to Oberwolfach
12 06.07.2023 Modular Galois representations Lukas Prader
13 13.07.2023 Taylor-Wiles systems for the Hecke algebra Guido Kings
14 20.07.2023 Universal deformation rings and Taylor-Wiles systems Han-Ung Kufner 



Winter Semester 22/23

AG Seminar - Main conjecture for totally real fields

No Date Title / Abstract Speaker
1 19.10.2022 The p-adic L-function and the main conjecture Julio de Mello Bezerra
2 26.10.2022 The p-adic L-function and the main conjecture (continuation) Julio de Mello Bezerra
3 02.11.2022 \Lambda-adic modular forms Guillermo Gamarra-Segovia
4 09.11.2022 \Lambda-adic modular forms (continuation) Guillermo Gamarra-Segovia
5 16.11.2022 \Lambda-adic Eisenstein series Lukas Prader
6 23.11.2022 \Lambda-adic Eisenstein series (continuation) Lukas Prader
7 30.11.2022 \Lambda-adic cusp forms Zhenghang Du
8 07.12.2022 Galois representations associated to \Lambda-adic forms Chiara Sabadin
9 14.12.2022 Galois representations associated to \Lambda-adic forms (continuation) Chiara Sabadin
10 21.12.2022 The Eisenstein ideal and stable lattices Han-Ung Kufner
11 11.01.2023 The Eisenstein ideal and stable lattices (continuation) Han-Ung Kufner
12 18.01.2023 The Eisenstein ideal and stable lattices (continuation) Han-Ung Kufner
13 01.02.2023 The Galois representation of a stable lattice Johannes Sprang
14 


References
  1. ↑ 1.0 1.1 1.2 G. Tamme. Hodge-Tate-Zerlegung für p-divisible Gruppen nach Fontaine, Diploma thesis, Universität Regensburg (2006)
  2. ↑ M. Morrow. p-Divisible Groups, Uni Bonn lecture notes from winter term 15/16
  3. ↑ P. Berthelot, A. Ogus. Notes on crystalline cohomology, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, vi+243 pp (1978).
  4. ↑ 4.0 4.1 4.2 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).
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