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|Hodge-Tate decomposition for p-divisible groups : p-divisible groups./ | |Hodge-Tate decomposition for p-divisible groups: p-divisible groups. / | ||
Briefly discuss the (co)tangent space and differentials for affine group schemes following [Tam06], 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 [Mor16], Proposition 4.12. For time reasons, you may focus on how to obtain a p-divisible group from a formal group. | Briefly discuss the (co)tangent space and differentials for affine group schemes following [Tam06], 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 [Mor16], Proposition 4.12. For time reasons, you may focus on how to obtain a p-divisible group from a formal group. | ||
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|23.10.2024 | |23.10.2024 | ||
|Hodge-Tate decomposition for p-divisible groups : The universal vector extension./ | |Hodge-Tate decomposition for p-divisible groups: The universal vector extension. / | ||
Cover [Tam06], 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. | Cover [Tam06], 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. | ||
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|30.10.2024 | |30.10.2024 | ||
|Hodge-Tate decomposition for p-divisible groups : The period pairing and Hodge-Tate decomposition. | |Hodge-Tate decomposition for p-divisible groups: The period pairing and Hodge-Tate decomposition. / | ||
Introduce the ring A1 inf following [Tam06], 2.1. Define the period pairing following section 3.1. Finally deduce the Hodge-Tate decomposition in section 3.2. | Introduce the ring A1 inf following [Tam06], 2.1. Define the period pairing following section 3.1. Finally deduce the Hodge-Tate decomposition in section 3.2. | ||
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|06.11.2024 | |06.11.2024 | ||
|Crystalline cohomology : Divided power structures. / | |Crystalline cohomology: 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 [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. | ||
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|13.11.2024 | |13.11.2024 | ||
|Crystalline cohomology : Crystalline cohomology. / | |Crystalline cohomology: Crystalline cohomology. / | ||
Introduce the crystalline site and topos following [BO78], §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. | Introduce the crystalline site and topos following [BO78], §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. | ||
|TBA | |TBA | ||
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|20.11.2024 | |20.11.2024 | ||
| | |Crystalline cohomology: 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. | |||
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|7 | |7 | ||
|27.11.2024 | |27.11.2024 | ||
| | |Crystalline cohomology: 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. | |||
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|04.12.2024 | |04.12.2024 | ||
| | |Crystalline 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. | |||
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|11.12.2024 | |11.12.2024 | ||
| | |Crystals associated to p-divisible groups: The crystals E(G0), E(G0) and D(G0) / | ||
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. | |||
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|10 | |10 | ||
|18.12.2024 | |18.12.2024 | ||
| | |Crystals associated to p-divisible groups: Proof of [Mes72], IV, Theorem 2.2. / | ||
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. | |||
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|11 | |11 | ||
| | |08.01.2025 | ||
| | |Kato’s explicit reciprocity law and applications: 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. | |||
|TBA | |TBA | ||
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|12 | |12 | ||
|15.01.2025 | |||
|Kato’s explicit reciprocity law and applications: 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. | |||
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|13 | |||
|22.01.2025 | |22.01.2025 | ||
| | |Kato’s explicit reciprocity law and applications: The explicit reciprocity law II. / | ||
Continuation of Talk 12. | |||
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| | |14 | ||
|29.01.2025 | |29.01.2025 | ||
| | |Kato’s explicit reciprocity law and applications:Relation to special L-values I. / | ||
By using the explicit reciprocity law, explain and prove [Kat93], Theorem 1.2.6. | |||
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| | |15 | ||
|05.02.2025 | |05.02.2025 | ||
| | |Kato’s explicit reciprocity law and applications: Relation to special L-values II. / | ||
Continuation of Talk 14. | |||
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Revision as of 10:37, 8 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 will come in september ]
| No | Date | Title / Abstract | Speaker |
|---|---|---|---|
| 1 | 16.10.2024 | Hodge-Tate decomposition for p-divisible groups: p-divisible groups. /
Briefly discuss the (co)tangent space and differentials for affine group schemes following [Tam06], 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 [Mor16], Proposition 4.12. For time reasons, you may focus on how to obtain a p-divisible group from a formal group. |
tba |
| 2 | 23.10.2024 | Hodge-Tate decomposition for p-divisible groups: The universal vector extension. /
Cover [Tam06], 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. |
tba |
| 3 | 30.10.2024 | Hodge-Tate decomposition for p-divisible groups: The period pairing and Hodge-Tate decomposition. /
Introduce the ring A1 inf following [Tam06], 2.1. Define the period pairing following section 3.1. Finally deduce the Hodge-Tate decomposition in section 3.2. |
tba |
| 4 | 06.11.2024 | Crystalline cohomology: 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. |
tba |
| 5 | 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∗ 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. |
TBA |
| 6 | 20.11.2024 | Crystalline cohomology: 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. |
tba |
| 7 | 27.11.2024 | Crystalline cohomology: 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. |
tba |
| 8 | 04.12.2024 | Crystalline 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. |
tba |
| 9 | 11.12.2024 | Crystals associated to p-divisible groups: The crystals E(G0), E(G0) and D(G0) /
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 | Crystals associated to p-divisible groups: Proof of [Mes72], IV, Theorem 2.2. /
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. |
tba |
| 11 | 08.01.2025 | Kato’s explicit reciprocity law and applications: 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. |
TBA |
| 12 | 15.01.2025 | Kato’s explicit reciprocity law and applications: 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. |
|
| 13 | 22.01.2025 | Kato’s explicit reciprocity law and applications: The explicit reciprocity law II. /
Continuation of Talk 12. |
|
| 14 | 29.01.2025 | Kato’s explicit reciprocity law and applications:Relation to special L-values I. /
By using the explicit reciprocity law, explain and prove [Kat93], Theorem 1.2.6. |
|
| 15 | 05.02.2025 | Kato’s explicit reciprocity law and applications: Relation to special L-values II. /
Continuation of Talk 14. |
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 |
