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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 | |
|---|---|---|---|---|
| I. Hodge-Tate decomposition for p-divisible groups | ||||
| 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 | |
| II. Crystalline cohomology | ||||
| 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 [3], §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 [3], §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 [3], §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. |
Guillermo Gamarra Segovia | |
| 8 | 04.12.2024 | Cohomology of crystals and de Rham cohomology.
Explain P. Berthelot and A. Ogus [3], 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 | |
| III. Crystals associated to p-divisible groups (optional) | ||||
| 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[4]. | ||||
| 9 | 11.12.2024 | The crystals E(G0), E(G0) and D(G0)
Explain the definition of the crystals E(G0), E(G0) and D(G0) following W. Messing [5], 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 [5], IV, Theorem 2.2.
Explain the proof of W. Messing [5], 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 [5], IV, 1.16 – 1.22. |
tba | |
| IV. Kato’s explicit reciprocity law and applications | ||||
| 11 | 08.01.2025 | BdR and the dual exponential map.
Explain K. Kato[6], 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 K. Kato[6], 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 K. Kato[6], 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
- ↑ 1.0 1.1 1.2 G. Tamme. Hodge-Tate-Zerlegung für p-divisible Gruppen nach Fontaine, Diploma thesis, Universität Regensburg (2006)
- ↑ M. Morrow. p-Divisible Groups, Uni Bonn lecture notes from winter term 15/16
- ↑ 3.0 3.1 3.2 3.3 3.4 P. Berthelot, A. Ogus. Notes on crystalline cohomology, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, vi+243 pp (1978).
- ↑ T. Tsuji. Explicit reciprocity law and formal moduli for Lubin-Tate formal groups, J. Reine Angew. Math. 569, 103-173 (2004).
- ↑ 5.0 5.1 5.2 5.3 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).
- ↑ 6.0 6.1 6.2 K. Kato. Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR, Colliot Thélène, Jean-Louis et al., Arithmetic algebraic geometry. Lectures given at the 2nd session of the Centro 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).
