SFB1085 - Higher Invariants - User contributions [en]

AG-Seminar (Kings)

2024-12-03T21:50:09Z

Kuh45866:

<hr> <hr>

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]

{| border="1"
!No
!Date
! Title / Abstract
!Speaker
|-
|colspan = 5| '''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<ref name="Tam06">G. Tamme. Hodge-Tate-Zerlegung für p-divisible Gruppen nach Fontaine, Diploma thesis, Universität
Regensburg (2006)</ref>, 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<ref name="Mor16"> M. Morrow. p-Divisible Groups, Uni Bonn lecture notes from winter term 15/16</ref>, 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<ref name="Tam06"></ref>, 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<ref name="Tam06"></ref>, 2.1. Define the period pairing following section 3.1. Finally deduce the Hodge-Tate decomposition in section 3.2.
|Julio de Mello Bezerra
|-
|colspan = 5| '''II. Crystalline cohomology'''
|-
|4
|06.11.2024
|'''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.
|Lukas Prader
|-
|5
|13.11.2024
|'''Crystalline cohomology.'''
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∗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 <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.
|Han-Ung Kufner
|-
|7
|27.11.2024
|'''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.
|Guillermo Gamarra Segovia
|-
|8
|04.12.2024
|'''Cohomology of crystals and de Rham cohomology.'''
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
|-
|colspan = 5 | ''' III. Kato’s explicit reciprocity law and applications'''
|-
|9
|11.12.2024
|'''BdR and the dual exponential map.'''
Explain K. Kato<ref name="Kat93">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).</ref>, 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
|-
|10
|18.12.2024
|'''The explicit reciprocity law 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.
|Han-Ung Kufner
|-
|11
|08.01.2025
|'''The explicit reciprocity law II.'''
Continuation of Talk 10.
|Han-Ung Kufner
|-
|12
|15.01.2025
|'''Relation to special L-values I.'''
By using the explicit reciprocity law, explain and prove K. Kato<ref name="Kat93"></ref>, Theorem 1.2.6.
|Guido Kings
|-
|13
|22.01.2025
|'''Relation to special L-values II.'''
Continuation of Talk 12.
|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
|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 <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
|-
|15
|18.12.2024
|'''Proof of W. Messing <ref name="Mes72"></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.
|tba
|-
|}

<hr> <hr> 

Winter Semester 23/24

'''Oberseminar - Euler systems'''

'''Time and place:''' Thursday 10-12, SFB Lecture Hall.
[https://sfb-higher-invariants.app.uni-regensburg.de/images/8/85/OS-Euler-systems.pdf Programme]

{| border="1"
!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
|
|
|-
|}

<hr> <hr> 

Summer Semester 23

'''Oberseminar - Modular Galois representations'''

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

{| border="1"
!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 [https://www.mfo.de/occasion/2326/www_view 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
|}

<hr> <hr> 

Winter Semester 22/23

'''AG Seminar - Main conjecture for totally real fields'''

{| border="1"
!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

2023-10-18T09:40:55Z

Kuh45866:

WinterSemester 23/24

'''Oberseminar - Euler systems'''

'''Time and place:''' Thursday 10-12, SFB Lecture Hall.
[https://sfb-higher-invariants.app.uni-regensburg.de/images/8/85/OS-Euler-systems.pdf Programme]

{| border="1"
!No
!Dateme
! Title / Abstract
!Speaker
|-
|1
|19.10.2023
|Euler systems and main results.
|Zhenghang Du
|-
|2
|26.10.2023
|Example: cyclotomic units.
|Chiara Sabadin
|-
|3
|02.11.2023
|Example: elliptic curves with CM I.
|Christoph Fronhöfer
|-
|4
|09.11.2023
|Example: elliptic curves with CM II.
|TBA
|-
|5
|16.11.2023
|The derivative construction.
|Julio de Mello Bezerra
|-
|6
|23.11.2023
|Local properties of derivative classes.
|Guido Kings
|-
|7
|30.11.2023
|(continuation)
|TBA
|-
|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
|
|
|-
|12
|18.01.2024
|
|
|-
|13
|25.01.2024
|
|
|-
|14
|01.02.2024
|
|
|-
|}

<hr> <hr> 

SummerSemester 23

'''Oberseminar - Modular Galois representations'''

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

{| border="1"
!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 [https://www.mfo.de/occasion/2326/www_view 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
|}

<hr> <hr> 

WinterSemester 22/23

'''AG Seminar - Main conjecture for totally real fields'''

{| border="1"
!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
|
|
|
|-
|}