Oberseminar Algebra und Geometrie

Winter term 2024/25: Mixed and non-mixed topics

Chaired by I. Halupczok, H. Kammeyer and B. Klopsch.

Organized by Adrian Baumann and Margherita Piccolo

All talks take place on Fridays at 14:30 in 25.22.03.73.

This term, the oberseminar is divided in two parts. In the first part we will learn about local class field theory and in the second part we will have mixed topics, mainly with guest speakers. Please get it touch if you have a suggestion on whom to invite. Note also the changed time of the seminar.

If you want get announcements about the seminar, please get in touch with I. Halupczok so that your address is added to the mailing list.

Infos für Studierende

Im WiSe 2024-2025 wird die erste Hälfte des Seminars aus einer Vortragsreihe über lokale Klassenkörpertheorie bestehen. Die zweite Hälfte wird sich aus Gastvorträgen mit einem losen Bezug zur ersten Hälfte zusammensetzen. Das Oberseminar richtet sich an alle, die einen Einblick in aktuelle Forschung erhalten möchten, ist aber tendenziell eher für fortgeschrittene Studierende geeignet (ab Master). Besonders empfohlen wird die Teilnahme an Oberseminaren, wenn Sie sich vorstellen können, zu promovieren. Wenn Sie intessiert sind, können Sie sich einfach (ohne Anmeldung) ins Seminar reinsetzen - gerne auch nur zu einzelnen Vorträgen, die Sie interessieren.

Die Vorträge in diesem Oberseminar sind auf englisch. Üblicherweise nehmen an Oberseminaren auch viele Doktorand:innen, Postdocs und Professor:innen teil. Wahrscheinlich werden Sie nicht alles verstehen; das passiert aber auch den fortgeschritteneren Teilnehmer:innen, und auch, wenn man nicht alles verstanden hat, hat man doch am Ende oft einen interessanten Einblick in ein neues Thema erhalten.

Local class field theory

Understanding the absolute Galois group of an arbitrary base field has turned out to be a very hard problem. The case of $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ in particular constitutes an open problem to this day. However, it turns out, that understanding just the maximal abelian extension of a given base field is much more feasible, especially if the base field is a local field. This can be done by means of group cohomology, since $\text{Gal}(\bar{K}/K)^{\text{ab}} = H^{-2}(\text{Gal}(\bar{K}/K), \mathbb{Z})$ (this is sometimes also denoted by $H_1(\text{Gal}(\bar{K}/K), \mathbb{Z})$). We will develop the means to better understand this cohomology group and give an explicit description in terms of the units of the base field for this local case. We will follow the exposition given by Neukirch [N] focusing on Part I and II. Depending on the number of guest talks and the eagerness of the participants, there is the option of a seventh talk describing the results in the global case.

Schedule

• 18.10.: Doris Grothusmann: Group cohomology 1

  Define the cohomology groups $H^i(G,A)$ as in Definition (2.1). Mention $H^{-i}(G,A) = H_i(G,A)$ for $i \geq 1$ as well as the difference to conventional cohomology in degrees 0 and $-1$. State Theorem 3.2 (no proof required). Give explicit descriptions of $H^0(G, A)$, $H^{-1}(G,A)$, $H^1(G,A)$, $H^1(G, \mathbb{Q}/\mathbb{Z})$ and $H^{-1}(G, \mathbb{Z})$ (Theorem 3.19.). (Source: Part I Sections 2 and 3.)

• 25.10.: Fabian Rodatz: Group cohomology 2

  Define the cup product as in Definition (5.1) and state its important properties (excluding those referring to inflation and restriction). Describe the cohomology of cyclic groups and define the Herbrand quotient. Prove its given properties. (Source: Part I Sections 5 and 6.)

• 1.11.: No Seminar (holiday)
• 8.11.: Clotilde Gauthier: Group cohomology 3

  Define inflation, restriction and corestriction and prove their basic properties (Theorem 4.6, Theorem 4.14). Sketch the proof of Tate's Theorem (Theorem 7.3). (Source: Part I Sections 4 and 7.)

• 15.11.: Jan Hennig: Abstract class field theory

  Give the definition of a class field formation (Defintion 1.3). Explain, why it fulfils the requirements of Tate's Theorem (Theorem 1.7) and deduce Theorem 1.9. Sketch the proof of Theorem 1.15. (Source: Part II Section 1.)

• 22.11.: Giada Serafini: Local class field theory in the unramified case

  Recount Theorem 2.2 and Theorem 2.3 (no proof required, if you are interested, this will most likely be shown in this semesters special lecture). Show that $T^\times$ is a class formation for the maximal unramified extension $T/K_0$ of a local field $K_0$ (Theorem 4.6). Recount definitions related to local fields if necessary. (Source: Part II Sections 2, 3, and 4.)

• 29.11.: Benjamin Klopsch: Local class field theory in the general case

  Show that, even in the ramified case, $\Omega^\times$ is a class formation for an algebraic closure $\Omega$ of a local field $K_0$ (Theorem 5.6). Deduce Theorem 5.7 and Theorem 5.9. State Theorem 5.13 (no proof required). (Source: Part II Section 5.)

References:

[N] J. Neukirch, Class Field Theory, Spinger-Verlag, Berlin, 2013. link

[G] G. Gras, Class field theory, Spirnger-Verlag, Belin, 2003. link

[M] J. S. Milne, Class field theory, Course notes, 2020. link

Mixed Topics