坂内健一研究室
発表記録 | ||
|---|---|---|
| 2012/09/28 | 山田 | |
| 2012/10/05 | 平川 | |
| 2012/10/12 | 川口 | |
| 2012/10/19 | 山田 | |
| 2012/10/26 | 平川 | |
| 2012/11/02 | 川口 | |
| 2012/11/09 | 山田 | |
| 2012/11/16 | 平川 | |
| 2012/11/30 | 川口 | |
| 2012/12/07 | 山田 | |
テキスト：J.-P. Serre, Local Fields, GTM 67, Springer
Cohomology論の準備 | ||
|---|---|---|
| 準備 | Tate Cohomology, Hilbert Theorem 90, Brauer群 | 山田 |
Chapter XII：Brauer Group of a Local Field | ||
| §12.1 | Existence of an unramified Splitting field | 平川 |
| §12.2 | Existence of an unramified Splitting field (Direct Proof) | |
| §12.3 | Determination of the Brauer group | |
Chapter XIII：Local Class Field Theory | ||
| §13.1 | The Group $\widehat{\mathbb{Z}}$ and its cohomology | 川口 |
| §13.2 | Quasi-Finie fields | |
| §13.3 | The Brauer Group | 山田 |
| §13.4 | Class Formation | |
| §13.5 | Dwork's Theorem | 平川 |
Chapter XIV：Local Symbols and Existence Theorem | ||
| §14.1 | General Definition of Local Symbols | 平川 |
| §14.3 | Computations of the Symbol $(a,b)_v$ for the Field $\mathbb{Q}_p$ | 川口 |
| §14.5 | The symbols $[a,b)$ | 山田 |
| §14.6 | The Existence Theorem | |
| §14.7 | Example: The maximal abelian extension of $\mathbb{Q}_p$ | |
Chapter XV：Ramification | ||
| §15.1 | Kernel and Cokernel of an Additive (resp. Multiplicative) Polynomial | 平川 |
| §15.2 | The Norm Groups | |
| §15.3 | Explicit Computations | |
Chapter XI：Class Formation | ||
| §11.1 | The Notion of Formation | 川口 |
| §11.2 | Class Formations | |
| §11.3 | Fundamental Classes and Reciprocity Isomorphisms | |
| §11.4 | Abelian Extensions and Norm Groups | |
| §11.5 | The Existence Theorem | |
発表記録 | ||
|---|---|---|
| 2012/04/10 | 川口 | 伊達 |
| 2012/04/17 | 山田 | 平川 |
| 2012/04/24 | 松本 | 伊達 |
| 2012/05/08 | 山田 | |
| 2012/05/15 | 平川 | 松本 |
| 2012/05/22 | 山田 | |
| 2012/05/29 | 川口 | 伊達 |
| 2012/06/05 | 東京数論幾何週間 | |
| 2012/06/12 | 松本 | 平川 |
| 2012/06/19 | 山田 | 川口 |
| 2012/06/26 | 松本 | |
| 2012/07/03 | 伊達 | 平川 |
| 山田 | ||
| 2012/07/10 | 川口 | 伊達 |
| 2012/07/17 | 平川 | 山田 |
| 2012/07/31 | 川口 | |
| 2012/08/01 | 平川 | 山田 |
テキスト：J.-P. Serre, Local Fields, GTM 67, Springer
Chapter I : Discrete Valutaion Rings and Dedekind Domains | ||
|---|---|---|
| §1.1 | Definition of Discrete Valutaion Ring | 川口 |
| §1.2 | Characterizations of Discrete Valutaion Rings | 伊達 |
| §1.3 | Dedekind Domains | 山田 |
| §1.4 | Extensions | 平川 |
| §1.5 | The Norm and Inclusion Homomorphism | 松本 |
| §1.6 | Example: Simple Extensions | 伊達 |
| §1.7 | Galois Extensions | 川口 |
| §1.8 | Frobenius Substitution | 山田 |
Chapter II : Completion | ||
| §2.1 | Absolute Values and the Topology Defined by a Discrete Valuation | 平川 |
| §2.2 | Extensions of a Complete Field | 松本 |
| §2.3 | Extension and Completion | 伊達 |
| §2.4 | Structure of Complete Discrete Valutaion Rings I Equal Characterstic Case | 山田・松本 |
| §2.5 | Structure of Complete Discrete Valutaion Rings II Unequal Characterstic Case | 平川 |
| §2.6 | Witt Vectors | 山田 |
Chapter III : Discriminant and Different | ||
| §3.1 | Lattices | 川口 |
| §3.2 | Discriminant of a Lattice with Respect to Bilinear Form | |
| §3.3 | Discriminant and Different of a Separable Extension | 伊達 |
| §3.4 | Elementary Properties of the Different and Discriminant | |
| §3.5 | Unramified Extensions | 松本 |
| §3.6 | Computation of Different and Discriminant | 平川 |
| §3.7 | A Differential Characterisation of the Different | |
Chapter IV : Ramification Groups | ||
| §4.1 | Definition of the Ramificaion Groups; First Properties | 山田 |
| §4.2 | The Quotients $G_i/G_{i+1}$, $i \geq 0$ | 川口・伊達 |
| §4.3 | The Functions $\phi$ and $\psi$; Herband's Theorem | 平川 |
| §4.4 | Example: Cyclotomic Extensions of the Field $\mathbb{Q}_p$ | 山田 |
Chapter V : The Norm | ||
| §5.1 | Lemmas | 川口 |
| §5.2 | The Unramified Case | |
| §5.3 | The Cyclic of Prime Order Totally Ramified Case | 松本 |
| §5.4 | Extension of a Residue Field in a Totally Ramified Extension | 平川 |
| §5.5 | Multiplicative Polynomials and Additive Polynomials | 山田 |
| §5.6 | The Galois Totally Ramified Case | |
| §5.7 | Application: Proof of the Hasse-Arf Theorem | |