講演内容
講演アブストラクト (Abstracts)
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プレスクール 水澤 靖 (名古屋工業大学)
サマースクール本編へ向けた準備講演。円分体のガロワ理論、p 進リー拡大、岩澤加群、アルティンの L 関数、非可換環の局所化など、サマースクールの本講演で用いられる諸概念を概観し、整理することを目的とする。
可換拡大の岩澤理論の代数的側面について 藤井 俊 (金沢工業大学)
本講演では、まず導入として岩澤理論の起源である Zp 拡大の一般論を解説し、後の講演で用いられる概念、用語の紹介を行う。
次いで、非可換岩澤理論で扱われる「分岐付岩澤加群」が、どのような文脈で岩澤理論に現れるのかについて解説をする。
On algebraic aspects of Iwasawa theory for abelian extensions
Satoshi Fujii (Kanazawa Institute of Technology)
In this lecture I will first explain general theory on Zp-extensions of algebraic number fields, which is the origin of Iwasawa theory. Then I introduce several concepts and terminologies concerning it which shall be used throughout this lecture series.
Under these preparations I would like to introduce the notion of the “Iwasawa module with ramification,” and explain how this notion appears in the classical Iwasawa theory.
参考文献(References): [SS03] 全般, [Iw73]の前半部分, [Wa97] の13章.
p 進 L 関数の Stickelberger 構成 三浦 崇 (鶴岡工業高等専門学校)
p 進 L 関数は Dirichlet L 関数の負の整数点での値を p 進補間するような p 進正則関数として定義される.p 進 L 関数の構成については色々な方法が知られているが,岩澤健吉は Stickelberger 元を用いて p 進測度 (あるいは岩澤冪級数) を構成し,それが Kubota-Leopoldt の p 進 L 関数と一致することを証明した.
本講演では,Deligne-Ribet, Cassou-Noguès, Barsky によって証明された,総実代数体の部分ゼータ関数の特殊値に関するある二つの性質 C(p) (特殊値の間の合同式) と D(p) (特殊値の分母に関する条件) を認めて,岩澤による Stickelberger 構成に沿って総実代数体の p 進 L 関数を構成する.
Stickelberger construction of p-adic L-functions
Takashi Miura (National Institute of Technology, Tsuruoka College)
The so-called “p-adic L-function” is defined as a p-adic holomorphic (or meromorphic) function which interpolates values of the Dirichlet L-function at negative integers p-adically. There are lots of well-known constructions of the p-adic L-function, but among them I will focus on Iwasawa's construction in this lecture. Kenkichi Iwasawa constructed a certain p-adic (pseudo-)measure (which is also called an Iwasawa power series in some literature) from Stickelberger elements, and then he proved that it coincided with the p-adic L-function of Kubota and Leopoldt.
In this lecture I will construct the p-adic L-functions for totally real number fields following Iwasawa's Stickelberger construction. In the construction I admit (without proofs) two conditions concerning special values of the partial zeta functions of totally real number fields— the condition C(p) (congruences among special values) and the condition D(p) (p-integrality of special values)—, which were verified by Deligne-Ribet, Cassou-Noguès and Barsky.
参考文献 (References): [Coa77], [Se78], [La90], [Wa97].
アーベル拡大での岩澤主予想 岡野 恵司 (都留文科大学)
本講演では,Mazur-Wiles,Wiles により証明された総実代数体上の古典的岩澤主予想について,簡単な解説を行う. 主予想の意味づけや古典的主予想の主張の言いかえを,直感的にわかりやすいように解説することで,非可換化への橋渡しを行う予定である.
Iwasawa main conjecture for abelian extensions
Keiji Okano (Tsuru University)
This lecture gives a brief survey on the classical Iwasawa main conjecture over totally real number fields, which was verified by Mazur-Wiles and Wiles. I would like to intermediate between the classical and non-commutative main conjectures by giving an intuitive and clear explanation on the meaning of the main conjectures and a reinterpretation of the classical main conjecture via K-theoretic language.
参考文献 (References): [SS03] の八森氏,尾﨑氏の稿,[CSSV12] の Coates-Kim の稿,[Kak13]
K 理論からの準備: 局所化完全列について 齋藤 翔 (名古屋大学)
非可換岩澤主予想の定式化における重要な要素の一つは、代数的 K 理論の利用である。この講演では、K 理論の局所化完全系列の手短かな紹介を与える。この系列から帰結する境界準同型の全射性は、非可換岩澤主予想の定式化において有限生成 Λ 加群の構造定理に代わる役割を担う。後続する講演においては0次および1次の K 群のみが用いられるが、概念的に簡潔な捉え方によって幾何的な背景を明確化すべく、本講演は高次代数的 K 理論の枠組みを用いて行われる。
Preliminaries from algebraic K-theory: on the localisation exact sequences
Sho Saito (Nagoya University)
One of the crucial ingredients in the formulation of the noncommutative Iwasawa main conjecture is the use of algebraic K-theory. In this talk I will give a short introduction to the localisation exact sequence of K-theory, the subjectivity of whose resulting boundary map serves as a noncommutative substitute for the structure theorem of finitely generated Λ-modules. Although in the later talks only the zeroth and first K-groups are used, I chose to give the talk in the language of whole algebraic K-theory spaces, aiming to provide a simple and conceptual perspective, which hopefully clarifies its background geometric contexts.
参考文献 (References): [Wal85], [TT90], [Schli11]
非可換岩澤主予想の定式化 森澤 貴之 (東京理科大学)
本講演では, 総実代数体の非可換岩澤主予想の定式化を行う. また, そのために必要となる許容 p 進リー拡大, 標準オーレ集合, セルマー複体, p 進ゼータ関数などを定義する.
Formulation of non-commutative Iwasawa main conjecture
Takayuki Morisawa (Tokyo University of Science)
In this lecture I will formulate the non-commutative Iwasawa main conjecture for totally real number fields. To this end, I will define several concepts including admissible p-adic Lie extension, the canonical Ore set, Selmer complexes and the notion of (non-commutative) p-adic zeta functions.
参考文献 (References): [CFKSV05], [CSSV12] の Sujatha の稿
深谷-加藤の補題と1次元コンパクト p 進リー群への帰着 村上 和明 (慶應義塾大学)
本講演では総実代数体上の許容な p 進リー拡大における岩澤主予想が、1次元の許容な p 進リー拡大における岩澤主予想に帰着されることを示す。またその過程で必要な深谷-加藤の補題の証明の概略を行う。
Fukaya-Kato's lemma and reduction to 1-dimensional compact p-adic Lie groups
Kazuaki Murakami (Keio University)
We shall reduce in this lecture the proof of the Iwasawa main conjecture for admissible p-adic Lie extensions over totally real number fields to the proof of the Iwasawa main conjecture for 1-dimensional admissible p-adic Lie extensions. We also sketch the proof of so-called Fukaya-Kato's lemma, which is necessary for the reduction step.
参考文献 (References): [FuKa06] の Proposition 1.5.1, [CSSV12] の Sujatha の稿, [Kak13] の Section 4.1.
非可換岩澤主予想の証明の方針: Burns-加藤の手法 原 隆 (東京電機大学)
総実代数体の p 進リー拡大の非可換岩澤主予想は、そこに含まれる可換な p 進リー拡大達に対する (Andrew Wiles によって証明された) 岩澤主予想を言わば《貼り合わせる》ことによってなされます。本講演では、後の講演の導入としての役割も兼ね併せて、David Burns により発案され加藤和也により洗練された岩澤主予想の《貼り合わせ》の手法について、特にその傑出したアイデアの部分に焦点を当てて解説します。
議論や計算が繁雑な部分は全て後の北島さん、大下さんの講演にお任せして、ここではアイデアの根幹がわかるように、枝葉は全て切り落としてなるべく平易に解説することを心掛けるつもりです。
A strategy for the proof of non-commutative Iwasawa main conjecture: the technique of Burns and Kato
Takashi Hara (Tokyo Denki University)
One may prove the non-commutative Iwasawa main conjecture for a p-adic Lie extension of a totally real number field by “patching” the classical Iwasawa main conjectures (verified by Andrew Wiles) for abelian p-adic Lie extensions contained in the p-adic Lie extension under consideration. As an introduction of the following lectures given by Kitajima and Ohshita, I will explain in this lecture the “patching” argument of Iwasawa main conjectures which was first proposed by David Burns and then elaborated by Kazuya Kato.
I will try to give an easy and clear explanation postponing all the complicated arguments and calculations to Kitajima and Ohshita's lectures, for the audience to understand the heart of Burns and Kato's outstanding ideas.
参考文献 (References): [Kato], [原1], [原2], [Bu10a], [Hara11], [Kak13] など
整対数準同型について 野村 次郎 (慶應義塾大学)
本講演では非可換岩澤主予想の証明の重要な道具となっている整対数準同型写像の定義を振り返り、基本的な性質を確認する。
On integral logarithmic homomorphisms
Jiro Nomura (Keio University)
In this lecture I will review the definition and several basic properties of integral logarithmic homomorphisms, which shall play crucial roles in the proof of the non-commutative Iwasawa main conjecture.
参考文献 (References): [Ol88], [OT88] など
非可換岩澤主予想の証明: 代数的側面について 北島 孝浩 (慶應義塾大学)
本講演では, 総実代数体の非可換岩澤主予想の証明の代数的側面として,岩澤代数とその局所化の K1 群のノルム像の決定について解説する.
ある種のトレースの像を決定し(加法的理論), Oliver-Taylor の整対数準同型によりノルムの像の計算(乗法的理論)に “翻訳” する.
The proof of non-commutative Iwasawa main conjecture: on algebraic aspects
Takahiro Kitajima (Keio University)
As algebraic aspects of the proof of the non-commutative Iwasawa main conjecture for totally real number fields, I will explain how to determine the images of norm maps of the K1-groups of the Iwasawa algebra and its canonical Ore localisation in this lecture.
I will first determine the image of certain trace maps (additive theory), and then “translate” it into the calculation of the norm maps (multiplicative theory) by using Oliver-Taylor's integral logarithms.
参考文献 (References): [CSSV12] の Schneider-Venjakob の稿,[Kak13],[OT88].
非可換岩澤主予想の証明: 解析的側面について 大下 達也 (愛媛大学)
本講演では,北島さんの講演に引き続いて, Kakde による総実代数体の非可換岩澤主予想の証明の紹介を行う.
前講演での議論 (すなわち,岩澤代数とその標準 Ore 局所化の K1' 群のノルム像の計算と,Burns–加藤の手法による貼り合わせ) により,総実代数体の非可換岩澤主予想の証明は,Deligne–Ribet の p 進ゼータ関数の間のいくつかの合同式の証明に帰着された.
本講演では,Hilbert 保型形式 (特に,Hilbert Eisenstein 級数と q 展開原理) という新たな道具立を駆使して,問題となっている p 進ゼータ関数の間の合同式を証明し,非可換岩澤主予想の証明を完結させる.
The proof of non-commutative Iwasawa main conjecture: on analytic aspects
Tatsuya Ohshita (Ehime University)
Subsequently to Kitajima's lecture, I will give an introduction of Kakde's proof of the non-commutative Iwasawa main conjecture for totally real number fields.
Due to the arguments in the previous lectures (namely, the calculations of the images of norm maps of K1'-groups of the Iwasawa algebra and its Ore localisation, and the patching arguments due to Burns–Kato's method), the proof of non-commutative Iwasawa main conjecture for totally number fields is reduced to several congruences among Deligne–Ribet's p-adic zeta functions.
In this lecture, I first introduce a new ingredient —theory on Hilbert modular forms (especially Hilbert Eisenstein series and q-expansion principle)— and complete the proof of the non-commutative Iwasawa main conjecture by verifying all the required congruences among p-adic zeta functions.
参考文献 (References): [CSSV12] の Kakde の稿,[Kak13],[DRi80].
講演レジュメ (合同式について), ヒルベルト保型形式についての補足, 練習問題
Zp-拡大の非アーベル岩澤理論 尾﨑 学 (早稲田大学)
(準備中)
Non-abelian Iwasawa theory for Zp-extensions
Manabu Ozaki (Waseda University)
(to be announced)
保型形式の q 展開原理について 佐久川 憲児 (大阪大学)
大下さんの講演に於いて総実代数体上の非可換岩澤主予想の証明が完結される予定であるが, その証明の中で用いられる最も大きな道具の一つは, Hilbert 保型形式に対する q 展開原理であった. q 展開原理とは, 大雑把にいうと「保型形式は一つのカスプに於ける q 展開で一意的に決まる」, という主張である.
本講演では楕円保型形式の場合に限って, q 展開原理の証明の概略を与える.
On the q-expansion principle for modular forms
Kenji Sakugawa (Osaka University)
The q-expansion principle for Hilbert modular forms is one of the key ingredient for the proof of non-abelian Iwasawa main conjecture over totally real fields. Roughly speaking, the statement of the q-expansion principle is that “a modular form is uniquely determined by its q-expansion at a cusp.”
In this talk, we will give a sketch of the proof of q-expansion principle for elliptic modular forms.
参考文献 (References): [Katz73], [KM85], [DRi80]
楕円曲線の非可換岩澤理論 越智 禎宏 (東京電機大学)
楕円曲線の非可換岩澤理論について概観する.特に GL2 拡大,PGL2 拡大,False Tate Curve 拡大の場合について,基礎的事実から Mordell-Weil rank の振る舞いや p 進 L 関数などに関する結果を解説したい.
Non-commutative Iwasawa theory for elliptic curves
Yoshihiro Ochi (Tokyo Denki University)
I am gonna give an overview of noncommutative Iwasawa theory for elliptic curves. I'll begin with some basic facts and discuss recent results on the Mordell-Weil ranks and p-adic L-functions in the case of GL2-extensions, PGL2-extensions and so-called false Tate curve extensions.
参考文献 (References): [八森04], [深谷11], [CGRR99], [Gr01], [CFKSV05], [Ven05], [FuKa06], [Hach07], [Ven07]
講演レジュメ (準備中)
Fröhlich 予想と同変岩澤理論 野村 次郎 (慶應義塾大学)
非可換岩澤主予想の証明においては整対数準同型写像が重要な役割を果たしたが、整対数写像はそれ以外の場面でも利用されてきた重要な対象である。本講演では、非可換岩澤理論以外で整対数準同型が利用された例として、Fröhlich 予想 (Martin J. Taylor によって証明された) を紹介する。
さらに、その証明と Ritter-Weiss による同変岩澤主予想(非可換岩澤主予想)の証明の類似性を整対数準同型写像という観点で観察する。
Fröhlich conjecture and equivariant Iwasawa theory
Jiro Nomura (Keio University)
As we have already seen, integral logarithmic homomorphisms play crucial roles in the proof of the non-commutative Iwasawa main conjecture. The notion of integral logarithms is, however, also important and utilised in many situations except for non-commutative Iwasawa theory. In this lecture I will introduce Fröhlich's conjecture (which was verified by Martin J. Taylor) as an example for which integral logarithms were effectively utilised.
Then I will observe the similarity between Taylor's proof of Fröhlich conjecture and Ritter-Weiss' proof of their equivariant Iwasawa main conjecture (the non-commutative Iwasawa main conjecture) focusing on the usages of integral logarithms.
参考文献 (References): [CSSV12] の Chinburg-Pappas-Taylor の稿, [Ta81], [Fr83], [RW11]
同変玉河数予想入門 佐野 昂迪 (慶應義塾大学)
同変玉河数予想は、モチーフの L 関数の値に関する最も一般的で精密な予想であり、Dirichlet と Dedekind による古典的な類数公式の膨大な一般化である。
本講演では、デターミナント加群に関する準備をした後、Tate モチーフに対する同変玉河数予想を中心に、可換係数の場合、非可換係数の場合の順に解説する。岩澤主予想との関係についても触れる。
Introduction to equivariant Tamagawa number conjecture
Takamichi Sano (Keio University)
The equivariant Tamagawa number conjecture is the most general and precise conjecture on values of the L-functions associated to motives, and a vast generalisation of the classical class number formula due to Dirichlet and Dedekind.
After a brief review on determinant modules, I will explain in this lecture the equivariant Tamagawa number conjecture mainly for Tate motives. I first deal with commutative-coefficient cases, and then explain non-commutative-coefficient cases. I'm also planning to discuss the relation between the equivariant Tamagawa number conjecture and the Iwasawa main conjecture.
参考文献 (References): [BlKa90], [Kato93a], [Kato93b], [FoPR94], [BuFl96], [Ki11], [BuFl01], [BuFl03], [HK02], [FuKa06], [Ve07] など。
Ritter-Weiss の同変岩澤理論について 原 隆 (東京電機大学)
今回のサマースクールの前半では [CFKSV05] に基づく総実代数体の岩澤主予想の定式化が紹介された後、Mahesh Ramesh Kakde の方法に則った主予想の証明の概略についての解説がなされている筈です。しかし、Kakde と同時期に (プレプリントの公表時期としては Kakde に先んじて) Jürgen Ritter と Alfred Weiss が総実代数体の岩澤主予想の証明を公開していました。彼等の手法は Kakde のものとはまた異なった戦略に則ったものであり、Kakde の手法と比較検討することは大変重要で意義深いことですが、彼等の証明は彼等自身による岩澤主予想の定式化 (野村さんの講演でも登場した『同変岩澤理論』) の枠組みに沿って行われているため、遺憾ながら両者を比較する前の段階で既に膨大なエネルギーを消費するものとなってしまっているのが現状だと思われます。
そこで本講演では、前半で [CFKSV05] のスタイルの主予想の定式化と Ritter-Weiss の同変岩澤理論に於ける主予想の定式化を簡単に比較し、それぞれの定式化が実質的に同じものであることを確認します。その後、(日本ではあまり紹介されてこなかったと思われる) Ritter と Weiss による主予想の証明方針を、特に Kakde の証明と違う点に重点を置きつつ時間の許す限り概観したいと考えております。
以上の様な趣旨の講演ですので、他の講演と比べると幾分 “ディープな” 内容になるかと思われますが、折角の機会ですので非可換岩澤主予想の世界を最後まで余すところなくお楽しみいただければ幸いです。
On Ritter-Weiss's equivariant Iwasawa theory
Takashi Hara (Tokyo Denki University)
In the first half of this summer school, we studied the formulation of the non-commutative Iwasawa main conjecture for totally number fields following [CFKSV05], and then overviewed the proof of the non-commutative Iwasawa main conjecture due to Mahesh Ramesh Kakde. The proof of the non-commutative Iwasawa main conjecture for totally real number fields was, however, also announced in the arXiv preprint of Jürgen Ritter and Alfred Weiss in advance of Kakde's proof. Ritter and Weiss's proof is based on a strategy rather different from Kakde's, and thus it is very important and interesting to compare their methods. But Ritter and Weiss's proof is written in the language of their equivariant Iwasawa theory (which has already appeared in Nomura's second lecture), and unfortunately it seems to be difficult to compare them (or even to understand both proofs simultaneously) due to this language barrier.
Under these circumstances, I will first compare briefly the formulation in the style of [CFKSV05] to the formulation in the style of Ritter-Weiss's equivariant Iwasawa theory, and then observe that these two formulations essentially coincide. Then I will try to introduce Ritter-Weiss's strategy toward the proof of the non-commutative Iwasawa main conjecture (which, I think, has not been introduced in Japan so seriously) as much as possible, focusing on the difference from the proof of Kakde.
参考文献 (References): [CSSV12] の Venjakob の稿, [RW02a], [RW02b], [RW04], [RW05], [RW06], [RW08a], [RW08b], [RW08c], [RW10], [RW11] など
参考文献について (On References)
- 代数的整数論の基礎知識 (Basics on algebraic number theory) [数論I], [数論II], [CaFr86], [Se86], [Neu92]
- (主に有限群の) 表現論の基礎知識 (Basics on representation theory of finite groups) [Se67], [CR81], [CR87]
- 代数体 (特に有理数体) の岩澤理論の一般論 (Generality on Iwasawa theory for number fields, especially for the rational number fields) [SS03], [数論II], [La90], [Wa97], [CS06]
- 代数体の岩澤類数公式 (Iwasawa's class number formula for number fields) [Iw59], [Se58-59]
- 総実代数体の円分 p 進 L 関数 (Cyclotomic p-adic L-functions for totally real number fields) [Iw72], [Coa77], [Se78], [Bar78], [CaN79], [DRi80]
- 総実代数体の円分岩澤主予想 (The cyclotomic Iwasawa main conjecture for totally real number fields) [MaWil84], [Wi90]
- 非可換岩澤主予想 (設定、定式化等) (Non-commutative Iwasawa main conjecture — settings and formulation) [CFKSV05], [Ven05], [FK06], [Hach07], [CSSV12], [Kak13]
- 非可換岩澤理論の日本語での概説 (Survey articles on non-commutative Iwasawa theory in Japanese) [八森04], [原10], [深谷11], [原12]
- 古典的 K 理論 (classical algebraic K-theory) [Bass68], [Sw68], [Mi71]
- 高次代数的 K 理論 (Higher algebraic K-theory) [Wal85], [TT90], [Schli11]
- 岩澤代数の K 理論 (K-theory for Iwasawa algebras) [Kato05], [SV10], [SV11]
- 整対数準同型 (Integral logarithmic homomorphisms) [Ol88], [OT88]
- Fröhlich 予想,Chinburg 予想とその周辺 (Fröhlich's conjecture, Chinburg's conjecture and related topics) [Ta81], [Fr83], [Ch85], [GRW99], [RW02a]
- Burns-加藤の手法 (Burns-Kato's method) [Kato], [Bu10a]
- 総実代数体の非可換岩澤主予想 (Non-commutative Iwasawa main conjecture for totally real number fields) [Kato], [Hara10], [Hara11], [RW08c], [RW11], [Kak11], [Kak13]
- 非アーベル岩澤理論 (Non-abelian Iwasawa theory) [Oz07]
- モジュライ空間の幾何学と幾何的保型形式 (Geometry of moduli spaces and geometric modular forms) [DRa73], [Katz73], [KM85], [Katz78], [DRi80], [Ra78]
- 楕円曲線の (非可換) 岩澤理論 ((Non-)commutative Iwasawa theory for elliptic curves) [CGRR99], [Gr01], [Hach07]
- (同変) 玉河数予想: 可換な係数の場合 ((Equivariant) Tamagawa number conjecture in commutative coefficients) [BlKa90], [Kato93a], [Kato93b], [FoPR94], [BuFl96], [Ki11]
- 同変玉河数予想: 非可換係数の場合 (Equivariant Tamagawa number conjecture in non-commutative coefficients) [BuFl01], [BuFl03], [HK02], [FuKa06], [Ve07]
- (標数 p の) 幾何学的非可換岩澤主予想 (Geometric non-commutative Iwasawa theory in characteristic p) [Bu10b], [Wit10], [Wit13]
- Ritter-Weiss の同変岩澤理論 (Ritter-Weiss' equivariant Iwasawa theory) [RW02a], [RW02b], [RW04], [RW05], [RW06], [RW08a], [RW08b], [RW08c], [RW10], [RW11]
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