講演内容: 第33回 (2026年度) 整数論サマースクール『関数体の数論』

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参考文献

可換環論の基礎 [AM69], [Mat70], [GW06]
代数的整数論の基礎 [数論I], [数論II], [CF86], [Ser68], [Neu92]
楕円曲線の基礎 [Sil86], [ST92]
p 進 Hodge 理論の基礎 [辻99], [中村10]
関数体の数論の基礎 [Gos98], [Ros02], [Tha04]
Carlitz 加群 [Car35], [Gos98], [Ros02], [Pap23], [Hay74]
Drinfeld 加群 [Dri74], [Gos98], [Ros02], [Pap23], [Poo95], [Tak82], [Poo97], [Ish24]
Anderson の t-モチーフ, A-モチーフ [And86], [Gar02], [HJ20]
Hodge–Pink 理論 [HJ20], [HK20]
Taelman 類群の数理 [Tae12], [KO25+]

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[数論I] 加藤和也，黒川信重，斎藤毅著，数論I —Fermat の夢と類体論，岩波書店 (2005).
[数論II] 黒川信重，栗原将人，斎藤毅著，数論II —岩澤理論と保型形式，岩波書店 (2005).
[辻99] 辻雄，Introduction to the theory of Fontaine on p-adic Galois representations，数理解析研究所講究録，1097 (1999), 1–26.
[中村10] 中村健太郎，p 進表現入門，第17回 (2009年度) 整数論サマースクール『ℓ 進ガロア表現とガロア変形の整数論』報告集 (2010), 183–210.
[And86] Greg William Anderson, t-motives, Duke Math. J., 53 (2) (1986), 457–502.
[AM69] Michael Francis Atiyah and Ian Grant Macdonald, Introduction to Commutative Algebra, Addison–Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. ix + 128 pp.
[Car35] Leonard Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J., 1 (2) (1935), 137–168.
[CF86] John William Scott Cassels and Albrecht Fröhlich edt., Algebraic Number Theory, A Nato Advanced Study Institute W, Academic Press (1986).
[Dri74] Vladimir Gershonovich Drinfeld (Володи́мир Ге́ршонович Дрінфельд), Эллиптические модули, Математический сборник (новая серия), 94 (136), номер 4 (8) (1974), 594–627.
(English Translation: Elliptic modules, Mathematics of the USSR-Sbornik, 23 (4) (1974), 561–592.
[Gar02] Francis Gardeyn, A Galois criterion for good reduction of τ-sheves, J. Number Theory, 97 (2) (2002), 447–471.
[Gos98] David Goss, Basic Structures of Function Field Arithmetic, Springer–Verlag (1998).
[GW06] Shiro Goto (後藤四郎) and Kei-ichi Watanabe (渡辺敬一), Commutative Algebra, Grad. Texts in Math., 309, Springer Singapore, [2026], ©2026. xvii + 370 pp.
※ 後藤四郎，渡辺敬一著可換環論, 日本評論社 (2011) の英訳，5月5日刊行予定．
[Hay74] David Ryan Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc., 189 (1974), 77–91.
[HJ20] Urs Hartl and Ann-Kristin Juschka, Pink's theory of Hodge structures and the Hodge conjecture over function fields, in: t-motives: Hodge structures, transcendence and other motivic aspects, 31–182, EMS Ser. Congr. Rep., EMS Publishing House, Berlin, [2020], ©2020.
[HK20] Urs Hartl and Wansu Kim, Local shtukas, Hodge–Pink structures and Galois representations, in: t-motives: Hodge structures, transcendence and other motivic aspects,183–259, EMS Ser. Congr. Rep., EMS Publishing House, Berlin, [2020], ©2020.
[Ish24] Shun Ishii (石井竣), The 𝔭-primary uniform boundedness conjecture for Drinfeld modules, Int. J. Number Theory, 20 (5) (2024), 1233–1263.
[KO25+] Takenori Kataoka (片岡武典) and Yoshiaki Okumura (奥村喜晶), Iwasawa-type asymptotic formula for Taelman class groups of Drinfeld modules, preprint available at arXiv, arXiv identifier: arXiv:2509.06633[math.NT].
[Mat70] Hideyuki Matsumura (松村英之), Commutative Algebra, W. A. Benjamin, Inc., New York, 1970. xii+262 pp. paperbound.
[Neu92] Jürgen Neukirch, Algebraische Zahlentheorie, Springer–Verlag (1992).
(邦訳: 足立恒雄監修梅垣敦紀訳『代数的整数論』丸善出版, 2012)
[Poo95] Bjorn Poonen, Local height functions and the Mordell–Weil theorem for Drinfeld modules, Compos. Math., 97 (3) (1995), 349–368.
[Poo97] Bjorn Poonen, Torsion in rank 1 Drinfeld modules and the uniform boundedness conjecture, Math. Ann., 308 (1997), 571–586.
[Ros02] Michael Rosen, Number Theory in Function Fields, Springer–Verlag (2002).
[Ser68] Jean-Pierre Serre, Corps locaux, Hermann (1968).
(英訳: Local Fields, Springer–Verlag, 1979)
[Sil86] Joseph Hillel Silverman, The arithmetic of elliptic curves, Grad. Texts in Math., 106, Springer–Verlag, New York (1992), xii+400pp.
(邦訳: 鈴木治郎訳『楕円曲線の数論 —基礎概念からアルゴリズムまで』丸善出版, 2023)
[ST92] Joseph Hillel Silverman and John Tate, Rational Points on Elliptic Curves, Undergrad. Texts Math., Springer–Verlag, New York (1992), x+281pp.
(邦訳: ；足立恒雄，木田雅成，小松啓一，田谷久雄訳『楕円曲線論入門』丸善出版, 2012)
[Tae12] Lenny Taelman, Special L-values of Drinfeld modules, Ann. of Math., 175 (2012), 369–391.
[Tak82] Toyofumi Takahashi (高橋豊文), Good reduction of elliptic modules, J. Math. Soc. Japan, 34 (3) (1982), 475–487.
[Tha04] Dinesh Thakur, Function Field Arithmetic, World Scientific (2004).

第33回 (2026年度) 整数論サマースクール 『関数体の数論』 2026.8.24–2026.8.28

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参考文献

第33回 (2026年度) 整数論サマースクール『関数体の数論』 2026.8.24–2026.8.28