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過去の記録 ～04/19｜次回の予定｜今後の予定 04/20～
| 開催情報 | 水曜日 17:00～18:00 数理科学研究科棟(駒場) 117号室 |
|---|---|
| 担当者 | 今井 直毅，ケリー シェーン |
2010年12月01日(水)
16:30-18:45 数理科学研究科棟(駒場) 056号室
星裕一郎 氏 (京都大学数理解析研究所) 16:30-17:30
On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)
Galois theory for schemes (ENGLISH)
星裕一郎 氏 (京都大学数理解析研究所) 16:30-17:30
On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves (JAPANESE)
[ 講演概要 ]
In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.
For a hyperbolic curve X over a number field, are the following three conditions equivalent?
(A) For any prime number l, X is quasi-l-monodromically full.
(B) There exists a prime number l such that X is l-monodromically full.
(C) X is l-monodromically full for all but finitely many prime numbers l.
The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.
For an elliptic curve E over a number field, the following four conditions are equivalent:
(0) E does not admit complex multiplication.
(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.
(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.
(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.
In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).
Marco Garuti 氏 (University of Padova) 17:45-18:45In this talk, we will discuss the following problem posed by Makoto Matsumoto and Akio Tamagawa concerning monodromic fullness of hyperbolic curves.
For a hyperbolic curve X over a number field, are the following three conditions equivalent?
(A) For any prime number l, X is quasi-l-monodromically full.
(B) There exists a prime number l such that X is l-monodromically full.
(C) X is l-monodromically full for all but finitely many prime numbers l.
The property of being (quasi-)monodromically full may be regarded as an analogue for hyperbolic curves of the property of not admitting complex multiplication for elliptic curves, and the above equivalence may be regarded as an analogue for hyperbolic curves of the following result concerning the Galois representation on the Tate module of an elliptic curve over a number field proven by Jean-Pierre Serre.
For an elliptic curve E over a number field, the following four conditions are equivalent:
(0) E does not admit complex multiplication.
(1) For any prime number l, the image of the l-adic Galois representation associated to E is open.
(2) There exists a prime number l such that the l-adic Galois representation associated to E is surjective.
(3) The l-adic Galois representation associated to E is surjective for all but finitely many prime numbers l.
In this talk, I will present some results concerning the above problem in the case where the given hyperbolic curve is of genus zero. In particular, I will give an example of a hyperbolic curve of type (0,4) over a number field which satisfies condition (C) but does not satisfy condition (A).
Galois theory for schemes (ENGLISH)
[ 講演概要 ]
We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.
We discuss some aspects of finite group scheme actions: the Galois correspondence and the notion of Galois closure.
