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過去の記録 ～03/21｜次回の予定｜今後の予定 03/22～
| 開催情報 | 金曜日 13:30～15:00 数理科学研究科棟(駒場) ハイブリッド開催/117号室 |
|---|---|
| 担当者 | 權業 善範、中村 勇哉、田中 公 |
2024年06月21日(金)
13:30-15:00 数理科学研究科棟(駒場) ハイブリッド開催/056号室
Kien Nguyen Huu 氏 (Normandie Université/KU Leuven)
ON THE POWER SERIES OF DENEF AND LOESER'S MOTIVIC VANISHING CYCLES OF JET POLYNOMIALS (English)
Kien Nguyen Huu 氏 (Normandie Université/KU Leuven)
ON THE POWER SERIES OF DENEF AND LOESER'S MOTIVIC VANISHING CYCLES OF JET POLYNOMIALS (English)
[ 講演概要 ]
Let f be a non-constant polynomial in n variables over a field k of characteristic
0. Denef and Loeser introduced the notion of motivic vanishing cycles of f as an element in
the localization Mμˆ of the Grothendieck ring Kμˆ(Var ) of k-varieties with a good action of k0k
μˆ := lim μm by inverting the affne line equipped with the trivial action of μˆ, where μm
is the group scheme over k of mth roots of unity. In particular, if k is the field of complex
numbers then Denef and Loeser showed that their motivic vanishing cycles and the complex
φf [n − 1] has the same Hodge characteristic, where φf is the complex of vanishing cycles
in the usual sense. Motivated by the Igusa conjecture for exponential sums and the strong
monodromy conjecture, we introduce the notion of Poincaré series of Denef-Loeser's van-
ishing cycles of jet polynomials of f, where jet polynomials of f are polynomials appearing
naturally when we compute the jet schemes of f. By using Davison-Meinhardt's conjecture
which was proved by Nicaise and Payne in 2019, we can show that our Poincaré series is a
rational function over a quotient ring of Mμˆ by very natural relations. In particular, we can k
recovery Denef and Loeser's motivic vanishing cycles from our Poincaré series. Moreover, we can show that our Poincaré series owns a universal property in the sense that if k is a number field then the Igusa local zeta functions, the motivic Igusa zeta functions, the Poincaré series of exponential sums modulo pm of f can be obtained from our Poincaré se- ries by suitable specialization maps preserving the rationality. If time permits, I will present some initial consequences that have arisen during the study of our Poincaré series.
Let f be a non-constant polynomial in n variables over a field k of characteristic
0. Denef and Loeser introduced the notion of motivic vanishing cycles of f as an element in
the localization Mμˆ of the Grothendieck ring Kμˆ(Var ) of k-varieties with a good action of k0k
μˆ := lim μm by inverting the affne line equipped with the trivial action of μˆ, where μm
is the group scheme over k of mth roots of unity. In particular, if k is the field of complex
numbers then Denef and Loeser showed that their motivic vanishing cycles and the complex
φf [n − 1] has the same Hodge characteristic, where φf is the complex of vanishing cycles
in the usual sense. Motivated by the Igusa conjecture for exponential sums and the strong
monodromy conjecture, we introduce the notion of Poincaré series of Denef-Loeser's van-
ishing cycles of jet polynomials of f, where jet polynomials of f are polynomials appearing
naturally when we compute the jet schemes of f. By using Davison-Meinhardt's conjecture
which was proved by Nicaise and Payne in 2019, we can show that our Poincaré series is a
rational function over a quotient ring of Mμˆ by very natural relations. In particular, we can k
recovery Denef and Loeser's motivic vanishing cycles from our Poincaré series. Moreover, we can show that our Poincaré series owns a universal property in the sense that if k is a number field then the Igusa local zeta functions, the motivic Igusa zeta functions, the Poincaré series of exponential sums modulo pm of f can be obtained from our Poincaré se- ries by suitable specialization maps preserving the rationality. If time permits, I will present some initial consequences that have arisen during the study of our Poincaré series.
