広島大学理学部数学科 代数数理グループ


通常の講演時間はおよそ 1 時間半です。


712(金) 1620 分   於 広島大学理学部 A 棟 A210 号室 (曜日・場所にご注意ください)
内海 和樹 氏(立命館大学)
The Mordell-Weil lattices of an Inose surface arising from isogenous elliptic curves
$II^∗$ 型特異ファイバーを 2 個持つ楕円 K3 曲面は猪瀬曲面と呼ばれる. 猪瀬曲面はある直積型 Kummer 曲面の 2 重被覆として構成できることが知られており,従って,2 個の楕円曲線から構成できる. 猪瀬曲面の Mordell-Weil 格子は一般には自明だが,付随する楕円曲線の間に同種写像が存在すれば,その同種写像に由来する因子が楕円曲面としての切断となり,Mordell-Weil 格子は非自明になる. この楕円曲線間の同種写像に由来する Mordell-Weil 格子の元を Weierstrass 方程式の有理点として記述するための方法と具体的な計算例を紹介する. 特に, Cayley-Bacharach の定理によって同種写像の次数に制限なく計算できる方法を確立できたので, これまで知られていなかった次数 5, 6 の場合の計算例を紹介する.

711(木) 1620 分   於 広島大学理学部 A 棟 A202 号室 (曜日・場所にご注意ください)
Ivan Cheltsov 氏(University of Edinburgh)
K-stability of pointless del Pezzo surfaces and pointless Fano threefolds
In this talk we will discuss K-stability of pointless del Pezzo surfaces and smooth Fano 3-folds defined over a non-algebraically closed field of characteristic zero. This is a joint work in progress with Takashi Kishimoto (Saitama), Hamid Abban (Nottingham) and Frederic Mangolte (Marseille).


73(水) 1620(時間が変更となりました)   於 広島大学理学部 A 棟 A201 号室
De-Qi Zhang 氏 (National University of Singapore)
Algebraic and Arithmetic Dynamics via Equivariant Minimal Model Program
We report our recent progress via the Equivariant Minimal Model Program (EMMP) towards the algebraic and arithmetic dynamics: Kawaguchi-Silverman conjecture (KSC) about the equality of dynamical degree and arithmetic degree of an endomorphism of a projective variety, and the Zariski Dense Orbit conjecture (ZDO) of an endomorphism.

626(水) 1620(時間が変更となりました)   於 広島大学理学部 A 棟 A201 号室
山内 卓也 氏 (東北大学)
代数体$K$上定義された代数多様体$X/K$の中間次数の法$p$エタールコホモロジー $H^d(X/\bar{K},{\bf Z}/p{\bf Z})$, $d={\rm dim}X$に付随するガロア表現について:${\rm Aut}_K(X)$が豊富な場合.
素数$p$, 代数体$K$上定義された(非特異射影的かつ幾何的連結な)代数多様体$X/K$, および整数$0\le i\le 2d,\ d={\rm dim}(X)$に対して, $i$次の法$p$エタールコホモロジー$V_{i,p}:=H^i(X/\bar{K},{\bf Z}/p{\bf Z})$を考えることができる ($V_{i,p}$は有限次元${\bf Z}/p{\bf Z}$ベクトル空間である). 空間$V_{i,p}$には 代数体$K$の絶対ガロア群$G_K$が${\bf Z}/p{\bf Z}$線形に作用し, ${\bf Z}/p{\bf Z}$線形表現 $\rho_{X,i,p}:G_K\longrightarrow {\rm Aut}_{{\bf Z}/p{\bf Z}}(V_{i,p})$ を得る. ガロア対応により, この表現の核に対応する$\bar{K}$の固定体$K_{\rho_{X,i,p}}$は は$X$の対称性をある意味で計っているといえる.

本講演では, Xとして, 楕円曲線, Dwork族などのCalabi-Yau超曲面(族)を考えた場合 $X$の自己同型群の情報を援用して, 固定体$K_{\rho_{X,i,p}}$の数論的情報を 如何に抽出することが可能であるか説明する.

本研究は部分的に独立して, Alex Ghitza (メルボルン大), 都築暢夫氏(東北大) との共同研究に基づくこと付記する.

209(金)1500 分  於 広島大学先端研 N 棟 404N 号室
Dino Festi 氏 (The University of Padua)
Black holes, rationalizability and modularity
Physicists interested in high energy physics often encounter Feynman integrals presenting square roots in their argument. Exact solutions of these integrals are normally out of reach and so they are usually solved numerically. In order to achieve higher precision in the numeric evaluationit is necessary to find a change of the variables of the integral that makes the square root disappear. Deciding the existence of such a change of variable is an algebraic problem that can be naturally translated into investigating the unirationality of a variety. The original problem can be generalized to sets of square roots and to algebraic extensions of function fields. We will finally present a case coming from the study of two black holes, treated also using modularity results.

The content of this talk is the fruit of a series of joint works with Marco Besier, Andreas Hochenegger, and Bert van Geemen.

112(金)1500 分  於 B702 号室
松澤 陽介 氏(大阪公立大学)
Preimages question of self-morphisms on projective varieties over number fields
Pulling back an invariant subvariety by a self-morphism on projective variety, you will get a tower of increasing closed subsets. Working over a number field, we expect that the set of rational points (of bounded degree) contained in this increasing subsets eventually stabilizes. I will explain why this expectation seems to be reasonable and introduce several affirmative cases, such as the case of etale morphisms and morphisms on the product of two P^1. I will also present some counterexamples that occur when we drop some of the assumptions.

This work is based on a joint work with Matt Satriano and Jason Bell, and recent work with Kaoru Sano.

106(金)1500 分  於 B702 号室
Pho Duc Tai 氏(VNU University of Science, Hanoi)
On the arithmetic of Edwards curves
In this talk we recall the history of elliptic curves in Weierstrass normal form, then we will explain the construction of the arithmetic (point addition and point doubling) on elliptic curves in Edwards normal form. Later we discuss the geometry of Edwards curves and applications.

929(金)1500 分  於 B702 号室
Davide Cesare Veniani 氏(Stuttgart University)
Non-degeneracy of Enriques surfaces
Enriques' original construction of Enriques surfaces dates back to 1896. It involves a 10-dimensional family of sextic surfaces in the projective space which are non-normal along the edges of a tetrahedron. The question whether all Enriques surfaces arise through Enriques' construction has remained open for more than a century.

In two joint works with G. Martin and G. Mezzedimi, we have now settled this question in all characteristics by studying particular configurations of genus one fibrations, and two invariants called maximal and minimal non-degeneracy. The proof involves so-called `triangle graphs' and the distinction between special and non-special 3- sequences of half-fibers.

In this talk, I will present the classification of Enriques surfaces of low non-degeneracy and explain how this classification solves this long- standing problem.

84(金)1500 分  於 B702 号室
Simon Brandhorst 氏(Saarland University)
K3 surfaces of zero entropy (Joint work with Giacomo Mezzedimi)
Automorphisms of K3 surfaces come in 3 flavors: 1) The orbit of every point is finite. 2) There exists a point with an infinite orbit, but no orbit is Zariski dense. 3) There is a Zariski dense orbit. In the first and second case the automorphism has zero topological entropy while in the last case it is of positive entropy. We say that a surface has zero entropy if every of its automorphisms has zero entropy.

In this talk we classify K3 surfaces of zero entropy yet with infinite automorphism group, equivalently, which have a unique elliptic fibration whose Jacobian has infinite Mordell-Weil group.

728(金)1500 分  於 B702 号室
山田 裕史 氏(岡山大学)
円周上のベクトル場のなす無限次元リー環はWitt代数と呼ばれる. その1次元中心拡大が表題のVirasoro代数である. 40年ほど前に脇本實氏と私は,そのFock表現について 特異ベクトルが特別なシューア函数として表示されることに気がついた. 長方形のヤング図形に対応するシューア函数である. なぜ長方形が登場するのかという疑問には,脇本ー山田のプレプリントが 出回った直後にN. Wallachが明確に答えてくれた. 回はこのWallachの仕事を大雑把に紹介したい.

710(月)1600 分  於 B701 号室
木谷 裕紀 氏(大阪公立大学)

710(月)1435 分  於 B701 号室
安福 智明 氏(国立情報学研究所)

77(金)1500 分  於 B702 号室
末續 鴻輝 氏(国立情報学研究所)

623(金)1500 分  於 B702 号室
山下 貴央 氏(広島大学)
Yama Nim and a comply/constrain operator of combinatorial games
We introduce Yama Nim, a variation of Nim played on a two-dimensional semi-infinite game board, with terminal positions in the upper left corner. The player can move two or more up steps and one right step, or two or more left steps and one down step. If a player cannot move, they lose. We find the solution to this game. We also consider a comply/constrain operator on impartial rulesets. Applied to the rulesets A and B, on each turn the opponent proposes one of the rulesets and the current player complies, by playing a move in that ruleset. If the outcome table of the comply/constrain variation of A and B is the same as the outcome table of A, then we say that B is dominated by A. We show necessary and sufficient conditions of "A dominates B". Yama Nim is a good example that dominates classical rulesets such as Nim and Wythoff Nim.