広島大学理学部数学科 代数数理グループ

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.

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.

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.

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.

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.

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.

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.

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．

将棋，囲碁などの偶然の要素を含まない完全情報ゲームを組合せゲームと呼ぶ．組合せゲームに対する理論計算機科学的興味の一つがお互いが最善手を続けた場合どちらが勝つかという判定問題における計算量である．本講演では，「手札を出し切ったプレイヤが勝ち」である手札消費型組合せゲームをいくつか取り上げ，それぞれのゲームに対する勝者判定アルゴリズムを紹介する．

組合せゲーム理論の分野において，不偏ゲームは古くから活発に研究されており，不偏ゲームのもつ代数的構造は興味深いものとして知られている．本講演では，不偏ゲームと代数との関係について，いくつかの具体的なゲームに触れながら紹介する．また，時間が許せば最新の研究成果についても紹介する．

本講演では、第一に組合せゲーム理論について簡単に説明し、次いで全象ルールに関する研究を紹介する。本研究で扱う全象ルールとは、正規形非不偏ゲームの（TransfiniteやLoopyではない）ルールセットであって、任意の「ゲームの値」を取ることができるものである。このようなルールはこれまで一つしか見つかっていなかったが、講演者の研究で新たに三つの全象ルールを発見した。それらの全象ルールと全象性の証明手法について本講演では紹介する。さらに時間が許せば、聴講者の興味に応じて講演者の過去の研究について紹介する。

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.