Number Theory / Arithmetic Geometry Seminar
Title
Galois action on the etale fundamental group of the Fermat curve.
Date
October 7 (Mon), 2019, 16:30-17:30
Room
Room 110, RIMS
Speaker
Pries, Rachel (Colorado State University)
Abstract
If $X$ is a curve defined over a number field $K$, then we are motivated to understand the action of the absolute Galois group of $K$ on the etale fundamental group of $X$. When $X$ is the Fermat curve of degree $p$ and $K$ is the cyclotomic field generated by a pth root of unity, Anderson proved several theorems about this action on the etale homology of $X$. In earlier work, we made Anderson's results more explicit, when the homology has coefficients modulo $p$. More recently, we use a cup product in cohomology to determine the action on the lower central series of the fundamental group of the Fermat curve with coefficients modulo $p$. The proof involves some fun Galois theory and combinatorics. This is joint work with Davis and Wickelgren.
Date
March 11 and 12, 2019
Place
Speaker
Rasmussen, Christopher (Wesleyan University)
Title
Improvements on Bounds for Heavenly Abelian Varieties
Abstract
For a rational prime $\ell > 2$, a natural question arising
from the study of the arithmetic of certain Galois representations
attached to fundamental groups is to consider the collection of abelian
varieties over a fixed number field $K$ whose $\ell$-power torsion
generates an extension of $K(\mu_{\ell})$ which is both pro-$\ell$ and
unramified away from $\ell$. (We say such an abelian variety is heavenly
at $\ell$.) Conjecturally, the set of $K$-isomorphism classes of heavenly
varieties of fixed dimension $g$ is finite -- even when the prime $\ell$
is permitted to vary. In past joint work with Tamagawa, we have provided
a proof of the Conjecture under GRH, and settled the Conjecture
positively in a number of cases.
Such results imply the existence of a bound $L = L(K,g)$ such that $\ell
< L$ for heavenly abelian varieties over $K$ of dimension $g$. Explicit
formulas for such a bound $L$ exist in general, but are too weak to be
of practical use. In this talk, we review previous results and also
demonstrate a method for improving such bounds when one is willing to
restrict attention to a fixed field $K$ and dimension $g$.
Speaker
Rasmussen, Christopher (Wesleyan University)
Title
Algorithms for Solving S-Unit Equations
Abstract
Many enumerative problems in arithmetic geometry and number theory take the form ``Find, up to isomorphism, all objects $\mathcal{O}$ with arithmetic property $\mathcal{P}(S)$,'' where the data $S$ is a finite set of places of a fixed number field $K$. Such problems often encounter a common computational obstacle; namely the solution of the $S$-unit equation $x + y = 1$ over the ring of $S$-integers of $K$. In this talk, we describe joint work with several mathematicians to create an open-source implementation of such an algorithm for general choices of $K$ and $S$. (Emphasis of this discussion will be on the mathematical, rather than computational, aspects of the project.) In addition, we provide some new partial results to various problems, including Asymptotic Fermat's Last Theorem over certain cubic number fields.
Speaker
Sakugawa, Kenji (RIMS)
Title
On Jannsen's conjecture for modular forms
Abstract
Let M be a pure motive over Q and let M_p denotes its p-adic
etale realization for each prime number p. In 1987's paper, Jannsen
proposed a conjecture about a range of integers r such that the second
Galois cohomology of M(r)_p vanishes for any prime number p. Here, M(r)
denotes the rth Tate twist of M. When M is an Artin motive, Jannsen's
conjecture had already essentially proven a half by Soule aroud early
1980's. In this talk, we consider the case when M is a motive associated
to elliptic modular forms. I will explain my ongoing work about an
approach to the conjecture used the p-weighted fundamental groups of
modular curves.
(Sakugawa's lecture is given in Japanese.)
** We plan to have a meal for welcoming Prof. Rasmussen after the seminar of March 11. If you join the meal, could you inform Takao (e-mail: takao_at_kurims.kyoto-u.ac.jp) by March 4?
Date and Place
Lecture 1, October 1st (Mon), 10:00-12:00, Room 110
Lecture 2, October 1st (Mon), 16:30-18:30, Room 110
Lecture 3, October 5th (Fri), 13:30-15:30, Room 110
Lecture 4, October 5th (Fri), 16:30-18:30, Room 110
Lecture 5, October 12th (Fri), 16:30-18:30, Room 111, 2018
Speaker
Collas, Benjamin (University of Bayreuth)
Title
Homotopical Arithmetic Geometry of Stacks
Abstract
Moduli spaces of curves possess properties which make them ideal
spaces where to concretely study fundamental abstract theories of
arithmetic geometry: they give geometric Galois representations that can
be computed explicitly, furnish examples of anabelian spaces, and in genus
zero generate the category of mixed Tate motives. They also possess a dual
nature, being either considered as schemes or algebraic stacks.
The goal of this series of talks is to provide a basic introduction to
these aspects by covering various fundamental geometric and arithmetic
properties. It is intended for graduate students in algebraic geometry and
non-specialists researchers. Elementary notions will be either recalled or
illustrated with pictures/examples.
I- Algebraic & Deligne-Mumford Stacks (lectures 1 and 2)
Taking the functor of points for schemes as initial motivation, we
introduce the notion of stacks as lax functors in groupoids with descent
conditions and show how to recover Laumon-Moret-Bailly's original
definition. We present how the Artin and Deligne-Mumford algebraic
versions -- that admit topological coverings by schemes -- allow to
``push'' algebraic geometry properties in this context.
Keywords: diagrams of groupoids, Grothendieck topology and
etale/ffpf/smooth morphisms, examples of global quotient and inertia
stacks.
II - Moduli Problems & Moduli Spaces of Curves (lecture 3)
We present how the scheme-stack structures and the geometry of curves lead
to two solutions for building classifying spaces. Having introduced the
notion of functor of moduli, we present Gieseker and Deligne-Mumford
constructions of the moduli space of curves: the former follows Mumford
G.I.T-theory and give a quasi-projective scheme, the latter produces a
smooth algebraic Deligne-Mumford global stack with a nice stable
compactification.
Keywords: Hilbert scheme, explicit examples in low genus, stable
compactification, formal neighbourhood.
III - Fundamental Group & Arithmetic (lecture 4)
We follow Grothendieck construction of the etale fundamental group
that leads to Geometric Galois actions of the absolute Galois group of
rational on the geometric fundamental group of moduli stack of curves. We
adapt this approach in the case of Deligne-Mumford stacks and show how it
leads to a divisorial and a stack arithmetic of the spaces. Following the
seminal work of Ihara, Matsumoto and Nakamura, we present explicit results
and properties of the former, then recent similar results in the case of
cyclic inertia for the latter.
Keywords: properties of the etale fundamental group, explicit
computations in low dimensions, tangential base points and
representations.
IV - Motivic Theory for Moduli Stack of Curves (lecture 5)
We present recent progress on an ongoing project on the construction of a
category of motives for the moduli stacks of curves, whose main property
is to reflect the arithmetic properties of the cyclic stack inertia.
Having recalled briefly what the category of mixed motives should be, we
first present Morel-Voevodski stable/unstable motivic homotopy categories,
then how their homotopical-simplicial approach is well adapted to our
goal.
Keywords: Quillen model category, Artin-Mazur etale topological type,
Mixed Tate motives and loop space.
Emiliano Ambrosi's successive lectures
Title
p-adic cohomology theories with a view toward l-adic applications
Date
Lecture 1, August 24th (Thu), 13:30-15:00
Lecture 2, August 24th (Thu), 15:30-17:00
Lecture 3, August 28th (Mon), 13:30-15:00
Lecture 4, August 28th (Mon), 15:30-17:00
Lecture 5, September 1st (Fri), 13:30-15:00
Lecture 6, September 1st (Fri), 15:30-17:00, 2017
Room
Room 006, RIMS
Speaker
Emiliano Ambrosi 氏 (Ecole Polytechinique)
Abstract
Chieh-Yu Chang 氏連続講義
Title
On Hilbert's seventh problem and transcendence theory
Date
October 9th (Wed), 23rd (Wed) and 28th (Mon), 2013, 10:30-12:00
[change in the date]
October 9th (Wed) 10:30-12:00,
October 23rd (Wed) 10:30-12:00, 13:30-15:00, 2013
Room
Room 006, RIMS
Speaker
Chieh-Yu Chang 氏 (National Tsing Hua University)
Abstract
Hilbert's seventh problem is about the linear independence question of two logarithms of algebraic numbers, which was solved by Gelfond and Schneider in the 1930s. Later on, it was generalized to several logarithms of algebraic numbers by Baker in the 1960s and generalized to general abelian logarithms of algebraic points by Wuestholz in the 1980s. This phenomenon can be also asked for multiple zeta values, but it is still open. In the first talk, we will give a survey on the classical theory and report recent progress on the parallel questions for function fields in positive characteristic. Current methods and tools of transcendence theory using t-motives will be discussed in the second and third talks.
Mini-Workshop on Number Theory / Arithmetic Geometry
Date
Thursday, January 31, 2013
Room
Room 206, RIMS, Kyoto University
10:00 -- 10:20
Arata Minamide (RIMS, M2)
Elementary Anabelian Properties of Graphs
10:30 -- 10:50
Yang Yu (RIMS, M2)
Arithmetic Fundamental Groups and Geometry of Curves
over a Discrete Valuation Ring
11:00 -- 11:20
Takeshi Okada (RIMS, M2)
On Finiteness of Twists of Abelian Varieties
13:00 -- 13:30
Yu Iijima (RIMS, D1)
Galois Action on Mapping Class Groups
13:45 -- 14:45
Chia-Fu Yu (Academia Sinica)
Density of the Ordinary Locus in the Hilbert-Siegel Moduli Spaces
Title
Resolution of nonsingularities for Mumford curves.
Date
December 15 (Thu), 2011, 14:15-15:45
Room
Room 206, RIMS
Speaker
Emmanuel Lepage 氏 (Institut Mathematique de Jussieu)
Abstract
Let $X$ be a hyperbolic curve over $\overline Q_p$. I am interested in the following property: for every semistable model $\mathcal X$ of $X$ and every closed point $x$ of the special fiber there exists a finite covering $Y$ of $X$ such that the minimal semistable model $\mathcal Y$ of $Y$ above $\mathcal X$ has a vertical component above $x$. I will try to explain why hyperbolic Mumford curves satisfy this property. I will give anabelian appplications of this to the tempered fundamental group.
Mini-Workshop ``Rational Points on Modular Curves and Shimura Curves''
Date
Monday, October 26th, 2009
13:30--14:30
Keisuke Arai (Univ. Tokyo)
Points on $X_0^+(N)$ over quadratic fields
(joint work with F. Momose)
Abstract:
Momose (1987) studied the rational points on the modular curve
$X_0^+(N)$ for a composite number $N$. He showed that the rational
points on $X_0^+(N)$ consist of cusps and CM points under certain
conditions on a prime divisor $p$ of $N$. But $p=37$ was excluded.
For $37$ is peculiar because $X_0(37)$ is a hyperelliptic curve and
$w_{37}$ is not the hyperelliptic involution. We show that the
rational points on $X_0^+(37M)$ consist of cusps and CM points.
We also show that the $K$-rational points on $X_0^+(N)$ consist
of cusps and CM points for a quadratic field $K$ under certain
conditions (both $p=37$ and $p\ne 37$ allowed).
14:45--15:45
Fumio Sairaiji (Hiroshima International Univ.)
Takuya Yamauchi (Osaka Prefecture Univ.)
On rational torsion points of central $\mathbb{Q}$-curves
Abstract:
Let $E$ be a central $\mathbb{Q}$-curve over a polyquadratic field $k$.
In this talk we give an upper bound for prime divisors $p$ of the order
of the $k$-rational torsion subgroup of $E$. For example, $p$ is less
than or equal to 13, if the scalar restriction of $E$ from $k$ to
$\mathbb{Q}$ is of GL$_2$-type with real multiplications. Our result
is a generalization of the result of Mazur on elliptic curves over
$\mathbb{Q}$, and it is a precision of the upper bounds of Merel and
Oesterl\'{e}.
16:00--17:00
Pierre Parent (Univ. Bordeaux 1)
Rational points on Shimura curves
Abstract:
For $B$ a rational quaternion algebra, the Shimura curve associated with $B$
(or more precisely its quotient by certain Atkin-Lehner involutions) is a
moduli
space, in a certain sense, for abelian surfaces with potential
multiplication by
$B$. Proving that those curves almost never have rational points would
therefore allow a small step towards the conjecture, attributed to Coleman
and Mazur, which predicts the scarcity of endomorphism algebras for
abelian varieties of GL$_2$-type over $\mathbb{Q}$. We will present
a method to study such rational points, developped by A. Yafaev and myself,
and recently improved by F. Gillibert.
Marco Boggi 氏連続講演
Title
``Profinite curve complexes and the congruence subgroup problem for the mapping class group''
Date
September 27th (Thu) and 28th (Fri), 2007
Room
at Room 206, RIMS, Kyoto University
27th (Thu)
10:00--12:00 Boggi
lunch
14:00--16:00 Boggi
16:00-- free discussion
dinner
28th (Fri)
10:00--12:00 Boggi
lunch
14:00--16:00 free discussion
Abstract
別紙のとおり
Title
Arithmetic from Geometry on Elliptic Curves
Date
June 2 (Fri), 2006, 16:30-17:30
Room
Room 202, RIMS
Speaker
Christopher Rasmussen (Rice Univ.)
Abstract
One of the philosophies of arithmetic geometry made popular by Grothendieck was the notion that the structure of the absolute Galois group of $\mathbf{Q}$, could be determined from geometric (or even combinatoric) data. In a related vein, one finds that the arithmetic properties of a curve are sometimes determined by its geometry. Specifically, the structure of a curve as a cover of the projective line can have arithmetic consequences for the Jacobian of the curve. We will discuss this situation in the case of elliptic curves, where this connection between arithmetic and geometry can be seen very clearly.
Title
Arithmetic Algebraic Geometry Lecture（集中講義）
Date
May 8 (Mon)- May 19, 2006.
Room
こちらをご覧ください
Speaker
加藤和也（京大 理）
Abstract
Weil が 1949 年に提出した Weil 予想は、 1970 年代のはじめに最終的に証明されるまで、 代数幾何学の大きな進展の原動力となり、 とくに Grothendieck が Weil 予想の証明をめざして導入した エタールコホモロジーは、現在の整数論の重要な道具となった。 その経緯をふり返りつつ、エタールコホモロジーの解説をおこなう。
Title
Algebraic dynamical systems (preperiodic points, Mahler measures, equidistribution of small points)
Date
May 1 (Mon), 2006, 16:30-
Room
Room 202, RIMS
Speaker
Lucien Szpiro (City Univ. New York)
Abstract
Reference: (available at http://math.gc.cuny.edu/faculty/szpiro/People_Faculty_Szpiro.html)
--Joint papers with T. Tucker
--Joint paper with E. Ullmo and S. Zhang
Date
April 10 (Mon), 2006, 14:00-17:00
Room
Room 202, RIMS
(14:00-15:15)
Speaker
Michel Matignon (Univ. Bordeaux 1/Chuo Univ.)
Title
Wild monodromy groups and automorphisms groups of curves
(16:00-16:45)
Speaker
Barry Green (Univ. Stellenbosch/Chuo Univ.)
Title
Selected results on liftings of Galois covers of smooth curves from char. p to char. 0
Title
Mini-Workshop ``Arithmetic Geometry of Covers of Curves and Related Topics''
Date
September 12 (Mon), 13(Tue), 2005
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