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Number Theory Seminar
Seminar information archive ～07/26｜Next seminar｜Future seminars 07/27～
| Date, time & place | Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Naoki Imai, Shane Kelly |
Seminar information archive
2008/01/30
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Luc Illusie (Universite Paris-Sud 11)
Odds and ends on finite group actions and traces
Luc Illusie (Universite Paris-Sud 11)
Odds and ends on finite group actions and traces
[ Abstract ]
Suppose a finite group G acts on a scheme X separated and of finite type over a field k. This raises several questions about the traces of elements s of G (or more generally products sg, for g in the Galois group of k) on cohomology groups of various types associated with X/k (with compact support or no support, Betti if k = C, l-adic, rigid). Some were considered and solved long ago, others only recently. I will in particular discuss an equivariant generalization of a theorem of Laumon on Euler-Poincar¥'e characteristics.
Suppose a finite group G acts on a scheme X separated and of finite type over a field k. This raises several questions about the traces of elements s of G (or more generally products sg, for g in the Galois group of k) on cohomology groups of various types associated with X/k (with compact support or no support, Betti if k = C, l-adic, rigid). Some were considered and solved long ago, others only recently. I will in particular discuss an equivariant generalization of a theorem of Laumon on Euler-Poincar¥'e characteristics.
2008/01/23
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Weizhe Zheng (Universite Paris-Sud 11)
Integrality, Rationality, and Independence of l in l-adic Cohomology over Local Fields
Weizhe Zheng (Universite Paris-Sud 11)
Integrality, Rationality, and Independence of l in l-adic Cohomology over Local Fields
[ Abstract ]
I will discuss two problems on traces in l-adic cohomology over local fields with finite residue field. In the first part, I will describe the behavior of integral complexes of l-adic sheaves under Grothendieck's six operations and the nearby cycle functor. In the second part, I will talk about rationality and independence of l. More precisely, I will introduce a notion of compatibility for systems of l-adic complexes and explain the proof of its stability by the above operations, in a slightly more general context (equivariant under finite groups). The main tool in this talk is a theorem of de Jong on
alterations.
I will discuss two problems on traces in l-adic cohomology over local fields with finite residue field. In the first part, I will describe the behavior of integral complexes of l-adic sheaves under Grothendieck's six operations and the nearby cycle functor. In the second part, I will talk about rationality and independence of l. More precisely, I will introduce a notion of compatibility for systems of l-adic complexes and explain the proof of its stability by the above operations, in a slightly more general context (equivariant under finite groups). The main tool in this talk is a theorem of de Jong on
alterations.
2008/01/16
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Antoine Chambert-Loir (Universite de Rennes 1)
Equidistribution theorems in Arakelov geometry
Antoine Chambert-Loir (Universite de Rennes 1)
Equidistribution theorems in Arakelov geometry
[ Abstract ]
The proof of Bogomolov's conjecture by Zhang made a crucial use
of an equidistribution property for the Galois orbits of points of small
heights in Abelian varieties defined over number fields.
Such an equidistribution property is proved using a method invented
by Szpiro, Ullmo and Zhang, and makes use of Arakelov theory.
This equidistribution theorem takes place in the complex torus
associated to the Abelian variety. I will show how a similar
equidistribution theorem can be proven for the p-adic topology ;
we have to use Berkovich space. Thanks to recent results of Yuan
about `big line bundles' in Arakelov geometry, the situation
is now very well understood.
The proof of Bogomolov's conjecture by Zhang made a crucial use
of an equidistribution property for the Galois orbits of points of small
heights in Abelian varieties defined over number fields.
Such an equidistribution property is proved using a method invented
by Szpiro, Ullmo and Zhang, and makes use of Arakelov theory.
This equidistribution theorem takes place in the complex torus
associated to the Abelian variety. I will show how a similar
equidistribution theorem can be proven for the p-adic topology ;
we have to use Berkovich space. Thanks to recent results of Yuan
about `big line bundles' in Arakelov geometry, the situation
is now very well understood.
2007/12/05
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
中村健太郎 (東京大学大学院数理科学研究科)
Classification of two dimensional trianguline representations of p-adic fields
中村健太郎 (東京大学大学院数理科学研究科)
Classification of two dimensional trianguline representations of p-adic fields
[ Abstract ]
Trianguline representation is a class of p-adic Galois representations of p-adic fields. This was defined by P.Colmez by using ($\\varphi, \\Gamma$)-modules over Robba ring. In his study of p-adic local Langlands correspondence of GL_2(Q_p), he completely classified two dimensional trianguline representations of Q_p. On the other hand, L.Berger recently defined the category of B-pairs and established the equivalence between the category of B-pairs and the category of ($\\varphi,\\Gamma$)-modules over Robba ring. In this talk, we extend the Colmez's result by using B-pairs. We completely classify two dimensional trianguline representations of K for any finite extension of Q_p. We also talk about a relation between two dimensional trianguline representations and principal series or special series of GL_2(K).
Trianguline representation is a class of p-adic Galois representations of p-adic fields. This was defined by P.Colmez by using ($\\varphi, \\Gamma$)-modules over Robba ring. In his study of p-adic local Langlands correspondence of GL_2(Q_p), he completely classified two dimensional trianguline representations of Q_p. On the other hand, L.Berger recently defined the category of B-pairs and established the equivalence between the category of B-pairs and the category of ($\\varphi,\\Gamma$)-modules over Robba ring. In this talk, we extend the Colmez's result by using B-pairs. We completely classify two dimensional trianguline representations of K for any finite extension of Q_p. We also talk about a relation between two dimensional trianguline representations and principal series or special series of GL_2(K).
2007/11/21
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Christopher Rasmussen (京都大学数理解析研究所)
Abelian varieties with constrained torsion
Christopher Rasmussen (京都大学数理解析研究所)
Abelian varieties with constrained torsion
[ Abstract ]
The pro-$l$ Galois representation attached to the arithmetic fundamental group of a curve $X$ is heavily influenced by the arithmetic of certain classes of its branched covers. It is natural, therefore, to search for and classify these special covers in a meaningful way. When $X$ is the projective line minus three points, one finds that such covers are very scarce. In joint work with Akio Tamagawa, we formulate a conjecture to quanitify this scarcity, and present a proof for the conjecture in the case of genus one curves defined over $\\Q$.
The pro-$l$ Galois representation attached to the arithmetic fundamental group of a curve $X$ is heavily influenced by the arithmetic of certain classes of its branched covers. It is natural, therefore, to search for and classify these special covers in a meaningful way. When $X$ is the projective line minus three points, one finds that such covers are very scarce. In joint work with Akio Tamagawa, we formulate a conjecture to quanitify this scarcity, and present a proof for the conjecture in the case of genus one curves defined over $\\Q$.
2007/10/31
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Pierre Colmez (Ecole Polytechnique)
On the p-adic local Langlands correspondance for GL2(Qp)
Pierre Colmez (Ecole Polytechnique)
On the p-adic local Langlands correspondance for GL2(Qp)
2007/10/24
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
阿部知行 (東京大学大学院数理科学研究科)
l進層のSwan導手とunit-root
overconvergent F-isocrystalの特性サイクルについて
阿部知行 (東京大学大学院数理科学研究科)
l進層のSwan導手とunit-root
overconvergent F-isocrystalの特性サイクルについて
[ Abstract ]
今回の講演ではBerthelotによる数論的D加群の理論を用いることによってunit-root overconvergent F-isocrystalに対してSwan導手を定義し、Kato-Saitoにより幾何学的な手法を用いて定義されたSwan導手と比較する。応用として、特異点の解消の仮定のもとでKato-SaitoのSwan導手の整数性予想を導く。
今回の講演ではBerthelotによる数論的D加群の理論を用いることによってunit-root overconvergent F-isocrystalに対してSwan導手を定義し、Kato-Saitoにより幾何学的な手法を用いて定義されたSwan導手と比較する。応用として、特異点の解消の仮定のもとでKato-SaitoのSwan導手の整数性予想を導く。
2007/10/10
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
James Lewis (University of Alberta)
Abel-Jacobi Maps Associated to Algebraic Cycles I
James Lewis (University of Alberta)
Abel-Jacobi Maps Associated to Algebraic Cycles I
[ Abstract ]
This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.
This talk concerns the Bloch cycle class map from the higher Chow groups to Deligne cohomology of a projective algebraic manifold. We provide an explicit formula for this map in terms of polylogarithmic type currents.
2007/09/19
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Gereon Quick (Universitaet Muenster)
Etale cobordism
Gereon Quick (Universitaet Muenster)
Etale cobordism
[ Abstract ]
We define and study a new candidate of etale topological cohomology theories for schemes over a field of abritrary characteristic: etale cobordism. As etale K-theory is related to algebraic K-theory, etale cobordism is related to algebraic cobordism of Voevodsky and Levine/Morel. It shares some nice properties of topological theories, e.g. it is equipped with an Atiyah-Hirzebruch spectral sequence from etale cohomology. We discuss in particular a comparison theorem between etale and algebraic cobordism after inverting a Bott element and, finally, we give an outlook to further possible applications of this theory.
We define and study a new candidate of etale topological cohomology theories for schemes over a field of abritrary characteristic: etale cobordism. As etale K-theory is related to algebraic K-theory, etale cobordism is related to algebraic cobordism of Voevodsky and Levine/Morel. It shares some nice properties of topological theories, e.g. it is equipped with an Atiyah-Hirzebruch spectral sequence from etale cohomology. We discuss in particular a comparison theorem between etale and algebraic cobordism after inverting a Bott element and, finally, we give an outlook to further possible applications of this theory.
2007/09/12
15:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)
E. Lau (Univ. of Bielefeld) 15:00-15:45
Classification of p-divisible groups by displays and duality
T. Zink (Univ. of Bielefeld) 16:00-16:45
Applications of the theory of displays
E. Looijenga (Univ. of Utrecht) 17:00-18:00
Presentation of mapping class groups from algebraic geometry
E. Lau (Univ. of Bielefeld) 15:00-15:45
Classification of p-divisible groups by displays and duality
T. Zink (Univ. of Bielefeld) 16:00-16:45
Applications of the theory of displays
E. Looijenga (Univ. of Utrecht) 17:00-18:00
Presentation of mapping class groups from algebraic geometry
[ Abstract ]
A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.
A presentation of the mapping class group of a genus g surface with one hole is due to Wajnryb with later improvements due to M. Matsumoto. The generators are Dehn twists defined by 2g+1 closed curves on the surface. The relations involving only two Dehn twists are the familiar Artin relations, we show that those involving more than two can be derived from algebro-geometry considerations.
2007/08/27
16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Steven Zucker (Johns Hopkins大学)
The reductive Borel-Serre motive
Steven Zucker (Johns Hopkins大学)
The reductive Borel-Serre motive
2007/07/18
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
梶原 健 (横浜国立大学)
Tropical toric varieties
梶原 健 (横浜国立大学)
Tropical toric varieties
2007/07/11
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Andreas Rosenschon (University of Alberta)
Algebraic cycles on products of elliptic curves over p-adic fields
Andreas Rosenschon (University of Alberta)
Algebraic cycles on products of elliptic curves over p-adic fields
2007/06/27
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
Stephen Lichtenbaum (Brown University)
The conjecture of Birch and Swinnerton-Dyer is misleading
Stephen Lichtenbaum (Brown University)
The conjecture of Birch and Swinnerton-Dyer is misleading
[ Abstract ]
All values of zeta and L-functions at integral points should be given in terms of products and quotients of Euler characteristics, and the order of the zeroes and poles at these
points should be given by the sum and difference of the ranks of
corresponding finitely generated abelian groups.
All values of zeta and L-functions at integral points should be given in terms of products and quotients of Euler characteristics, and the order of the zeroes and poles at these
points should be given by the sum and difference of the ranks of
corresponding finitely generated abelian groups.
2007/05/09
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
宮崎 直 (東京大学大学院数理科学研究科)
$(g,K)$-module structures of principal series representations
of $Sp(3,R)$
宮崎 直 (東京大学大学院数理科学研究科)
$(g,K)$-module structures of principal series representations
of $Sp(3,R)$
2007/05/02
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
長谷川 泰子 (東京大学大学院数理科学研究科)
Cohen-Eisenstein series and modular forms associated to imaginary quadratic fields
長谷川 泰子 (東京大学大学院数理科学研究科)
Cohen-Eisenstein series and modular forms associated to imaginary quadratic fields
2007/04/25
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
津嶋 貴弘 (東京大学大学院数理科学研究科)
Localized Characteristic Class and Swan Class
津嶋 貴弘 (東京大学大学院数理科学研究科)
Localized Characteristic Class and Swan Class
2007/04/11
16:30-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)
斎藤 毅 (東京大学大学院数理科学研究科)
l進層の暴分岐と特性サイクル
斎藤 毅 (東京大学大学院数理科学研究科)
l進層の暴分岐と特性サイクル
2007/01/31
15:15-18:45 Room #117 (Graduate School of Math. Sci. Bldg.)
Dennis Eriksson (東大数理/Paris) 15:15-16:15
Towards a proof of a metrized Deligne-Riemann-Roch theorem
小林 真一 (名古屋大学多元数理) 16:30-17:30
CM楕円曲線の超特異点における2変数p進L関数
(A two variable p-adic L-function for CM elliptic curves at supersingular primes)
Frans Oort (Utrecht) 17:45-18:45
Irreducibility of strata and leaves in the moduli space of abelian varieties
Dennis Eriksson (東大数理/Paris) 15:15-16:15
Towards a proof of a metrized Deligne-Riemann-Roch theorem
小林 真一 (名古屋大学多元数理) 16:30-17:30
CM楕円曲線の超特異点における2変数p進L関数
(A two variable p-adic L-function for CM elliptic curves at supersingular primes)
Frans Oort (Utrecht) 17:45-18:45
Irreducibility of strata and leaves in the moduli space of abelian varieties
2006/12/20
16:30-18:45 Room #117 (Graduate School of Math. Sci. Bldg.)
Anna Cadoret (RIMS/JSPS) 16:30-17:30
On the profinite regular inverse Galois problem
An elementary perspective on modular representation theory
Anna Cadoret (RIMS/JSPS) 16:30-17:30
On the profinite regular inverse Galois problem
[ Abstract ]
Given a field $k$ and a (pro)finite group $G$, consider the
following weak version of the regular inverse Galois problem:
(WRIGP/$G$/$k$) \\textit{there exists a smooth geometrically
irreducible curve $X_{G}/k$ and a Galois extension $E/k(X_{G})$
regular over $k$ with group $G$.} (the regular inverse Galois
problem (RIGP/$G$/$k$) corresponding to the case
$X_{G}=\\mathbb{P}^{1}_{k}$). A standard descent argument shows that
for a finite group $G$ the (WRIGP/$G$/$k$) can be deduced from the
(RIGP/$G$/$k((T))$). For
profinite groups $G$, the (WRIGP/$G$/$k((T))$) has been proved for
lots of fields (including the cyclotomic closure of characteristic $0$
fields) but the descent argument no longer works.\\\\
\\indent Let $p\\geq 2$ be a prime, then a profinite group
$G$ is said to be \\textit{$p$-obstructed} if it fits in a profinite group extension
$$1\\rightarrow K\\rightarrow G\\rightarrow G_{0}\\rightarrow 1$$
with $G_{0}$ a finite group and $K\\twoheadrightarrow
\\mathbb{Z}_{p}$. Typical examples of such profinite groups $G$ are
universal $p$-Frattini covers of finite $p$-perfect groups or
pronilpotent projective groups.\\\\
\\indent I will show that the (WRIGP/$G$/$k$) - even under
its weaker formulation: (WWRIGP/$G$/$k$) \\textit{there exists a
smooth geometrically irreducible curve $X_{G}/k$ and a Galois
extension $E/k(X_{G}).\\overline{k}$ with group $G$ and field of
moduli $k$.} - fails for the whole class of $p$-obstructed profinite
groups $G$ and any field $k$ which is either a finitely generated
field of characteristic $0$ or a finite field of characteristic
$\\not= p$.\\\\
\\indent The proof uses a profinite generalization of the cohomological obstruction
for a G-cover to be defined over its field of moduli and an analysis of the constrainsts
imposed on a smooth geometrically irreducible curve $X$ by a degree $p^{n}$
cyclic G-cover $X_{n}\\rightarrow X$, constrainsts which are too rigid to allow the
existence of projective systems $(X_{n}\\rightarrow
X_{G})_{n\\geq 0}$ of degree $p^{n}$ cyclic G-covers
defined over $k$. I will also discuss other implicsations of these constrainsts
for the (RIGP).
Eric Friedlander (Northwestern) 17:45-18:45Given a field $k$ and a (pro)finite group $G$, consider the
following weak version of the regular inverse Galois problem:
(WRIGP/$G$/$k$) \\textit{there exists a smooth geometrically
irreducible curve $X_{G}/k$ and a Galois extension $E/k(X_{G})$
regular over $k$ with group $G$.} (the regular inverse Galois
problem (RIGP/$G$/$k$) corresponding to the case
$X_{G}=\\mathbb{P}^{1}_{k}$). A standard descent argument shows that
for a finite group $G$ the (WRIGP/$G$/$k$) can be deduced from the
(RIGP/$G$/$k((T))$). For
profinite groups $G$, the (WRIGP/$G$/$k((T))$) has been proved for
lots of fields (including the cyclotomic closure of characteristic $0$
fields) but the descent argument no longer works.\\\\
\\indent Let $p\\geq 2$ be a prime, then a profinite group
$G$ is said to be \\textit{$p$-obstructed} if it fits in a profinite group extension
$$1\\rightarrow K\\rightarrow G\\rightarrow G_{0}\\rightarrow 1$$
with $G_{0}$ a finite group and $K\\twoheadrightarrow
\\mathbb{Z}_{p}$. Typical examples of such profinite groups $G$ are
universal $p$-Frattini covers of finite $p$-perfect groups or
pronilpotent projective groups.\\\\
\\indent I will show that the (WRIGP/$G$/$k$) - even under
its weaker formulation: (WWRIGP/$G$/$k$) \\textit{there exists a
smooth geometrically irreducible curve $X_{G}/k$ and a Galois
extension $E/k(X_{G}).\\overline{k}$ with group $G$ and field of
moduli $k$.} - fails for the whole class of $p$-obstructed profinite
groups $G$ and any field $k$ which is either a finitely generated
field of characteristic $0$ or a finite field of characteristic
$\\not= p$.\\\\
\\indent The proof uses a profinite generalization of the cohomological obstruction
for a G-cover to be defined over its field of moduli and an analysis of the constrainsts
imposed on a smooth geometrically irreducible curve $X$ by a degree $p^{n}$
cyclic G-cover $X_{n}\\rightarrow X$, constrainsts which are too rigid to allow the
existence of projective systems $(X_{n}\\rightarrow
X_{G})_{n\\geq 0}$ of degree $p^{n}$ cyclic G-covers
defined over $k$. I will also discuss other implicsations of these constrainsts
for the (RIGP).
An elementary perspective on modular representation theory
2006/12/06
16:30-18:45 Room #117 (Graduate School of Math. Sci. Bldg.)
Vincent Maillot (Jussieu/京大数理研) 16:30-17:30
New applications of the arithmetic Riemann-Roch theorem
Don Blasius (UCLA) 17:45-18:45
Zariski Closures of Automorphic Galois Representations
Vincent Maillot (Jussieu/京大数理研) 16:30-17:30
New applications of the arithmetic Riemann-Roch theorem
Don Blasius (UCLA) 17:45-18:45
Zariski Closures of Automorphic Galois Representations
2006/11/01
16:30-18:45 Room #117 (Graduate School of Math. Sci. Bldg.)
G.Bayarmagnai (東大数理) 16:30-17:30
Essential dimension of some finite group schemes
Jacques Tilouine (パリ北大学) 17:45-18:45
Overconvergent Siegel modular forms
G.Bayarmagnai (東大数理) 16:30-17:30
Essential dimension of some finite group schemes
Jacques Tilouine (パリ北大学) 17:45-18:45
Overconvergent Siegel modular forms
[ Abstract ]
We recall what is known and what is conjectured on p-adic families of overconvergent Siegel modular forms. We show how this relates to a Fontaine-Mazur type conjecture on the classicality of certain overconvergent Siegel forms of genus 2. We explain few results known in this direction.
We recall what is known and what is conjectured on p-adic families of overconvergent Siegel modular forms. We show how this relates to a Fontaine-Mazur type conjecture on the classicality of certain overconvergent Siegel forms of genus 2. We explain few results known in this direction.
2006/10/25
17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)
平之内 俊郎 (九州大学)
Extensions of truncated discrete valuation rings ( 田口雄一郎先生との共同研究 )
平之内 俊郎 (九州大学)
Extensions of truncated discrete valuation rings ( 田口雄一郎先生との共同研究 )
[ Abstract ]
局所体の拡大とその付値環の或る商である"truncated" dvrの拡大の圏を比較する. 不分岐拡大と剰余体の拡大が一対一対応するのと同じ様に, 分岐に関する条件を加えれば,局所体と "truncated" dvr の拡大の圏が同値になる (Deligne).
今回は, 古典的な(上付き)分岐群の代わりにAbbes-斎藤による分岐群を用いて分岐に関する条件を与える. そして,この分岐群の Rigid 幾何的解釈を踏襲する事でDeligneの定理の剰余体が非完全な場合への一般化が得られる事を述べる.
局所体の拡大とその付値環の或る商である"truncated" dvrの拡大の圏を比較する. 不分岐拡大と剰余体の拡大が一対一対応するのと同じ様に, 分岐に関する条件を加えれば,局所体と "truncated" dvr の拡大の圏が同値になる (Deligne).
今回は, 古典的な(上付き)分岐群の代わりにAbbes-斎藤による分岐群を用いて分岐に関する条件を与える. そして,この分岐群の Rigid 幾何的解釈を踏襲する事でDeligneの定理の剰余体が非完全な場合への一般化が得られる事を述べる.
2006/10/18
16:30-18:45 Room #117 (Graduate School of Math. Sci. Bldg.)
Fabrice Orgogozo (東大数理・Ecole Polytechnique de Paris) 16:30-17:30
p-dimension of henselian fields: an application of Ofer Gabber's algebraization technique
Kim Minhyong (Purdue大学・京大数理研) 17:45-18:45
Fundamental groups and Diophantine geometry
Fabrice Orgogozo (東大数理・Ecole Polytechnique de Paris) 16:30-17:30
p-dimension of henselian fields: an application of Ofer Gabber's algebraization technique
Kim Minhyong (Purdue大学・京大数理研) 17:45-18:45
Fundamental groups and Diophantine geometry
2006/09/06
16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Bas Edixhoven (Univ. of Leiden)
Computation of the mod l Galois representations associated to Delta
Bas Edixhoven (Univ. of Leiden)
Computation of the mod l Galois representations associated to Delta
