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Seminars (NAG)

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Event When Speaker Title Presentation Material
NAGW01 27th July 2009
10:00 to 11:00
P Deligne Counting l-adic representations, in the function field case
We will explain some countings similar to the one Drinfeld did in 1981, and wonder what they mean.
NAGW01 27th July 2009
11:30 to 12:30
On the noncommutative Iwasawa Main Conjecture for CM-elliptic curves
We discuss under which assumptions the (commutative) 2-variable Main Conjecture for CM-elliptic curves (due to Rubin, Yager, Katz etc.) implies the non-commutative Main Conjecture as formulated together with Coates, Fukaya, Kato and Sujatha. Related Links •http://www.mathi.uni-heidelberg.de/~otmar/ - personal homepage
NAGW01 27th July 2009
14:00 to 15:00
Grothendieck's Section Conjecture and zero-cycles on varieties
After some background material on Grothendieck's Section Conjecture, we discuss an obstruction for the existence of splittings of the abelianized homotopy exact sequence for the étale fundamental group. As an application, we explain how to find examples for smooth projective curves over Q that have points everywhere locally but the homotopy exact sequence does not split. This is joint work with David Harari, with explicit examples contributed by Victor Flynn.
NAGW01 27th July 2009
15:30 to 16:30
On the p-adic local Langlands correspondence for $GL_2({\bf Q}_p)$
NAGW01 28th July 2009
10:00 to 11:00
J Coates Iwasawa theory of elliptic curves with complex multiplication
We shall discuss, and prove in the very simplest case, one of the conjectures made in a previous joint paper with Fukaya, Kato, Sujatha, and Venjakob about the dual Selmer group of elliptic curves over those p-adic Lie extensions of the base field F which contain the cyclotomic Zp-extension of F. The results discussed are joint work with Sujatha.
NAGW01 28th July 2009
11:30 to 12:30
Reciprocity maps and Selmer groups
This talk concerns certain homomorphisms that arise in the study of Galois cohomology with restricted ramification. Given a set S of primes of a number field containing all those above a given prime p, the S-reciprocity map is a homomorphism on S-units that interpolates values of a cup product with those S-units. We will discuss the properties of and connections between this and related homomorphisms, and study their application to Selmer groups of reducible representations. Finally, we will explore a connection with a conjecture of the author on the relationship between these maps for cyclotomic fields and a modular two-variable p-adic L-function, taken modulo an Eisenstein ideal.
NAGW01 28th July 2009
14:00 to 15:00
Obstructions to homotopy sections of curves over number fields
Grothendieck's section conjecture is analogous to equivalences between fixed points and homotopy fixed points of Galois actions on related topological spaces. We use cohomological obstructions of Jordan Ellenberg coming from nilpotent approximations to the curve to study the sections of etale pi_1 of the structure map. We will relate Ellenberg's obstructions to Massey products, and explicitly compute mod 2 versions of the first and second for P^1-{0,1,infty} over Q. Over R, we show the first obstruction alone determines the connected components of real points of the curve from those of the Jacobian.
NAGW01 28th July 2009
15:30 to 16:30
Potential automorphy of n dimensional Galois representations
I will discuss recent improvements in the potential automorphy theorems available for Galois representations of any dimension. In particular I will discuss the case of ordinary Galois representations and applications to elliptic modular forms, in particular the proof of the Sato-Tate conjecture for all elliptic modular new forms.
NAGW01 29th July 2009
10:00 to 11:00
On the birational p-adic section conjecture
I plan to explain Grothendieck's section conjecture, which relates rational points of (completions of) hyperbolic curves to conjugacy classes of sections of the canonical projection between fundamental groups. I will explain a few variants of this conjecture (birational, p-adic), and finally discuss the status of the art of the conjecture.
NAGW01 29th July 2009
11:30 to 12:30
On the section conjecture for universal curves over function fields
In this talk I will discuss a version of Grothendieck's Section Conjecture for the universal curve over the function field of the moduli space of curves type (g,n) with a level m structure.
NAGW01 30th July 2009
10:00 to 11:00
M Emerton p-adically completed cohomology and the p-adic Langlands program
Speaking at a general level, a major goal of the p-adic Langlands program (from a global, rather than local, perspective) is to find a p-adic generalization of the notion of automorphic eigenform, the hope being that every p-adic global Galois representation will correspond to such an object. (Recall that only those Galois representations that are motivic, i.e. that come from geometry, are expected to correspond to classical automorphic eigenforms). In certain contexts (namely, when one has Shimura varieties at hand), one can begin with a geometric definition of automorphic forms, and generalize it to obtain a geometric definition of p-adic automorphic forms. However, in the non-Shimura variety context, such an approach is not available. Furthermore, this approach is somewhat remote from the representation-theoretic point of view on automorphic forms, which plays such an important role in the classical Langlands program. In this talk I will explain a different, and very general, approach to the problem of p-adic interpolation, via the theory of p-adically completed cohomology. This approach has close ties to the p-adic and mod p representation theory of p-adic groups, and to non-commutative= Iwasawa theory. After introducing the basic objects (namely, the p-adically completed cohomology spaces attached to a given reductive group), I will explain several key conjectures that we expect to hold, including the conjectural relationship to Galois deformation spaces. Although these conjectures seem out of reach at present in general, some progress has been made towards them in particular cases. I will describe some of this progress, and along the way will introduce some of the tools that we have developed for studying p-adically completed cohomology, the most important of these being the Poincare duality spectral sequence. This is joint work with Frank Calegari.
NAGW01 30th July 2009
11:30 to 12:30
The cuspidalization of sections of arithmetic fundamental groups
All results presented in this talk are part of a joint work with Akio Tamagawa. We introduce the problem of cuspidalization of sections of arithmetic fundamental groups which relates the Grothendieck section conjecture to its birational analog. We exhibit a necessary condition for a section of the arithmetic fundamental group of a hyperbolic curve to arise from a rational point which we call the goodness condition. We prove that good sections of arithmetic fundamental groups of hyperbolic curves can be lifted to sections of the maximal cuspidally abelian Galois group of the function field of the curve (under quite general assumptions). As an application we prove a (geometrically pro-p) p-adic local version of the Grothendieck section conjecture under the assumption that the existence of sections of cuspidally (pro-p) abelian arithmetic fundamental groups implies the existence of tame points. We also prove, using cuspidalization techniques, that for a hyperbolic curve X over a p-adic local field and a set of points S of X which is dense in the p-adic topology every section of the arithmetic fundamental group of U=X\S arises from a rational point. As a corollary we deduce that the existence of a section of the absolute Galois group of a function field of a curve over a number field implies that the set of adelic points of the curve is non-empty.
NAGW01 30th July 2009
14:00 to 15:00
The structure of Galois groups and birational algebraic geometry
In the talk I will discuss the relation between recent results on the structure of Galois groups for functional fields and birational geometry of algebraic varieties. Most of the results discussed in the talk are obtained in joint work with Yuri Tschinkel
NAGW01 30th July 2009
15:30 to 16:30
Annihilating Tate-Shafarevic groups
We describe how main conjectures in non-commutative Iwasawa theory lead naturally to the (conjectural) construction of a family of explicit annihilators of the Bloch-Kato-Tate-Shafarevic Groups that are attached to a wide class of p-adic representations over non-abelian extensions of number fields. Concrete examples to be discussed include a natural non-abelian analogue of Stickelberger's Theorem (which is proved) and of the refinement of the Birch and Swinnerton-Dyer Conjecture due to Mazur and Tate. Parts of this talk represent joint work with James Barrett and Henri Johnston.
NAGW01 31st July 2009
10:00 to 11:00
Patching and a local-global principle for curves (joint with Julia Hartmann and Daniel Krashen)
Using patching, we establish a local-global principle for actions of algebraic groups that are defined over the function field of a curve over a complete discretely valued field. This result has applications to quadratic forms and to Brauer groups. In the case of quadratic forms, we obtain a result on the u-invariant of function fields, in particular reproving the theorem of Parimala and Suresh that the u-invariant of a one-variable p-adic function field is 8. Concerning Brauer groups, we obtain results on the period-index problem for such fields, in particular reproving a result of Lieblich. We also obtain local-global principles for quadratic forms and for Brauer groups. Our local-global principle for group actions holds in general for connected rational groups. In the disconnected case, the validity of the principle depends on the topology of a graph associated to the closed fiber of a model of the curve. The fundamental group of this graph is isomorphic to a certain quotient of the etale fundamental group; and the local-global principle holds even for disconnected rational groups if and only if the graph is a tree. In that case, the local-global principle for quadratic forms can be strengthened. In general, the cohomology of the graph determines the kernel of the local-global map on Witt groups.
NAGW01 31st July 2009
11:30 to 12:30
Tate-Shafarevich groups over anticyclotomic Z p extensions
Let E be an elliptic curve over Q with supersingular reduction at p and K an imaginary quadratic extension of Q. We analyze the structure of the p-primary part of the Tate-Shafarevich group of E over the anticyclotomic Z_p-extension K_\infty/K by viewing it as a module over Z_p[Gal(K_\infty/K)].
NAGW01 31st July 2009
14:00 to 15:00
A Beilinson Epsilon-factors for the period determinants
I will explain why the determinants of the period matrices of holonomic D-modules on curves admit a natural epsilon-factorization, as was conjectured by Deligne back in 1984. See http://xxx.lanl.gov/abs/0903.2674
NAGW01 31st July 2009
15:30 to 16:30
Galois theory and Diophantine geometry
NAG 3rd August 2009
17:00 to 18:00
Cohomology of algebraic varieties
My lecture will try to explain the miracle of the many ways to compute the cohomology of algebraic varieties, and associated structures.
NAG 4th August 2009
11:00 to 12:00
Anabelian geometry I
NAG 4th August 2009
14:00 to 15:00
S Corry The pro-$p$ Hom-form of the birational anabelian conjecture
In this talk we will indicate a proof of the pro-$p$ Hom-form of Grothendieck's birational anabelian conjecture for function fields over sub-$p$-adic fields. The proof uses Kummer Theory and projective geometry to deduce the result from Mochizuki's Theorem in the case of transcendence degree 1. This is joint work with Florian Pop.
NAG 5th August 2009
11:00 to 12:00
Anabelian geometry II
NAG 6th August 2009
11:00 to 12:00
Anabelian geometry III
NAG 7th August 2009
11:00 to 12:00
Anabelian geometry IV
NAG 10th August 2009
14:30 to 15:30
P Deligne Multiple zeta values 1
NAG 10th August 2009
16:00 to 17:00
P Deligne Multiple zeta values II
NAG 11th August 2009
11:00 to 12:00
Anabelian geometry V
NAG 11th August 2009
14:00 to 15:00
A Vdovina Fundamental groups of CW-complexes from different points of view
NAG 12th August 2009
11:00 to 12:00
Anabelian geometry VI
NAG 12th August 2009
14:00 to 15:00
A local analog of the Grothendieck conjectures
NAG 13th August 2009
11:00 to 12:00
Anabelian geometry VII
NAG 14th August 2009
11:00 to 12:00
Anabelian geometry VIII
NAG 17th August 2009
14:30 to 15:30
P Deligne Mulitple zeta values III
NAG 17th August 2009
16:00 to 17:00
P Deligne Multiple zeta values IV
NAG 18th August 2009
11:00 to 12:00
Anabelian geometry IX
NAG 18th August 2009
14:00 to 15:00
A local analog of the Grothendieck conjectures
NAG 19th August 2009
11:00 to 12:00
Anabelian geometry X
NAG 20th August 2009
11:00 to 12:00
Anabelian geometry XI
NAG 20th August 2009
14:30 to 15:30
Canonical embedded resolution of singularities for two-dimensional schemes
NAG 21st August 2009
11:00 to 12:00
Anabelian geometry XII
NAG 21st August 2009
14:30 to 15:30
P Deligne Multiple zeta values V
NAG 21st August 2009
16:00 to 17:00
P Deligne Multiple zeta values VI
NAG 21st August 2009
16:00 to 18:00
Discussion session
NAGW02 24th August 2009
10:00 to 11:00
Local-global principle for zero-cycles of degree one and integral Tate conjecture for 1-cycles
Shuji Saito showed that an integral version of the Tate conjecture for 1-dimensional cycles on a variety over a finite field essentially implies that the Brauer-Manin obstruction to the existence of a zero-cycle of degree 1 on varieties over a global function field (function field in one variable over a finite field) is the only obstruction. In this talk we describe some known results about integral versions of the Tate conjecture, and we give two applications, one of which comes from joint work with T. Szamuely.
NAGW02 24th August 2009
11:30 to 12:30
Galois Theory and Diophantine geometry 2
NAGW02 24th August 2009
14:00 to 15:00
H Furusho Double shuffle relation for associators
I will explain that Drinfel'd's pentagon equation implies the double shuffle relation.
NAGW02 24th August 2009
15:30 to 16:30
Local-Global principles for affine curves
There has been a lot of work recently on local-global principles for curves, from various viewpoints, including the Brauer-Manin obstruction, descent obstructions and adelic intersections in the Jacobian. We will discuss these and suggest generalizations for integral points on affine curves.
NAGW02 25th August 2009
09:30 to 10:30
A local analog of the Grothendieck conjecture for higher local fields
Suppose K is an N-dimensional local field where N is a non-negative integer. By definition, if N=0 then K is just a finite field, otherwise, K is a complete discrete valuation field and its residue field is an (N-1)-dimensional local field. Let G be the absolute Galois group of K. If N=1 then the structure of the topological group G depends only on very weak invariants of K and is not sufficient to recover uniquely the field K. The situation becomes totally different if we take into account the filtration of G by its ramification subgroups. Then the corresponding functor from the category of 1-dimensional local fields to the category of profinite groups with decreasing filtration is fully faithful. In the talk it will be discussed an analog of this statement for higher local fields and its relation to the Grothendieck conjecture in the context of global fields.
NAGW02 25th August 2009
11:00 to 12:00
Computing functional equations of l-adic polylogarithmic characters on Galois group
From a family of morphisms to the projective line minus 3 points satisfying Wojtkowiak-Zagier test, one can derive a functional equation of polylogarithms. In this talk, I present a joint work with Z.Wojtkowiak computing l-adic analogs of such functional equations without omitting lower degree terms and l-adic error terms.
NAGW02 25th August 2009
14:00 to 15:00
On comparison isomorphisms for smooth formal schemes
NAGW02 25th August 2009
15:30 to 16:30
A Chebotarev-type density theorem for divisors on algebraic varieties
Let Z/X be a finite branched Galois cover (with Galois group G) of normal, geometrically integral, projective varieties of dimension at least two over a field of characteristic zero. For each Weil prime divisor D on X, we can define the decomposition class C_D of D to be the conjugacy class of the decomposition group of any Weil prime divisor on Z mapping to D. Using the structure of the induced push-forward map on divisors, we derive density results on the set of prime divisors on X with a given decomposition class and explain some applications.
NAGW02 26th August 2009
10:00 to 11:00
Recovering function fields from their decomposition graphs
This is the "global theory" of our strategy to recovering function fields from their pro-l a-b-c (abelian-by-central) Galois theory. As an application we will show that some pro-l a-b-c form of Ihara's question / Oda-Matsumoto conjecture holds.
NAGW02 26th August 2009
11:30 to 12:30
Random braids, finite extensions of global fields, stable cohomology, and variations on Cohen-Lenstra
It is by now a common technique to construct and evaluate conjectures about number fields by means of analogous conjectures over function fields. For example, the Cohen-Lenstra conjecture about the statistics of ell-parts of class groups of quadratic imaginary fields can be thought of, on the function field side, as a conjecture about the statistics of the finite group Jac(X)[ell^infty](F_q), where X is a "random" hyperelliptic curve over F_q of large genus. In this setting, the Cohen-Lenstra conjecture is compatible with the heuristic that the image of Frobenius is a random symplectic matrix in Aut(T_ell Jac(X)). I will argue, somewhat against the grain of this conference, that in this setting the anabelian story may not be so different from the abelian story. In particular, one may productively study the statistics of non-abelian extensions of global fields by means of the heuristic that the action of Frobenius on pi_1^{et}(X/F_q) should act as a random element of (an appropriate subgroup of) Ihara's profinite braid group. To make this more concrete, I will explain a) how to prove a weak version of the Cohen-Lenstra conjecture over F_q(t) via a new theorem on stable cohomology of Hurwitz spaces (joint work with A. Venkatesh, C. Westerland) and b) how to frame a "non-abelian pro-p Cohen-Lenstra conjecture," describing the statistics of the Galois group of the maximal pro-p extension unramified away from a random set of primes ell_1, .. ell_r congruent to 1 mod p. (joint work with N. Boston, A. Venkatesh.)
NAGW02 27th August 2009
09:30 to 10:30
A Tamagawa Torsion of abelian schemes and rational points on moduli spaces (joint work with Anna Cadoret)
We show the following result supporting the uniform boundedness conjecture for torsion of abelian varieties: Let k be a field finitely generated over the rationals, X a smooth curve over k, and A an abelian scheme over X. Let l be a prime number and d a positive integer. Then there exists a non-negative integer N, such that, for any closed point x of X with [k(x):k] \leq d and any k(x)-rational, l-primary torsion point v of A_x, the order of v is \leq l^N. (Here, A_x stands for the fiber of the abelian scheme A at x.) As a corollary of this result, we settle the one-dimensional case of the so-called modular tower conjecture, posed by Fried in the context of the (regular) inverse Galois problem. The above result is obtained by combining geometric results (estimation of genus/gonality) and Diophantine results (Mordell/Mordell-Lang conjecture, proved by Faltings) for certain ``moduli spaces''. If we have time, we will also explain our recent progress on a variant of these geometric results, where the set of powers of the fixed prime l is replaced by the set of all primes. For an extension of the above results to more general l-adic representations, see Cadoret's talk on Friday.
NAGW02 27th August 2009
11:00 to 12:00
Brauer-Manin obstructions for sections of the fundamental group
We introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension of varieties over number fields. This obstruction is then shown in an example to yield a proof for the absence of sections.
NAGW02 27th August 2009
14:00 to 15:00
On the combinatorial cuspidalizations and the faithfulness of the outer Galois representations of hyperbolic curves
In this talk, I discuss the combinatorial anabelian geometry for nodally nondegenerate outer representations on the fundamental groups of hyperbolic curves. I plan to explain a result of a combinatorial version of the Grothendieck conjecture for nodally nondegenerate outer representations obtained in the joint work with Shinichi Mochizuki. As an application, we prove the injectivity portion of the combinatorial cuspidalization. We also generalize, by means of this injectivity result, the faithfulness proven by Makoto Matsumoto of the outer representation of the absolute Galois group on the profinite fundamental group of an affine hyperbolic curve over a certain field (e.g. number or p-adic local field) to the case where the given hyperbolic curve is proper.
NAGW02 27th August 2009
15:30 to 16:30
The anabelian geometry of hyperbolic curves over finite fields
I will discuss some recent results on the anabelian geometry of hyperbolic curves over finite fields (joint work with Akio Tamagawa)
NAGW02 28th August 2009
09:30 to 10:30
B Poonen Néron-Severi groups under specialization
This is joint work with Davesh Maulik and Claire Voisin. We prove that given a smooth proper family X --> B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. In fact, we give two proofs, and they show that the locus on the base where the Picard number jumps is "small" in two different senses. The first proof uses Hodge theory and the actions of geometric monodromy groups and Galois groups to show that the locus is small in a sense related to Hilbert irreducibility. The second proof uses the "p-adic Lefschetz (1,1) theorem" of Berthelot and Ogus to show that in a family of varieties with good reduction at p, the locus is nowhere p-adically dense. Finally, we prove analogous statements for cycles of higher codimension, under the assumption of the variational Hodge conjecture or a p-adic analogue conjectured by M. Emerton.
NAGW02 28th August 2009
11:00 to 12:00
The Birational Anabelian
NAGW02 28th August 2009
14:00 to 15:00
A uniform open image theorem for $\ell$-adic representations (joint work with Akio Tamagawa - R.I.M.S.)
In this talk, we extend some of the results presented in Tamagawa's talk to more general $\ell$-adic representations.\\ \indent Let $k$ be a finitely generated field of characteristic $0$, $X$ a smooth, separated, geometrically connected curve over $k$ with generic point $\eta$. A $\ell$-adic representation $\rho:\pi_{1}(X)\rightarrow \hbox{\rm GL}_{m}(\mathbb{Z}_{\ell})$ is said to be geometrically strictly rationnally perfect (GSRP for short) if $\hbox{\rm Lie}(\rho(\pi_{1}(X_{\overline{k}})))^{ab}=0$. Typical examples of such representations are those arising from the action of $\pi_{1}(X)$ on the generic $\ell$-adic Tate module $T_{\ell}(A_{\eta})$ of an abelian scheme $A$ over $X$ or, more generally, from the action of $\pi_{1}(X)$ on the $\ell$-adic etale cohomology groups $H^{i}(Y_{\overline{\eta}},\mathbb{Q}_{\ell})$, $i\geq 0$ of the geometric generic fiber of a smooth proper scheme $Y$ over $X$. Let $G$ denote the image of $\rho$. Any closed point $x$ on $X$ induces a splitting $x:\Gamma_{\kappa(x)}:=\pi_{1}(\hbox{\rm Spec}(\kappa(x))) \rightarrow\pi_{1}(X_{\kappa(x)})$ of the canonical restriction epimorphism $\pi_{1}(X_{\kappa(x)})\rightarrow \Gamma_{\kappa(x)}$ (here, $\kappa(x)$ denotes the field of definition of $x$) so one can define the closed subgroup $G_{x}:=\rho\circ x(\Gamma_{\kappa(x)})\subset G$ (up to inner automorphisms).\\ \indent The main result I am going to discuss is the following uniform open image theorem. \textit{Under the above assumptions, for any representation $\rho:\pi_{1}(X)\rightarrow \hbox{\rm GL}_{m}(\mathbb{Z}_{\ell})$ and any integer $d\geq 1$, the set $X_{\rho, d,\geq 3}$ of all closed points $x\in X$ such that $G_{x}$ has codimension $\geq 3$ in $G$ and $[\kappa(x):k]\leq d$ is finite. Furthermore, if $\rho:\pi_{1}(X)\rightarrow \hbox{\rm GL}_{m}(\mathbb{Z}_{\ell})$ is GSRP then the set $X_{\rho, d,\geq 1}$ of all closed points $x\in X$ such that $G_{x}$ has codimension $\geq 1$ in $G$ and $[\kappa(x):k]\leq d$ is finite and there exists an integer $B_{\rho,d}\geq 1$ such that $[G:G_{x}]\leq B_{\rho,d}$ for any closed point $x\in X\smallsetminus X_{\rho,d,\geq 1}$ such that $[\kappa(x):k]\leq d$.}\\
NAGW02 28th August 2009
15:30 to 16:30
Differences between Galois representations in automorphism and outer-automorphism groups of the fundamental group of curves
Fix a prime l. Let C be proper smooth geometrically connected curve over a number field K, and x be its L-rational point. Let Pi denotes the pro-l completion of the geometric fundamental group of C with geometric base point over x. We have two non-abelian Galois representations: rho_A : Gal_L -> Aut(Pi) rho_O : Gal_K -> Out(Pi). Ker(rho_A) is included in Ker(rho_O). Our question is whether they differ or not: more precisely, whether or not Ker(rho_A) = (Ker(rho_O) "intersection" Gal_L.) We show that, the equality does not hold in general, by showing: Theorem: Assume that g >=3, l divides 2g-2. Then, there are infinitely many pairs (C,K) with the following property. For any extension field L with [L:K] coprime to l, and for any x in C(L), the nonequality Ker(rho_A) "not equal to" (Ker(rho_O) "intersection" Gal_L) holds. This is in contrast to the fact that for the projective line minus three point and its canonical tangential base points, the equality holds (a result of Deligne and Ihara).
NAG 1st September 2009
11:00 to 12:00
P Schneider The algebraic theory of p-adic Lie groups I
NAG 2nd September 2009
11:00 to 12:00
P Schneider The algebraic theory of p-adic Lie groups II
NAG 3rd September 2009
11:00 to 12:00
P Schneider The algebraic theory of p-adic Lie groups III
NAG 7th September 2009
11:00 to 12:00
C Breuil On the p-adic Langlands programme for GL2(F) I: Construction of representation of GL(2)F over bar F p
NAG 8th September 2009
10:00 to 11:00
P Schneider The algebraic theory of p-adic Lie groups IV
NAG 8th September 2009
11:30 to 12:30
C Breuil On the p-adic Langlands programme for GL(2) F II: (phi, Gamma)-modules over barmathbb F p and tensor induction
NAG 9th September 2009
10:00 to 11:00
P Schneider The algebraic theory of p-adic Lie groups V
NAG 9th September 2009
11:30 to 12:30
C Breuil On the p-adic Langlands programme for GL(2)F III: Qp-locally analytic representations of GL(2)F and crystalline representations
NAG 10th September 2009
10:00 to 11:00
P Schneider The algebraic theory of p-adic Lie groups VI
NAG 10th September 2009
11:30 to 12:30
C Breuil On the p-adic Langlands programme for GL(2)F IV: The Hodge filtration and speculation on extra components
NAG 15th September 2009
10:00 to 11:00
The algebraic theory of p-adic Lie groups VII
NAG 15th September 2009
11:30 to 12:30
Iwasawa theory of elliptic curves over p-adic Lie extensions of dimension greater than 1 - I
NAG 16th September 2009
10:00 to 11:00
P Schneider The algebraic theory of p-adic Lie groups VIII
NAG 16th September 2009
11:30 to 12:30
S Ramdorai Iwasawa theory of elliptic curves over p-adic Lie extensions of dimension greater than 1 - II
NAG 17th September 2009
10:00 to 11:00
P Schneider The algebraic theory of p-adic Lie groups IX
NAG 17th September 2009
11:30 to 12:30
S Ramdorai Iwasawa theory of elliptic curves over p-adic Lie extensions of dimension greater than 1 - III
NAG 21st September 2009
11:00 to 12:00
Iwasawa theory of elliptic curves over p-adic Lie extensions of dimension greater than 1 - IV
NAG 22nd September 2009
11:00 to 12:00
The algebraic theory of p-adic Lie groups X
NAG 23rd September 2009
11:00 to 12:00
The algebraic theory of p-adic Lie groups XI
NAG 24th September 2009
11:00 to 12:00
The algebraic theory of p-adic Lie groups XII
NAG 5th October 2009
14:00 to 15:00
Motivic fundamental groups and diophantine geometry - I
NAG 5th October 2009
15:30 to 16:30
Motivic fundamental groups and diophantine geometry - II
NAG 6th October 2009
11:00 to 12:00
L Schneps Relations between multi-zeta values and Grothendieck-Teichmueller theory
NAG 7th October 2009
14:00 to 15:00
Motivic fundamental groups and diophantine geometry - III
NAG 7th October 2009
15:30 to 16:30
Motivic fundamental groups and diophantine geometry - IV
NAG 12th October 2009
14:00 to 15:00
Motivic fundamental groups and diophantine geometry - V
NAG 12th October 2009
15:30 to 16:30
Motivic fundamental groups and diophantine geometry - VI
NAG 16th October 2009
14:00 to 15:00
Motivic fundamental groups and diophantine geometry - VII
NAG 16th October 2009
15:30 to 16:30
Motivic fundamental groups and diophantine geometry - VIII
NAG 29th October 2009
14:00 to 16:00
P-ADIC analysis
NAGW05 30th October 2009
11:00 to 12:00
T Berger Modularity of Galois representations
NAGW05 30th October 2009
13:45 to 14:45
F Diamond Modularity lifting theorems
NAGW05 30th October 2009
15:15 to 16:15
Potential modularity of residual representations
NAGW05 30th October 2009
16:30 to 17:30
J Manoharmayum L-functions and applications
NAG 5th November 2009
14:00 to 15:00
Constructing dessin d'enfants on curves
NAG 9th November 2009
14:00 to 15:30
M Kakde The main conjectures of Iwasawa theory I
NAG 9th November 2009
16:00 to 17:00
M Kakde The main conjectures of Iwasawa theory II
NAG 16th November 2009
14:00 to 15:30
M Kakde The main conjectures of Iwasawa theory III
NAG 16th November 2009
16:00 to 17:00
M Kakde The main conjectures of Iwasawa theory IV
NAG 18th November 2009
14:00 to 15:00
M Kakde The main conjectures of Iwasawa theory V
NAG 18th November 2009
16:00 to 17:00
M Kakde The main conjectures of Iwasawa theory VI
NAG 19th November 2009
11:00 to 12:00
Topological methods in Grothendieck-Teichmueller theory
NAG 23rd November 2009
14:00 to 15:00
Periods of Tate motives, examples, Galois side
NAG 25th November 2009
14:00 to 15:00
Work of Chenevier and Chenevier-Clozel
NAG 26th November 2009
14:00 to 15:00
Work of Chenevier and Cheniever-Clozel II
NAG 26th November 2009
16:00 to 17:00
Euler characteristics of fine Selmer groups
NAG 30th November 2009
14:00 to 15:00
BD Kim Iwasawa theory for supersingular primes: for a larger class of abelian varieties and multi-variable Iwasawa theory
NAG 2nd December 2009
11:00 to 12:00
On the abelian fundamental group scheme of a family of varieties
NAG 2nd December 2009
14:00 to 15:00
Bigness and modularity lifting
NAG 3rd December 2009
14:00 to 16:00
P-adic analysis
NAG 4th December 2009
11:00 to 12:00
The Galois module structure of p-adic realisations of Picard 1-motives and applications
NAGW04 14th December 2009
10:00 to 11:00
Simple cuspidals and the local Langlands correspondence for GL(n)
Class field theory is the case n=1 of the Langlands correspondence. The case of general n may be considered as a kind of non-abelian class field theory. However the Langlands correspondence relates two equally mysterious sides. On the example of simple cuspidal representations for GL(n) over a non-Archimedean local field (pointed out by Gross and Reeder), we shall see that it is not so easy, but still possible, to determine the corresponding Galois representations (joint work in progress with Bushnell).
NAGW04 14th December 2009
11:30 to 12:30
Iwasawa algebras and enveloping algebras
I will compare and contrast the representation theory of Iwasawa algebras and of universal enveloping algebras.
NAGW04 14th December 2009
14:00 to 15:00
A Noncommutative Iwasawa Main Conjecture for Varieties over Finite Fields
We discuss a noncommutative Iwasawa Main Conjecture for $\ell$-adic Lie coverings of separated schemes of finite type over a finite field of characteristic $p$ different from $\ell$.
NAGW04 14th December 2009
15:30 to 16:30
Arithmetic invariants of Eisenstein type arising from fundamental groups of once punctured elliptic curves
We discuss behaviors of certain arithmetic invariants (introduced by Bloch, Tsunogai) for fundamental groups of once-punctured elliptic curves. Anabelian geometry of tangential basepoints on M(1,1) and M(1,2) will also be brought into the view.
NAGW04 15th December 2009
10:00 to 11:00
Weights and Hasse principles for higher-dimensional fields
We present a Hasse principle for higher-dimensional fields which proves a conjecture of K. Kato. In addition to earlier results we also treat the case of p-torsion in positive characteristic p, asssuming resolution of singularities. Due to recent results on resolution we obtain unconditional results for low dimension. The principal tool is the consideration of weights on cohomology, as initiated in Deligne's proof of the Weil conjectures. The consideration of these weights is less standard for p-torsion in characteistic p.
NAGW04 15th December 2009
11:30 to 12:30
Arthur's theory of automorphic forms on classical groups (a survey)
As requested by the organizers, I will try to give the flavour of Arthur's results describing completely the automorphic forms on orthogonal or symplectic groups. In view of the exhaustive results on Galois representations associated to forms on GL(n), I will try to explain the consequences for temperedness (or not) of cohomological representations of these groups, and possibly on the Galois representations appearing in the related Shimura varieties. This is a purely expository lecture.
NAGW04 15th December 2009
14:00 to 15:00
V Dokchitser Parity of ranks of elliptic curves I
I will explain why both the Birch-Swinnerton-Dyer conjecture and the Shafarevich-Tate conjecture imply that the parity of the rank of an elliptic curve over a number field can be expressed as a sum of (computable) local invariants, and describe some arithmetic consequences.
NAGW04 15th December 2009
15:30 to 16:30
D Rössler Conjectures on the logarithmic derivatives of Artin L-functions
We shall present a general conjecture relating the logarithmic derivatives of Artin L-functions to arithmetic intersection numbers on certain Shimura varieties. This is joint work with V. Maillot.
NAGW04 16th December 2009
09:00 to 10:00
On the pro-l abelian-by-central I/OM
After presenting the classical I/OM (Ihara problem / Oda-Matsumoto conjecture), I will present its abelian-by-central variant, and show its connection with a Programme initiated by Bogomolov. I will finally sketch a proof of the abelian-by-central I/OM, which is a much stronger assertion than the classical I/OM.
NAGW04 16th December 2009
10:15 to 11:15
K 1 of some noncommutative p adic group rings
In this talk we will compute K_1 of some p-adic group rings which reduces, by a strategy of Burns and Kato, the noncommutative main conjecture for totally real number fields to commutative main conjecture for totally real number fields (a theorem of Wiles assuming Iwasawa invariant mu to be 0) and certain congruences between abelian p-adic zeta functions.
NAGW04 16th December 2009
11:45 to 12:45
Diophantine geometry and Galois theory 9
NAGW04 17th December 2009
10:00 to 11:00
On the p-adic section conjecture
The talk presents results from joint work with Florian Pop, in which we found that sections of the fundamental group of a hyperbolic curve over a p-adic field all meet certain geometric requirements.
NAGW04 17th December 2009
11:30 to 12:30
On the mixed Hodge polynomials of character varieties of Riemann surfaces
I will describe a calculation of the number of points over finite fields of the varieties of the title (parameterizing generic representations of the fundamental group of a Riemann surface into GL_n). Since the varieties are polynomial count this calculation yields their E-polynomial (a geometric invariant). The results are best expressed in terms of a generating function involving the Macdonald polynomials of combinatorics. We conjecture that a natural deformation of these formulas in fact gives the full mixed Hodge polynomial of the varieties. This is joint work with T. Hausel and E. Letellier.
NAGW04 17th December 2009
14:00 to 15:00
C Popescu The Galois module structure of p-adic realisations of Picard 1-motives and applications
NAGW04 17th December 2009
15:30 to 16:30
Y Flicker Counting local systems with local principal unipotent monodromy
We compute, jointly with P. Deligne, the number of equivalence classes of irreducible rank n ell-adic local systems on the geometric X-S, namely n-dimensional ell-adic representations of pi_1(geometrix(X-S)), invariant under the Frobenius, whose local monodromy at each point of S is a single Jordan block of rank n. Here X is a smooth projective absolutely irreducible curve over the finite field of cardinality q, S a finite set of closed points of X of cardinality N>1, ell a prime with (ell,q)=1, and n>1 an integer.
NAGW04 18th December 2009
09:00 to 10:00
Relation between the \'etale and the algebraic fundamental groups
Joint work with V. Mehta. We show the relation between those two groups. Over the complex numbers, we know that if the 'etale fundamental group is trivial, so is the proalgebraic one. Among other things, we show the corresponding statement in char. p>0 over X projective smooth (Gieseker conjecture).
NAGW04 18th December 2009
10:15 to 11:15
Congruences between derivatives of Artin L-functions
NAGW04 18th December 2009
11:45 to 12:45
Parity of ranks of elliptic curves II
In a follow-up to Vladimir's talk, I will discuss local formulae relating root numbers of elliptic curves to their Tamagawa numbers and periods. In particular, I will explain the proof of the parity conjecture for p-infinity Selmer ranks of elliptic curves with a p-isogeny and related results.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons