# Workshop Programme

## for period 24 - 28 August 2009

### Anabelian Geometry

24 - 28 August 2009

Timetable

 Monday 24 August 08:30-09:55 Registration 09:55-10:00 Welcome - Ben Mestel 10:00-11:00 Colliot-Thélène, JL (Paris-Sud 11) Local-global principle for zero-cycles of degree one and integral Tate conjecture for 1-cycles Sem 1 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. 11:00-11:30 Coffee 11:30-12:30 Kim, M (UCL) Galois Theory and Diophantine geometry 2 Sem 1 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Furusho, H (Nagoya) Double shuffle relation for associators Sem 1 I will explain that Drinfel'd's pentagon equation implies the double shuffle relation. 15:00-15:30 Tea 15:30-16:30 Voloch, F (Texas at Austin) Local-Global principles for affine curves Sem 1 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. 16:30-17:30 Discussion 17:30-18:30 Welcome Wine Reception 18:45-19:30 Dinner at Wolfson Court (Residents Only)
 Tuesday 25 August 09:30-10:30 Abrashkin, V (Durham) A local analog of the Grothendieck conjecture for higher local fields Sem 1 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. 10:30-11:00 Coffee 11:00-12:00 Nakamura, H (Okayama) Computing functional equations of l-adic polylogarithmic characters on Galois group Sem 1 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. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Iovita, A (Concordia) On comparison isomorphisms for smooth formal schemes Sem 1 15:00-15:30 Tea 15:30-16:30 Holschbach, A (Pennsylvania) A Chebotarev-type density theorem for divisors on algebraic varieties Sem 1 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. 17:00-18:00 Discussion 18:45-19:30 Dinner at Wolfson Court (Residents Only)
 Wednesday 26 August 10:00-11:00 Pop, F (Pennsylvania) Recovering function fields from their decomposition graphs Sem 1 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. 11:00-11:30 Coffee 11:30-12:30 Ellenberg, J (Wisconsin-Madison) Random braids, finite extensions of global fields, stable cohomology, and variations on Cohen-Lenstra Sem 1 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.) 12:30-13:30 Lunch at Wolfson Court 18:45-19:30 Dinner at Wolfson Court (Residents Only)
 Thursday 27 August 09:30-10:30 Tamagawa, A (Kyoto) Torsion of abelian schemes and rational points on moduli spaces (joint work with Anna Cadoret) Sem 1 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. 10:30-11:00 Coffee 11:00-12:00 Stix, J (Heidelberg) Brauer-Manin obstructions for sections of the fundamental group Sem 1 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. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Hoshi, Y (Kyoto) On the combinatorial cuspidalizations and the faithfulness of the outer Galois representations of hyperbolic curves Sem 1 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. 15:00-15:30 Tea 15:30-16:30 Saidi, M (Exeter) The anabelian geometry of hyperbolic curves over finite fields Sem 1 I will discuss some recent results on the anabelian geometry of hyperbolic curves over finite fields (joint work with Akio Tamagawa) 17:00-18:00 Discussion 19:30-23:00 Conference Dinner at St John's College
 Friday 28 August 09:30-10:30 Poonen, B (MIT) Néron-Severi groups under specialization Sem 1 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. 10:30-11:00 Coffee 11:00-12:00 Koenigsmann, J (Oxford) The Birational Anabelian Sem 1 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Cadoret, A (Bordeaux 1) A uniform open image theorem for $\ell$-adic representations (joint work with Akio Tamagawa - R.I.M.S.) Sem 1 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 $[\kappa(x):k]\leq d$.}\\ 15:00-15:30 Tea 15:30-16:30 Matsumoto, M (Hiroshima) Differences between Galois representations in automorphism and outer-automorphism groups of the fundamental group of curves Sem 1 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). 18:45-19:30 Dinner at Wolfson Court (Residents Only)