We establish the Langlands functoriality conjecture for the transfer from the generic spectrum of GSp(4) to GL(4) and give a criterion for the cuspidality of its image. We apply this to prove results toward the generalized Ramanujan conjecture for generic representations of GSp(4).

We prove an almost optimal lower bound for the so called last ‘geometric’ minimum for the height of a subvariety V of a power of the multiplicative group ${\mathbb G}_m^n$. More precisely, one shows that if $\boldsymbol x$ is a point of $V(\overline{\mathbb Q})$ which does not belong to a translate of a sub-torus of ${\mathbb G}_m^n$ lying in V, of height less than a function essentially linear in the inverse of the degree of V, then $\boldsymbol x$ belongs to a finite ‘exceptional’ subset of V. Thus, one proves ‘up to an $\varepsilon$’ the sharpest conjectures that can be formulated on this problem. Previously, the best known result in this direction was due to the second author and Philippon, who provided lower bounds which were inverse monomial in the degree of V (Minorations des hauteurs normalisées des sous-variétés des tores, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 489–543; Errata29 (2000)).

We propose a conjecture refining the Stark conjecture St($K/k,S$) in the function-field case. Of course St($K/k,S$) in this case is a theorem due to Deligne and independently to Hayes. The novel feature of our conjecture is that it involves two- rather than one-variable algebraic functions over finite fields. As the conjecture is formulated in the language and framework of Tate's thesis, we have powerful standard techniques at our disposal to study it. We build a case for our conjecture by: (i) proving a parallel result in the framework of adelic harmonic analysis, which we dub the adelic Stirling formula; (ii) proving the conjecture in the genus-zero case; and (iii) explaining in detail how to deduce St($K/k,S$) from our conjecture. In the genus-zero case the class of two-variable algebraic functions thus arising includes all the solitons over a genus-zero global field previously studied and applied by the author, collaborators and others. Ultimately the inspiration for this paper comes from striking examples given some years ago by Coleman and Thakur.

Let X be a smooth projective curve of genus $g > 1$ over an algebraically closed field of positive characteristic. This paper is a study of a natural stratification, defined by the absolute Frobenius morphism of X, on the moduli space of vector bundles. In characteristic two, there is a complete classification of semi-stable bundles of rank 2 which are destabilized by Frobenius pull-back. We also show that these strata are irreducible and obtain their respective dimensions. In particular, the dimension of the locus of bundles of rank two which are destabilized by Frobenius is $3g-4$. These Frobenius destabilized bundles also exist in characteristics two, three and five with ranks 4, 3 and 5, respectively. Finally, there is a connection between (pre)-opers and Frobenius destabilized bundles. This allows an interpretation of some of the above results in terms of pre-opers and provides a mechanism for constructing Frobenius destabilized bundles in large characteristics.

We introduce the notion of the boundary motive of a scheme X over a perfect field. By definition, it measures the difference between the motive ${\mathop{M_{\rm gm}}\nolimits} (X)$ and the motive with compact support ${\mathop{M_{\rm gm}^{\rm c}}\nolimits}(X)$, as defined and studied by Voevodsky et al. in Cycles, transfers, and motivic homology theories (Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)). We develop three tools to compute the boundary motive in terms of the geometry of a compactification of X: co-localization; invariance under abstract blow-up; and analytical invariance. We then prove auto-duality of the boundary motive of a smooth scheme X. As a formal consequence of this, and of co-localization, we obtain a fourth computational tool, namely localization for the\break boundary motive. In a sequel to this work (J. Wildeshaus, On the boundary motive of a Shimura variety, Prépublications du Laboratoire d'Analyse, Géométrie et Applications de l'Université Paris 13, no. 2004-23), these tools will be applied to Shimura varieties.

Let $\mathbbm{k}$ be a (topological) field of characteristic 0. To any representation of a given Hopf algebra $\mathfrak{B}_n(\mathbbm{k})$, one can associate (using a Drinfeld associator) a representation $\widehat{\Phi}(\rho)$ of the braid group over the field $\mathbbm{k}((h))$ of Laurent series. We investigate the dependence on $\Phi$ of $\widehat{\Phi}(\rho)$ for a certain class of representations (so-called GT-rigid representations) and from this dependence deduce (continuous) projective representations of the Grothendieck–Teichmüller group $GT_1(\mathbbm{k})$; in particular, for $\mathbbm{k} = \mathbbm{Q}_l$ we obtain representations of the absolute Galois group of $\mathbbm{Q}(\mu_{l^{\infty}})$. In most situations, these projective representations can be decomposed into linear characters, as we do for the representations of the Iwahori–Hecke algebra of type A. In this case, moreover, we express $\widehat{\Phi}(\rho)$ when $\Phi$ is even and obtain unitary matrix models for the representations of the Iwahori–Hecke algebra. The representations of this algebra corresponding to hook diagrams have noteworthy properties under the action of $GT_1(\mathbbm{k})$.

A fundamental object in the theory of arithmetic surfaces is the Green's function associated to the canonical metric. Previous expressions for the canonical Green's function have relied on general functional analysis or, when using specific properties of the canonical metric, the classical Riemann theta function. In this article, we derive a new identity for the canonical Green's function involving the hyperbolic heat kernel. As an application of our results, we obtain bounds for the canonical Green's function through covers and for families of modular curves.

The number of flags in a complete fan, or more generally in an Eulerian poset, is encoded in the cd-index. We prove the non-negativity of the cd-index for complete fans, regular CW-spheres and Gorenstein* posets.

We define nef line bundles ${\mathcal L}_r$ on a projective variety X with the property that, for a curve $C \subset X$, the intersection ${\mathcal L}_r.C$ is zero, if and only if the restriction morphism ${\rm Hom}(\pi_1(X),{\rm U}(r)) \to {\rm Hom}(\pi_1(C),{\rm U}(r))$ has finite image up to conjugation. This yields a rational morphism $\smash[b]{\xymatrix{X \ar@{-->}[r]^-{{\rm alb}_r} & {\rm Alb}_r(X)}}$ contracting those curves C with ${\mathcal L}_r.C=0$. For $r=1$ this is the Stein factorization of the Albanese morphism.

We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let $u_1=x^{-2}y\min x$, and $u_{n+1} = [xu_nx\min,yu_ny\min]$. The main result states that a finite group G is solvable if and only if for some n the identity $u_n(x,y)\equiv 1$ holds in G. We also develop a new method to study equations in the Suzuki groups. We believe that, in addition to the main result, the method of proof is of independent interest: it involves surprisingly diverse and deep methods from algebraic and arithmetic geometry, topology, group theory, and computer algebra (SINGULAR and MAGMA).

In this article, we prove that a free divisor in a three-dimensional complex manifold must be Euler homogeneous in a strong sense if the cohomology of its complement is the hypercohomology of its logarithmic differential forms. Calderón-Moreno et al. conjectured this implication in all dimensions and proved it in dimension two. We prove a theorem that describes in all dimensions a special minimal system of generators for the module of formal logarithmic vector fields. This formal structure theorem is closely related to the formal decomposition of a vector field by Kyoji Saito and is used in the proof of the above result. Another consequence of the formal structure theorem is that the truncated Lie algebras of logarithmic vector fields up to dimension three are solvable. We give an example that this may fail in higher dimensions.

We introduce the notion of the Bernstein–Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible) using the theory of V-filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by multiplier ideals coincides essentially with the restriction of the V-filtration. This implies a relation between the roots of the Bernstein–Sato polynomial and the jumping coefficients of the multiplier ideals, and also a criterion for rational singularities in terms of the maximal root of the polynomial in the case of a reduced complete intersection. These are generalizations of the hypersurface case. We can calculate the polynomials explicitly in the case of monomial ideals.

We show that complete conformally flat manifolds of dimension $n\geq 3$ with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line; or are globally conformally equivalent to ${\mathbb R}^n$ or to a spherical spaceform ${\mathbb S}^n/\Gamma$. This extends previous results due to Cheng, Noronha, Chen, Zhu and Zhu.