Let k be a field and X = Spec (k[t,t−1]). Katz proved that a differential equations with coefficients in k((t−1)) is uniquely extended to a special algebraic differential equation on X when k is of characteristic 0. He also proved that a finite extension of k((t−1)) is uniquely extended to a special covering of X when k is of any characteristic. These theorems are called canonical extension or Katz correspondence. We shall prove a p-adic analogue of canonical extension for quasi-unipotent overconvergent isocrystals. As a consequence, we can show that the local index of a quasi-unipotent overconvergent is equal to its Swan conductor.

We use some fundamental work of Bernstein to study parabolic induction in reductive p-adic groups. In particular, we determine when parabolic induction from a component of the Bernstein decomposition of a Levi subgroup to the corresponding component of the full group is an equivalence of categories.

The local intersection cohomology of a point in the Baily–Borel compactification (of a Hermitian locally symmetric space) is shown to be canonically isomorphic to the weighted cohomology of a certain linear locally symmetric space (an arithmetic quotient of the associated self-adjoint homogeneous cone). Explicit computations are given for the symplectic group in four variables.

We recall the main result of L. Caporaso, J. Harris, and B. Mazur's 1997 paper of ‘Uniformity of rational points.’ It says that the Lang conjecture on the distribution of rational points on varieties of general type implies the uniformity for the numbers of rational points on curves of genus at least 2. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.

Given three distinct primitive complex characters χ1,χ2,χ3 satisfying some technical conditions, we prove that the triple product of twisted L-functions L(f·χ1,1/2) L(f·χ2,1/2) L(f·χ3,1/2) does not vanish for a positive proportion of weight 2 primitive forms for Γ0(q), when q goes to infinity through the set of prime numbers. This result, together with some variants, implies the existence of quotients of J0(q) of large dimension satisfying the Birch–Swinnerton-Dyer conjecture over cyclic number fields of degree less than 5.

We present an elementary proof that given a general collection of d points in Pn the linear system of cubics singular on each point has the expected codimension except when n=4 and d=7. In that case the cubic is unique. This, together with previous work of the author, gives a proof of the Alexander–Hirschowitz interpolation theorem.

To any finite group $\Gamma \subset {SL}_2({\mathbb {C}})$ and each element τ in the center of the group algebra of Γ, we associate a category, ${\mathcal {C}}\! oh({\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},{\mathbb {P}}^1).$ It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category ${\mathcal {C}}\! oh({\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},{\mathbb {P}}^1)$ should be thought of as the category of coherent sheaves on a ‘noncommutative projective space’, ${\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},$ equipped with a framing at ${\mathbb {P}}^1$, the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of ${\mathcal {C}}\! oh({\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }},{\mathbb {P}}^1)$ and the Nakajima quiver variety arising from Γ via the McKay correspondence. We apply the above isomorphism to deduce a generalization of the Crawley-Boevey and Holland conjecture, saying that the moduli space of `rank 1' projective modules over the deformed preprojective algebra is isomorphic to a particular quiver variety. This reduces, for $\Gamma =\{ 1\}$, to the recently obtained parametrisation of the isomorphism classes of right ideals in the first Weyl algebra, $\sf {A_1}$, by points of the Calogero–Moser space, due to Cannings and Holland and Berest and Wilson. Our approach is algebraic and is based on a monadic description of torsion free sheaves on ${\mathbb {P}}^2_{_{\! \Gamma ^{\! },^{\! }\tau }}$. It is totally different from the one used by Berest and Wilson, involving $\tau $-functions.

We give the fermionic character formulas for the spaces of coinvariants obtained from level k integrable representations of $\widehat {\frak sl}_2$. We establish the functional realization of the spaces dual to the coinvariant spaces. We parameterize functions in the dual spaces by rigged partitions, and prove the recursion relations for the sets of rigged partitions.

We find explicit free resolutions for the D-modules Dfs and D[s] fs / D[s] fs+1, where f is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper.

An investigation of morphisms that coincide topologically is used to generalize to all characteristics and partly reprove Tamagawa's theorem on the Grothendieck conjecture in anabelian geometry for affine hyperbolic curves. The theorem now deals with π1tame of curves over a finitely generated field and its effect on the sets of isomorphisms. Universal homeomorphisms are formally inverted.

We factor the virtual Poincaré polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as $t^{2u} (t^2-1)^r Q_{G/H}(t^2)$ for a polynomial $Q_{G/H}$ with nonnegative integer coefficients. Moreover, we show that $Q_{G/H}(t^2)$ divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.

In Invent. Math.126 (2000), pp. 513–545, we gave a proof of Lang's conjecture on Abelian varieties leading to an effective bound for the number of translates involved. We show here that the method can be extended to give a similar statement for the ‘Mordell–Lang plus Bogomolov’ theorem proven by B. Poonen and independently by S. Zhang. We deal in detail with tori for which effective results have been obtained by J.-H. Evertse and H. P. Schlickewei; we improve on these mainly by providing polynomial bounds in the degree instead of doubly exponential ones. We also state a theorem for Abelian varieties. In both cases the strategy of proof is based on the approach of Mumford and Vojta–Faltings–Bombieri together with an effective Bogomolov property and therefore does not rely on either equidistribution nor subspace theorem arguments.

We show that for all finite fields Fq, there exists a curve C over Fq of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over Fq.