We study the transcendence degree (of the values at algebraic points) of the coordinate functions of a given one parameter subgroup on a simple g dimensional T-module. In such a situation, we obtain a lower bound for this transcendence degree depending on g and on the growth order of the parameter subgroup. As a particular case, one gets the finite characteristic analogue of a result of Siegel on the algebraic independence for the values of the classical Bessel function.

Résumé.  Nous obtenons une minoration du degré de transcendance d'un ensemble de valeurs en des points algébriques des fonctions coordonnées d'un sous-groupe à un paramètre d'un T-module simple, en fonction de l'ordre de croissance de ces dernières et de la dimension du T-module. Un cas particulier de ce résultat permet d'obtenir en caractéristique finie un analogue d'un résultat de Siegel sur l'indépendance algébrique des valeurs de la fonction de Bessel usuelle.

This paper is principally concerned with the action of the absolute Galois group on a family of dessins d'enfants i.e. isomorphism classes of coverings of the projective line unramified outside three points. More precisely, we prove a generalisation of a conjecture proposed by Yu. Kotchetkov in 1997. The main tool used in this work is a correspondence between dessins d'enfants and ribbon graphs arising from the theory of Strebel differentials.

In this paper, we investigate the action of the ${\Bbb Q}$-cohomology of the compact dual $\widehat {X}$ of a compact Shimura Variety $S(Γ)$ on the ${\Bbb Q}$-cohomology of $S(Γ)$ under a cup product. We use this to split the cohomology of $S(Γ)$ into a direct sum of (not necessarily irreducible) ${{\Bbb Q}}$-Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups ${\rm U}(p,q)$ arising from Hermitian forms over CM fields, the Mumford–Tate groups associated to certain holomorphic cohomology classes on $S(Γ)$ are Abelian. As another application, we show that all classes of Hodge type (1,1) in H2 of unitary four-folds associated to the group ${\rm U}(2,2)$ are algebraic.

Let M be a hyperbolic surface and Γ(M) its extended mapping class group. We show that Γ(M) is isomorphic to the automorphism group of the following graph G(M). The set of vertices of G(M) is the set S(M) of nonseparating simple closed geodesics of M. Two vertices u and v of S(M) are related by an edge if u and v intersect exactly once in M. The graph G(M) can be thought of as a combinatorial model for M.

The rank 4 locus of a general skew-symmetric 7 × 7 matrix gives the Pfaffian variety in P20 which is not defined as a complete intersection. Intersecting this with a general P6 gives a Calabi–Yau manifold. An orbifold construction seems to give the 1-parameter mirror-family of this. However, corresponding to two points in the 1-parameter family of complex structures, both with maximally unipotent monodromy, are two different mirror-maps: one corresponding to the general Pfaffian section, the other to a general intersection of G(2,7) ⊂ P20 with a P13. Apparently, the Pfaffian and G(2,7) sections constitute different parts of the A-model (Kähler structure related) moduli space, and, thus, represent different parts of the same conformal field theory moduli space.

We develop a localized intersection theory for arithmetic schemes on the model of Fulton's intersection theory. We prove a Lefschetz fixed point formula for arithmetic surfaces, and give an application to a conjecture of Serre on the existence of Artin's representations for regular local rings of dimension 2 and unequal characteristic.

Let K be a function field over finite field ${\Bbb F}_q$ and let ${\Bbb A}$ be a ring consisting of elements of K regular away from a fixed place ∞ of K. Let φ be a Drinfeld ${\Bbb A}$-module defined over an ${\Bbb A}$-field L. In the case where L is a finite ${\Bbb A}$-field, we study the characteristic polynomial $P$φ(X) of the geometric Frobenius. A formula for the sign of the constant term of $P$φ(X) in terms of ‘leading coefficient’ of φ is given. General formula to determine signs of other coefficients of $P$φ(X) is also derived. In the case where L is a global ${\Bbb A}$-field of generic characteristic, we apply these formulae to compute the Dirichlet density of places where the Frobenius traces have the maximal possible degree permitted by the ‘Riemann hypothesis’.

We prove a conjecture of Duke on the number of elliptic curves over the rationals of bounded height which have exceptional primes.

In this paper we study dynamics on the Fatou set of a rational function φ ∈ $\overline {\Bbb{Q}}_p(z)$. Using a notion of ‘components’ of the Fatou set defined by Benedetto, we state and prove an analogue of Sullivan's No Wandering Domains Theorem for p-adic rational functions which have no wild recurrent Julia critical points.

Serre's nonnegativity conjecture for intersection multiplicities has recently been proven by O. Gabber. In this paper we investigate Serre's positivity conjecture using the methods which he developed. We show in particular that the positivity conjecture has implications for properties of symbolic powers of prime ideals in regular local rings.

Let $k = {\Bbb F}_q(t)$ be the rational function field with finite constant field and characteristic $p \geq 3$, and let K/k be a finite separable extension. For a fixed place v of k and an elliptic curve E/K which has ordinary reduction at all places of K extending v, we consider a canonical height pairing $\langle \phantom {x},\phantom {x}\! \rangle _v \colon E(K^{\rm {sep}}) \times E(K^{\rm {sep}}) \to {\Bbb C}_{v}^{\times }$ which is symmetric, bilinear and Galois equivariant. The pairing $\langle \phantom {x},\phantom {x}\! \rangle _\infty$ for the “infinite” place of k is a natural extension of the classical Néron–Tate height. For v finite, the pairing $\langle \phantom {x},\phantom {x}\! \rangle _v$ plays the role of global analytic p-adic heights. We further determine some hypotheses for the nondegeneracy of these pairings.

For a commutative Noetherian ring A of finite Krull dimension containing the field of rational numbers, an Abelian group called the Euler class group is defined. An element of this group is attached to a projective A-module of rank = dimA and it is shown that the vanishing of this element is necessary and sufficient for P to split off a free summand of rank 1. As one of the applications of this result, it is shown that for any n-dimensional real affine domain, a projective module of rank n (with trivial determinant), all of whose generic sections have n generated vanishing ideals, necessarily splits off a free direct summand of rank 1.

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.

We show that for any positive integer τ there exist on $4{\Bbb CP}$2, the connected sum of four complex projective planes, twistor spaces whose algebraic dimensions are two. Here, τ appears as the order of the normal bundle of C in S, where S is a real smooth half-anti-canonical divisor on the twistor space and C is a real smooth anti-canonical divisor on S. This completely answers the problem posed by Campana and Kreussler. Our proof is based on the method developed by Honda, which can be regarded as a generalization of the theory of Donaldson and Friedman.

This paper is concerned with the arithmetic of curves of the form vp=us(1−u), where p is a prime with $p$ ≥ 5 and s is an integer such that 1 ≤ s ≤ p−2. The Jacobians of these curves admit complexion by a primitive p-th root of unity ζ. We find explicit rational functions on these curves whose divisors are p-multiples of divisors representing (1-ζ)2 - and (1-ζ)3-division points on the corresponding Jacobians. This also gives an effective version of a theorem of Greenberg.