The Albanese map of a 3-fold of general type whose canonical map is composed with a pencil

Received: 15 October 1999; in final form: 13 June 2000 / Published online: 29 April 2002     

Sur les modules de torsion des courbes elliptiques

Sans résumé

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Re?u le 1 janvier 1997     

On maximal curves in characteristic two

The genus g of an q-maximal curve satisfies g=g ₁≔q(q−1)/2 or . Previously, q-maximal curves with g=g ₁ or g=g ₂, q odd, have been characterized up to q-isomorphism. Here it is shown that an q-maximal curve with genus g ₂, q even, is q-isomorphic to the non-singular model of the plane curve ∑ _{i =1}} ^t y ^{q /2 i} =x ^{q +1}, q=2 ^t , provided that q/2 is a Weierstrass non-gap at some point of the curve. Received: 3 December 1998     

On the flatness of models of certain Shimura varieties of PEL-type

Consider a PEL-Shimura variety associated to a unitary group that splits over an unramified extension of . Rapoport and Zink have defined a model of the Shimura variety over the ring of integers of the completion of the reflex field at a place lying over p, with parahoric level structures at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced. Received: 11 September 2000 / Published online: 24 September 2001     

Transcendance de l'invariant modulaire en caractéristique finie

Transcendence of the modular invariant in finite characteristic Abstract. Let J be the Fourier expansion at infinity of the modular invariant j associated to a Drinfeld module of rank 2 defined on the algebraic closure of and t an element of the completion of . Then at least one of the two elements t, J(t) is transcendental over .

Received: October 22, 1997; in final form January 21, 1998     

Irregularity of certain algebraic fiber spaces

Let X, Y be smooth complex projective varieties, and be a fiber space whose general fiber is a curve of genus g. Denote by q _f the relative irregularity of f. It is proved that , if f is not generically trivial; moreover, if either a) f is non-constant and the general fiber is either hyperelliptic or bielliptic or b) q(Y)= 0, then , and the bound is best possible. A classification of fiber surfaces of genus 3 with q _f = 2 is also given in this note. Received: 19 March 1997 / Revised version: 29 October 1997     

Generalized Shioda–Inose structures on K3 surfaces

In this note, we study the action of finite groups of symplectic automorphisms on K3 surfaces which yield quotients birational to generalized Kummer surfaces. For each possible group, we determine the Picard number of the K3 surface admitting such an action and for singular K3 surfaces we show the uniqueness of the associated abelian surface. Received: 9 April 1998 / Revised version: 17 July 1998     

Heights for line bundles on arithmetic varieties

Let X be an arithmetic variety and L be an element of the Néron-Severi group of its generic fiber X _K . Then there are only finitely many line bundles on X, generically belonging to L, such that the degrees of on the irreducible components of the special fibers of X and the height of are bounded. The concept of a height used here is recalled. Several elementary properties of this height are proven. Received: 9 March 1996     

A procedure to calculate torsion¶of elliptic curves over ℚ

We present an algorithm which uses the analytic parameterization of elliptic curves to rapidly calculate torsion subgroups, and calculate its running time. This algorithm is much faster than the “traditional” Lutz–Nagell algorithm used by most computer algebra systems to calculate torsion subgroups. Received: 7 August 1997 / Revised version: 28 November 1997     

On height functions on Jacobian surfaces

The canonical height on an abelian variety is useful and important for the study of the Mordell-Weil group. But it is difficult to calculate the canonical height in general. We give an effective method to calculate the canonical height on a Jacobian surface. As an application, we verify the Birch-Swinnerton-Dyer conjecture for the Jacobian surface of a twisted modular curve. Received: 15 July 1996 / Revised version: 19 January 1997     

Isogénie minimale entre modules de Drinfel'd

Résumé. Nous majorons le degré d'une isogénie minimale entre deux modules de Drinfel'd. Il s'agit d'un analogue d'un résultat démontré tout d'abord sur les courbes elliptiques, puis généralisé aux variétés abéliennes par Masser et Wüstholz. Comme dans le cas abélien, la majoration dépend uniquement de la hauteur de l'un des modules et du degré d'un corps de définition commun aux deux modules. Cette dépendance est polyn?miale.

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We give a bound for the degree of a minimal isogeny between two Drinfel'd modules. This result is an anlogue of a theorem first proved on elliptic curves and then extended to abelian varieties by Masser and Wüstholz. This upper bound, as in the abelian case depends only on the height of one of the modules and on the degree of a field over which both modules are defined. We get a polynomial bound.

Received: 4 October 1994 / in final form: 30 July 1998     

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form Y ²=f ₃ X ⁶+f ₂ X ⁴+f ₁ X ²+f ₀, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα : ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ. Received: 27 November 1998 / Revised version: 4 June 1999