Imposing Singular Points and Many Nodes to Curves on Surfaces

 Let S be a smooth projective surface. Here we study the conditions imposed to curves of a fixed very ample linear system by a general union of types of singularities τ when most of connected components of τ are ordinary double points. This problem is related to the existence of “good” families of curves on S with prescribed singularities, most of them being nodes, and to the regularity of their Hilbert scheme. Received 6 July 2000; in revised form 16 June 2001     

Weil Pairing for Drinfeld Modules

As is well-known, there exists a Weil pairing for elliptic curves which is a perfect bilinear form from the m-torsion of the elliptic curve E to the m-th roots of unity. In this paper we will show how Andersons paper [1] gives rise to an analogue of this pairing for Drinfeld modules.The author was supported by NWO Grant 613.007.040. The author would like to thank G. Böckle and S. J. Edixhoven for their comments.     

Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points

 We construct algebraic curves C defined over a finite prime field such that the number of -rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1.     

A special integral basis for a plane model of the Drinfeld modular curve X1(<?) mod T

In this work we get some properties of the plane model C ₁(?) (Section 3) of the reduction mod T of the Drinfeld modular curve X ₁(?). The main result is the explicit presentation of a special integral basis for this model (4.5). This integral basis is close to be minimal. Received: 23 October 1998     

Explicit valuations of division polynomials¶of an elliptic curve

In this paper, we estimate valuations of division polynomials and compute them explicitely at singular primes. We show that ν_?(ψ _m (M)) is asymptotically equal to ν_?(m) for a non-torsion point M such that M mod ? is non-zero and non-singular, and it is asymptotically equal to c ₁ m ¹ for some constant c ₁ for a non-torsion point M such that M mod ? is either singular or zero. Furthermore, we show that the common factors of φ _m (M) and ψ _m ²(M) have valuations at ? asymptotically equal to c ₂ m ² for some constant c ₂ when M mod ? is singular, which is a generalization of M. Ayad's result. Received: 10 July 1997 / Revised version: 11 May 1998     

Rational cuspidal plane curves of type (d,d-4) with

In the present paper we classify rational cuspidal plane curves with maximal multiplicity deg C - 4 and at least three cusps and where (V,D) is the minimal (SNC) resolution of (ℙ²,C). Received: 28 August 1998     

A note on Wronskians and the ABC theorem in function fields of prime characteristic

We provide a sharp bound for the order sequence of Wronskians. We also give another proof of the truncated second main theorem over function fields which is a generalization of the ABC theorem due to Mason, Voloch, Brownawell and Masser, Noguchi and the author. Received: 9 June 1998 / Revised version: 24 September 1998     

Sum-Difference Sequences and Catalan Numbers

 Let be a sequence of natural numbers > 1, and set . The sequence is called admissible if a _i divides for all i. It is known that the admissible sequences are counted by the Catalan numbers. We present a proof of this fact which, in turn, leads to some interesting combinatorial and number-theoretic questions. Received 12 May 1997; in revised form 9 June 1997     

Integral and modular representations attached to the Selmer group of an elliptic curve with complex multiplication

Let E be an elliptic curve with complex multiplication over the ring of integers of an imaginary quadratic field K. Denote by p an odd prime that splits into in and by the unique -extension of K totally ramified above . It is well-known that the Selmer group attached to any finite extension of is analogous to the minus part of the p-class group of divisors of the cyclotomic - extensions of CM number fields. One of the most striking examples of this analogy is the existence of a translation formula à la Kida for the codimension of the Selmer group at the top of the tower. In this article we carry on the analogy with the presentations of results similar to those proven by Gold and Madan in the cyclotomic case (see [8]), which were the continuation of Kida's work. More precisely, we describe the -structure of the Selmer group when G is a cyclic group of order p or . In addition, we study the modular representation of G on the subgroup of points of order p of the Selmer group, when G is cyclic of order . Received December 3, 1997     

On the Distribution of Coefficients of Logarithmic Derivativesof L-Functions Attached to Certain Arithmetical Semigroups

 Under the Riemann Hypothesis for the classical Riemann zeta function, there exist infinitely many arithmetically non-isomorphic arithmetical semigroups with the property that one of the associated L-functions vanishes at . Moreover, there are no restrictions in the distribution of prime divisors of a given norm except an obvious one concerning the order of magnitude. Received 22 December 1997 in revised form 12 May 1998     

Algebraic points of low degree¶on the Fermat curve of degree seven

We determine all algebraic points of degree at most five over Q on the Fermat curve of degree seven. Received: 26 February 1998 / Revised version: 1 June 1998     

Rational points of the group of components¶of a Néron model

The structure of component groups of Néron models has been investigated on several occasions. Here we admit non-separably closed residue fields and are interested in the subgroup of rational points or, in other terms, in the subgroup of geometrically connected components of a Néron model. We consider Néron models of abelian varieties and of algebraic tori and give detailed computations in the case of Jacobians of curves. Received: 4 May 1998 / Revised version: 12 October 1998     

Cycles for Rational Maps of Good Reduction Outside a Prescribed Set

Let K be a number field and S a fixed finite set of places of K containing all the archimedean ones. Let R _S be the ring of S-integers of K. In the present paper we study the cycles in for rational maps of degree ≥2 with good reduction outside S. We say that two ordered n-tuples (P ₀, P ₁,… ,P _n−1) and (Q ₀, Q ₁,… ,Q _n−1) of points of are equivalent if there exists an automorphism A ∈ PGL₂(R _S ) such that P _i = A(Q _i ) for every index i∈{0,1,… ,n−1}. We prove that if we fix two points , then the number of inequivalent cycles for rational maps of degree ≥2 with good reduction outside S which admit P ₀, P ₁ as consecutive points is finite and depends only on S and K. We also prove that this result is in a sense best possible.     

Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-Shifts

 Let α be an expansive automorphisms of compact connected abelian group X whose dual group is cyclic w.r.t. α (i.e. is generated by for some ). Then there exists a canonical group homomorphism ξ from the space of all bounded two-sided sequences of integers onto X such that , where σ is the shift on . We prove that there exists a sofic subshift such that the restriction of ξ to V is surjective and almost one-to-one. In the special case where α is a hyperbolic toral automorphism with a single eigenvalue and all other eigenvalues of absolute value we show that, under certain technical and possibly unnecessary conditions, the restriction of ξ to the two-sided beta-shift is surjective and almost one-to-one. The proofs are based on the study of homoclinic points and an associated algebraic construction of symbolic representations in [13] and [7]. Earlier results in this direction were obtained by Vershik ([24]–[26]), Kenyon and Vershik ([10]), and Sidorov and Vershik ([20]–[21]). (Received 27 October 1998; in revised form 17 May 1999)     

Enlargements and Coverings by Rees Matrix Semigroups

 We formulate a general condition, called an enlargement, under which a semigroup T is covered by a Rees matrix semigroup over a subsemigroup. (Received 1 February 1999; in revised form 19 May 1999)