On improved asymptotic bounds for codes from global function fields

We analyse a binary cyclotomic sequence constructed via generalized cyclotomic classes by Bai et al. (IEEE Trans Inforem Theory 51: 1849–1853, 2005). First we determine the linear complexity of a natural generalization of this binary sequence to arbitrary prime fields. Secondly we consider k-error linear complexity and autocorrelation of these sequences and point out certain drawbacks of this construction. The results show that the parameters for the sequence construction must be carefully chosen in view of the respective application.      

Punctured distributions of global function fields

We obtain some criteria for elements in the universal ordinary punctured even (odd) distributions on global function fields to be torsion and then compute the torsion subgroups of the level groups of the universal punctured ordinary even (odd) distributions of global function fields. Supported by Korea Research Foundation grant (KRF-2002-070-C00003)     

Euler systems in global function fields

In this paper, the construction of Euler systems of cyclotomic units in a general global function fields is explained. As an application, an analogue of Gras’ conjecture in a global function field is proved.     

Capitulation problem for global function fields

Let q be a power of an odd prime number p, k=\mathbbF_q(t){p, k=\mathbb{F}_{q}(t)} be the rational function field over the finite field \mathbbF_q.{\mathbb{F}_{q}.} In this paper, we construct infinitely many real (resp. imaginary) quadratic extensions K over k whose ideal class group capitulates in a proper subfield of the Hilbert class field of K. The proof of the infinity of such fields K relies on an estimation of certain character sum over finite fields.     

Unimodular quadratic forms over global function fields

The isometry problem is studied for unimodular quadratic forms over the Hasse domains of global function fields. Over the polynomial ring k[x] the problem reduces to classification of forms over k; but examples are provided showing that in general no such reduction occurs, even when the underlying ring is Euclidean. Connections with the structure of the ideal class group are given, and a complete invariant for the isometry class is found in the ternary isotropic case.     

Double complex and gamma-monomials in global function fields

We investigate Γ-monomials of positive characteristic Γ-functions in the global function fields by using Anderson's double complex method. The results are the generalizations of those in the rational function field case.     

Common divisors of elliptic divisibility sequences over function fields

Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points RE(k(T)), write x_R=A_R/D_R² with relatively prime polynomials A_R(T),D_R(T)k[T]. The sequence {D_nR}_{n 1} is called the elliptic divisibility sequence of R. Let P,QE(k(T)) be independent points. We conjecture that deg (gcd(D_nP, D_mQ)) is bounded for m, n 1, and that gcd(D_nP, D_nQ) = gcdD_P, D_Q) for infinitely many n 1. We prove these conjectures in the case that j(E)k. More generally, we prove analogous statements with k(T) replaced by the function field of any curve and with P and Q allowed to lie on different elliptic curves. If instead k is a finite field of characteristic p and again assuming that j(E)k, we show that deg (gcd(D_nP, D_nQ)) is as large as for infinitely many n0 (mod p).Mathematics Subject Classification (2000): Primary: 11D61; Secondary: 11G35Acknowledgements. I would like to thank Gary Walsh for rekindling my interest in the arithmetic properties of divisibility sequences and for bringing to my attention the articles [1] and [3], and David McKinnon for showing me his article [14]. I also want to thank Zeev Rudnick for his helpful comments concerning the first draft of this paper, especially for Remark 5, for pointing out [7], and for letting me know that he described conjectures similar to those made in this paper at CNTA 7 in 2002.     

Cyclotomic units and Stickelberger ideals of global function fields

In this paper, we define the group of cyclotomic units and Stickelberger ideals in any subfield of the cyclotomic function field. We also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of Stickelberger ideals in the integral group ring.

     

Mass formula of division algebras over global function fields

In this paper we give two proofs of the mass formula for definite central division algebras over global function fields, due to Denert and Van Geel. The first proof is based on a calculation of Tamagawa measures. The second proof is based on analytic methods, in which we establish the relationship directly between the mass and the value of the associated zeta function at zero.     

Class fields,Dirichlet characters, and extended genus fields of global function fields

We study the extended genus field of an abelian extension of a rational function field. We follow the definition of Anglès and Jaulent, which uses the class field theory. First, we show that the natural definition of extended genus field of a cyclotomic function field obtained by means of Dirichlet characters is the same as the one given by Anglès and Jaulent. Next, we study the extended genus field of a finite abelian extension of a rational function field along the lines of the study of genus fields of abelian extensions of rational function fields. In the absolute abelian case, we compare this approach with the one given by Anglès and Jaulent.     

Diophantine equations over global function fields I: The Thue equation

We solve completely Thue equations in function fields over arbitrary finite fields. In the function field case such equations were formerly only solved over algebraically closed fields (of characteristic zero and positive characteristic). Our method can be applied to similar types of Diophantine equations, as well.     

Rational places in extensions and sequences of function fields of Kummer type

In this paper we prove results on the number of rational places in extensions of Kummer type over finite fields and give sufficient conditions for non-trivial lower bounds on the number of rational places at each step of sequences of function fields over a finite field, that we call (a, b)-sequences. In the case of a prime field, we apply these results to the study of rational places in certain sequences of function fields of Kummer type.     

Toroidalization of generating sequences in dimension two function fields of positive characteristic

We give a characteristic free proof of the main result of [L. Ghezzi, H.T. Hà, O. Kashcheyeva, Toroidalization of generating sequences in dimension two function fields, J. Algebra 301 (2) (2006) 838-866. ArXiv:math.AC/0509697.] concerning toroidalization of generating sequences of valuations in dimension two function fields. We show that when an extension of two-dimensional algebraic regular local rings R⊂S satisfies the conclusions of the Strong Monomialization theorem of Cutkosky and Piltant, the map between generating sequences in R and S has a toroidal structure.     

Abelian subgroups of any order in class groups of global function fields

Let be a finite field with q elements, and T a transcendental element over . In this paper, we construct infinitely many real function fields of any fixed degree over with ideal class numbers divisible by any given positive integer greater than 1. For imaginary function fields, we obtain a stronger result which shows that for any relatively prime integers m and n with m,n>1 and relatively prime to the characteristic of , there are infinitely many imaginary fields of fixed degree m such that the class group contains a subgroup isomorphic to .     

The sum of two S-units being a perfect power in global function fields

Let x ₁ and x ₂ be integers divisible only by some fixed primes. Is it possible that x ₁+x ₂ is a perfect power? Special cases of the equation x ₁+x ₂ = y ^k were formerly considered over ?. In this paper we develop an algorithm to solve this equation over global algebraic function fields. Our method is illustrated by two explicit examples.