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)     

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.     

Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields

We show that, for all characteristic p global fields k and natural numbers n coprime to the order of the non-p-part of the Picard group Pic⁰(k) of k, there exists an abelian extension L/k whose local degree at every prime of k is equal to n. This answers in the affirmative in this context a question recently posed by Kisilevsky and Sonn. As a consequence, we show that, for all n and k as above, the n-torsion subgroup Br_n(k) of the Brauer group Br(k) of k is algebraic, answering a question of Aldjaeff and Sonn in this context.     

The distribution of class groups of function fields

For any sufficiently general family of curves over a finite field F_q and any elementary abelian ?-group H with ? relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[?](F_q)≅H. In doing so, we prove a conjecture of Friedman and Washington.     

Higher Heegner points on elliptic curves over function fields

Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a -tower of finite extensions of k, and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of Cornut and Vatsal.     

Higher rank subgroups in the class groups of imaginary function fields

Let F be a finite field and T a transcendental element over F. In this paper, we construct, for integers m and n relatively prime to the characteristic of F(T), infinitely many imaginary function fields K of degree m over F(T) whose class groups contain subgroups isomorphic to (Z/nZ)^m. This increases the previous rank of m−1 found by the authors in [Y. Lee, A. Pacelli, Class groups of imaginary function fields: The inert case, Proc. Amer. Math. Soc. 133 (2005) 2883-2889].     

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.     

Class number indivisibility for quadratic function fields

Let M?5. For any odd prime power q and any prime ??q, we show that there are at least pairwise coprime D∈Fq[T] which are square-free and of odd degree ?M, such that ? does not divide the class number of the complex quadratic functions fields .     

Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L₁ be a quadratic geometric extension of K and let L₂ be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L₁ and L₂ have the same m-rank.     

On primitive roots of tori: The case of function fields

We generalize Bilharz's Theorem for to all one-dimensional tori over global function fields of finite constant field. As an application, we also derive an analogue, in the setting of function fields, of a theorem (Chen-Kitaoka-Yu, Roskam) on the distribution of fundamental units modulo primes. Received: 16 October 2000 / Published online: 2 December 2002 Research partially supported by National Science Council, Rep. of China.     

Arithmetic equivalence for function fields, the Goss zeta function and a generalisation

A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual (characteristic zero) zeta function by the positive characteristic zeta function introduced by Goss. We prove that for function fields whose characteristic exceeds their degree, equality of the Goss zeta function is the same as Gaßmann equivalence (a purely group theoretical property), but this statement can fail if the degree exceeds the characteristic. We introduce a ‘Teichmüller lift’ of the Goss zeta function and show that equality of such is always the same as Gaßmann equivalence.     

Arithmetic actions on cyclotomic function fields

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.     

The Scholz theorem in function fields

The Scholz theorem in function fields states that the l-rank difference between the class groups of an imaginary quadratic function field and its associated real quadratic function field is either 0 or 1 for some prime l. Furthermore, Leopoldt's Spiegelungssatz (= the Reflection theorem) in function fields yields a comparison between the m-rank of some subgroup of the class group of an imaginary cyclic function field L₁ and the m-rank of some subgroup of the class group of its associated real cyclic function field L₂ for some prime number m; then their m-ranks also equal or differ by 1. In this paper we find an explicit necessary condition for their m-ranks (respectively l-ranks) to be the same in the case of cyclic function fields (respectively quadratic function fields). In particular, in the case of quadratic function fields, if l does not divide the regulator of L₂, then their l-ranks are the same, equivalently if their l-ranks differ by 1, then l divides the regulator of L₂.     

Newtonian and Schinzel quadratic fields

We consider a quadratic extension of a global field and give the maximal length of a Newton sequence, that is, a simultaneous ordering in Bhargava’s sense or a Schinzel sequence, that satisfies the condition of the Brownin-Schinzel problem. In the case of a number field , we show that the maximal length of a Schinzel sequence is 1, except in seven particular cases, and explicitly compute the maximal length of a Schinzel sequence in these special cases. We show that Newton sequences are also finite, except for at most finitely many cases, all real, and such that . For , we show that the maximal length of a Newton sequence is 1, except in five particular cases, and again explicitly compute the maximal length in these special cases. In the case of a quadratic extension of a function field F_q(T), we similarly show that, unless the ring of integers is isomorphic to some function field (in which case there are obviously infinite Newton and Schinzel sequences), the maximal length of a Schinzel sequence is finite and in fact, equal to q. For imaginary extensions, Newton sequences are known to be finite (unless the ring of integers is isomorphic to some function field) and we show here that the same holds in the real case, but for finitely many extensions.     

Non-crossed products over function fields

Noncrossed product division algebras are constructed over all function fields and iterated power series fields over global fields, using Hilbert's Irreducibility Theorem and the construction of [B]. Minimum indexes obtained are p ² for odd p and 2³ otherwise. Examples are obtained with large index to exponent ratio. Received: 12 February 2001 / Revised version: 26 November 2001     

The Rubin-Stark conjecture for a special class of function field extensions

We prove a strong form of the Brumer-Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K/k contained in the compositum k_p∞?k_p·k_∞ of the maximal pro-p abelian extension k_p/k and the maximal constant field extension k_∞/k of k, which happens to sit inside the maximal abelian extension k^ab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321-367).     

Constructing function fields¶with many rational places via the Carlitz module

We consider certain decomposition fields in extensions of ?q (Z) by the Carlitz module and give formulas for their genera and numbers of rational places, suitable for automatic computations. By extensive calculations we found some function fields which have more rational places than the known examples of the respective genus. Received: 7 September 2001/ Revised version: 19 October 2001     

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     

Calculating formula and divisibility for relative class numbers of abelian function fields

In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K/k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k.     

The class number one problem for some totally complex quartic number fields

We prove that there are 95 non-isomorphic totally complex quartic fields whose rings of algebraic integers are generated by an algebraic unit and whose class numbers are equal to 1. Moreover, we prove Louboutin's Conjecture according to which a totally complex quartic unit εu generally generates the unit group of the quartic order Z[εu].