On primitive roots for Carlitz modules

Let be the polynomial ring over a finite field. We prove that for every element a of a global -field of finite -characteristic the set of places for which a is a primitive root under the Carlitz action possesses a Dirichlet density. We also give a criterion for this density to be positive. This is an analogue of Bilharz’ version of the primitive roots conjecture of Artin, with replaced by the Carlitz module.     

A generalization of the primitive normal basis theorem

The primitive normal basis theorem asks whether every finite field extension has a primitive normal basis of this extension. The proof of this problem has recently been completed by Lenstra and Schoof (1987) [6], and another proof is given by Cohen and Huczynska (2003) [3]. We present a more general result, where the primitive element generating a normal basis is replaced by a primitive element generating the finite Carlitz module. Such generators always exist except for finitely many cases which might not exist.     

The strict Waring problem for polynomial rings

We prove among several results that under mild conditions any polynomial in Fq[t] is a strict sum of k⁴kth powers improving on an exponential (k²²k+1) bound of Car-Effinger-Hayes.     

A generating function of higher-dimensional Apostol-Zagier sums and its reciprocity law

We introduce higher-dimensional Dedekind sums with a complex parameter z, generalizing Zagier's higher-dimensional Dedekind sums. The sums tend to Zagier's higher-dimensional Dedekind sums as z→∞. We show that the sums turn out to be generating functions of higher-dimensional Apostol-Zagier sums which are defined to be hybrids of Apostol's sums and Zagier's sums. We prove reciprocity law for the sums. The new reciprocity law includes reciprocity formulas for both Apostol and Zagier's sums as its special case. Furthermore, as its application we obtain relations between special values of Hurwitz zeta function and Bernoulli numbers, as well as new trigonometric identities.     

On primitive roots for rank one Drinfeld modules

Let k be a global function field over a finite field and let A be the ring of the elements in k regular outside a fixed place ∞. Let K be a global A-field of finite A-characteristic and let ? be a rank one Drinfeld A-module over K. Given any α∈K, we show that the set of places P of K for which α is a primitive root modulo P under the action of ? possesses a Dirichlet density. We also give conditions for this density to be positive.     

On the existence of primitive completely normal bases of finite fields

Let ${\mathbb{F}}_q$ be the finite field of characteristic p with q elements and ${\mathbb{F}}_{q^n}$ its extension of degree n. We prove that there exists a primitive element of ${\mathbb{F}}_{q^n}$ that produces a completely normal basis of ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$, provided that $n=p^{?}m$ with $(m,p)=1$ and $q>m$.     

Gauss sums for symplectic groups over a finite field

For a nontrivial additive character and a multiplicative character of the finite field withq elements, the Gauss sums (trg) overgSp(2n,q) and (detg)(trg) overgGSp(2n, q) are considered. We show that it can be expressed as a polynomial inq with coefficients involving powers of Kloosterman sums for the first one and as that with coefficients involving sums of twisted powers of Kloosterman sums for the second one. As a result, we can determine certain generalized Kloosterman sums over nonsingular matrices and generalized Kloosterman sums over nonsingular alternating matrices, which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.Supported in part by Basic Science Research Institute program, Ministry of Education of Korea, BSRI 95-1414 and KOSEF Research Grant 95-K3-0101 (RCAA)Dedicated to my father, Chang Hong Kim     

On the value set of the Ramanujan function

We give lower bounds on the number of distinct values of the Ramanujan function τ(n), n ≦ x, and on the number of distinct residues of τ(n), n ≦ x, modulo a prime ℓ. We also show that for any prime ℓ the values τ(n), n ≦ ℓ⁴, form a finite additive basis modulo ℓ. Received: 6 October 2004     

On reciprocity

We prove a reciprocity formula between Gauss sums that is used in the computation of certain quantum invariants of 3-manifolds. Our proof uses the discriminant construction applied to the tensor product of lattices.     

Dedekind symbols with plus reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols), and the symbols are determined uniquely by their reciprocity laws, up to an additive constant. For Dedekind symbols D and F, we can consider two kinds of reciprocity laws: D(p,q)−D(q,−p)=R(p,q) and F(p,q)+F(q,−p)=T(p,q). The first type, which we call minus reciprocity laws, have been studied extensively. On the contrary, the second type, which we call plus reciprocity laws, have not yet been investigated. In this note we study fundamental properties of Dedekind symbols with plus reciprocity law F(p,q)+F(q,−p)=T(p,q). We will see that there is a fundamental difference between Dedekind symbols with minus and plus reciprocity laws.     

How many points have the same Engel and Sylvester expansions?

In this paper, we prove that the set of points in (0,1] with the same Engel and Sylvester expansions is of Hausdorff dimension .     

On Fourier coefficients of Eisenstein series and Niebur Poincaré series of integral weight

We give a Katok-Sarnak type correspondence for Niebur type Poincaré series and Eisenstein series on the three-dimensional hyperbolic space.     

Irregularities in the distributions of primes in function fields

We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985) 221-225] and Shiu [D.K.L. Shiu, Strings of congruent primes, J. London Math. Soc. 61 (2000) 359-373] concerning the distribution of primes in short intervals.     

On the behaviour of p-adic L-functions

Text

Let Lp(s,χ) denote a Leopoldt-Kubota p-adic L-function, where p>2 and χ is a nonprincipal even character of the first kind. The aim of this article is to study how the values assumed by this function depend on the Iwasawa λ-invariant associated to χ. Assuming that λ?p−1, it turns out that Lp(s,χ) behaves, in some sense, like a polynomial of degree λ. The results lead to congruences of a new type for (generalized) Bernoulli numbers.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=5aaB1d6fZDs.     

On the p-adic L-function of Hilbert modular forms at supersingular primes

In this paper, I discuss the construction of the p-adic L-function attached to a Hilbert modular form f, supersingular or ordinary, which turns out to be the non-archimedean Mellin transform of an h-admissible measure. And h is explicitly given. As a special case, when the Fourier coefficient of f at p|p is zero, plus/minus p-adic L-functions are furthermore defined as bounded functions, and they interpolate special values of L(f,χ,s) for cyclotomic characters χ. This can be used to formulate Iwasawa main conjecture for supersingular elliptic curve defined over a totally real field.     

Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold

We present simple trace formulas for Hecke operators Tk(p) for all p>3 on Sk(Γ₀(3)) and Sk(Γ₀(9)), the spaces of cusp forms of weight k and levels 3 and 9. These formulas can be expressed in terms of special values of Gaussian hypergeometric series and lend themselves to recursive expressions in terms of traces of Hecke operators on spaces of lower weight. Along the way, we show how to express the traces of Frobenius of a family of elliptic curves equipped with a 3-torsion point as special values of a Gaussian hypergeometric series over Fq, when . As an application, we use these formulas to provide a simple expression for the Fourier coefficients of η⁸(3z), the unique normalized cusp form of weight 4 and level 9, and then show that the number of points on a certain threefold is expressible in terms of these coefficients.     

On string topology of classifying spaces

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.     

On Gauss's proof of Seeber's Theorem

Summary Gauss proved Seeber's Theorem, that the determinant of a reduced positive definite ternary quadratic form is at least half the product of its diagonal coefficients, by means of two determinantal identities whose origin has remained unclear. We examine Gauss's method from a general standpoint, as a method whereby, in certain circumstances, a polynomial in several variables may be shown to be non-negative on a convex polytope by representing it as a positive multilinear combination of the linear forms which determine the polytope. We show that Gauss's identities may be obtained in this manner and that the two identities can in fact be replaced by a simpler single identity which also gives Oppenheim's precise minimum for the determinant.