The divisibility modulo 24 of Kloosterman sums on GF(2m)GF(2^m), mm even

In a recent work by Charpin, Helleseth, and Zinoviev Kloosterman sums $K(a)$ over a finite field \F_{2^m} were evaluated modulo 24 in the case $m$ odd, and the number of those $a$ giving the same value for $K(a)$ modulo 24 was given. In this paper the same is done in the case $m$ even. The key techniques used in this paper are different from those used in the aforementioned work. In particular, we exploit recent results on the number of irreducible polynomials with prescribed coefficients.Comment: 15 pages, submitted; an annoying typo corrected in the abstrac

Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm

Let \F_q ($q=p^r$) be a finite field. In this paper the number of irreducible polynomials of degree $m$ in \F_q[x] with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is obtained improving the bound by Wan if $m$ is small compared to $q$. As a corollary, sharp bounds are obtained for the number of elements in \F_{q^3} with prescribed trace and norm over \F_q improving the estimates by Katz in this special case. Moreover, a characterization of Kloosterman sums over \F_{2^r} divisible by three is given generalizing the earlier result by Charpin, Helleseth, and Zinoviev obtained only in the case $r$ odd. Finally, a new simple proof for the value distribution of a Kloosterman sum over the field \F_{3^r}, first proved by Katz and Livne, is given.Comment: 21 pages; revised version with somewhat more clearer proofs; to appear in Acta Arithmetic

The Finnish national bibliography 2022

How many books are published in Finland each year? What are the most common genres of music? Are books today shorter or longer than a hundred years ago? The answers to these questions and more can be found in this publication which presents themes and trends in the Finnish national imprint through the years

Beyond East-West : marginality and national dignity in Finnish identity construction

Since the end of the Cold War it has become common for Finnish academics and politicians alike to frame debates about Finnish national identity in terms of locating Finland somewhere along a continuum between East and West (e.g., Harle and Moisio 2000). Indeed, for politicians properly locating oneself (and therefore Finland) along this continuum has often been seen as central to the winning and losing of elections. For example, the 1994 referendum on EU membership was largely interpreted precisely as an opportunity to relocate Finland further to the West (Jakobson 1998, 111; Arter 1995). Indeed, the tendency to depict Finnish history in terms of a series of ‘westernising’ moves has been notable, but has also betrayed some of the politicised elements of this view (Browning 2002). However, this framing of Finnish national identity discourse is not only sometimes politicised, but arguably is also too simplified and results in blindness towards other identity narratives that have also been important through Finnish history, and that are also evident (but rarely recognised) today as well. In this article we aim to highlight one of these that we argue has played a key role in locating Finland in the world and in formulating notions of what Finland is about, what historical role and mission it has been understood as destined to play, and what futures for the nation have been conceptualised as possible and as providing a source of subjectivity and national dignity. The focus of this article is therefore on the relationship between Finnish nationalism and ideas of ‘marginality’ through Finnish history

A note on evaluations of some exponential sums

1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form $S(a,p^{α}+1) := ∑_{x∈_q} χ(ax^{p^{α}+1})$ where χ is a non-trivial additive character of the finite field $_q$, $q = p^e$ odd, and $a ∈ *_q$. In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of $p^{α}+1$. The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain sums of the form $∑_{x∈_q} χ(ax^{p^{α}+1} + bx)$