Generalization of a theorem of Carlitz

AbstractWe generalize Carlitzʼ result on the number of self-reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula holds for the number of irreducible polynomials obtained by a fixed quadratic transformation. Our main tools are a combinatorial argument and Hurwitz genus formula

Rademacher-Carlitz Polynomials

We introduce and study the \emph{Rademacher-Carlitz polynomial} \RC(u, v, s, t, a, b) := \sum_{k = \lceil s \rceil}^{\lceil s \rceil + b - 1} u^{\fl{\frac{ka + t}{b}}} v^k where $a, b \in \Z_{>0}$, $s, t \in \R$, and $u$ and $v$ are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view \RC(u, v, s, t, a, b) as a polynomial analogue (in the sense of Carlitz) of the \emph{Dedekind-Rademacher sum} \r_t(a,b) := \sum_{k=0}^{b-1}\left(\left(\frac{ka+t}{b} \right)\right) \left(\left(\frac{k}{b} \right)\right), which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz polynomials, we show how they are the only nontrivial ingredients of integer-point transforms $$\sigma(x,y):=\sum_{(j,k) \in \mathcal{P}\cap \Z^2} x^j y^k$$ of any rational polyhedron $\mathcal{P}$, and we derive a novel reciprocity theorem for Dedekind-Rademacher sums, which follows naturally from our setup

Arithmetic of characteristic p special L-values (with an appendix by V. Bosser)

Recently the second author has associated a finite \F_q[T]-module $H$ to the Carlitz module over a finite extension of \F_q(T). This module is an analogue of the ideal class group of a number field. In this paper we study the Galois module structure of this module $H$ for `cyclotomic' extensions of \F_q(T). We obtain function field analogues of some classical results on cyclotomic number fields, such as the $p$-adic class number formula, and a theorem of Mazur and Wiles about the Fitting ideal of ideal class groups. We also relate the Galois module $H$ to Anderson's module of circular units, and give a negative answer to Anderson's Kummer-Vandiver-type conjecture. These results are based on a kind of equivariant class number formula which refines the second author's class number formula for the Carlitz module.Comment: (v2: several corrections in section 9; v3: minor corrections, improved exposition; v4: minor corrections; v5 minor corrections

A matrix generalization of a theorem of Fine

In 1947 Nathan Fine gave a beautiful product for the number of binomial coefficients $\binom{n}{m}$, for $m$ in the range $0 \leq m \leq n$, that are not divisible by $p$. We give a matrix product that generalizes Fine's formula, simultaneously counting binomial coefficients with $p$-adic valuation $\alpha$ for each $\alpha \geq 0$. For each $n$ this information is naturally encoded in a polynomial generating function, and the sequence of these polynomials is $p$-regular in the sense of Allouche and Shallit. We also give a further generalization to multinomial coefficients.Comment: 9 pages; publication versio