Let $k=\mathbb F_q$. We classify, up to pre- and post-composition by $k$-M\"obius transformations, all separable $k$-indecomposable rational functions $f\in k(X)$ of degree greater than one whose Galois closure has genus zero. The classification is valid in arbitrary characteristic and includes exact arithmetic conditions and class counts over the prescribed field. Semilinear Frobenius descent determines the finite-field forms and which geometric decompositions descend to $k$. For every separable $f\in k(X)$ of degree greater than one with Galois closure of genus zero and every $m\geqslant1$, we prove that $f$ permutes $\mathbf P^1(\mathbb F_{q^m})$ if and only if it is exceptional over $\mathbb F_{q^m}$, meaning that it permutes $\mathbf P^1(L)$ for infinitely many finite extensions $L/\mathbb F_{q^m}$. No indecomposability assumption or lower bound on $q$ is needed. The argument combines fixed-point averaging with ramification on the Galois-closure curve. If the full constant field is $\mathbb F_{q^d}$, these properties depend only on $\gcd(m,d)$. For each classified family we determine the permutation extension degrees explicitly and characterize polynomial representatives.
Let $k=\mathbb F_q$. We reconstruct every $k$-indecomposable $g\in k(X)$ of degree greater than one whose normal closure has genus one. Such a map is automatically separable and, after independent degree-one changes of the source and target coordinates over $k$, arises from a separable equivariant isogeny between ellip...
Let $k=\mathbb F_q$. We classify, up to two-sided $k$-M\"obius equivalence, all separable indecomposable tame exceptional maps $\mathbf P^1_k\to\mathbf P^1_k$ whose geometric Galois closure has genus one. The resulting arithmetic fixed-field forms are encoded by Frobenius-stable affine elliptic quotient data recovered...
Let $q$ be a prime power and $\mathbb{F}_{q^2}$ be the finite fields of $q^2$ elements. The Hermitian function field $H=\mathbb{F}_{q^2}(x,y)$ defined by $y^q+y=x^{q+1}$ is a well-known maximal function field with the largest possible genus. Let $A(P_\infty)$ be the decomposition group of the infinity place $P_\infty$...
We investigate the equal-characteristic case of the Moon--Taguchi bounded-conductor finiteness conjecture for mod $p$ Galois representations over global function fields. We first establish a conditional finiteness theorem: for any global function field $K$ of characteristic $p$ and any integer $n \ge 1$, there are only...
Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the conjecture of Birch and Swinnerton-Dyer asserts that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ as $x \to \infty$. In this paper...
Let $p \geq 5$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E/K$ be an elliptic curve with good reduction. Let $F_n$ denote the $n$-division polynomial of $E$. Silverman proved that if the reduction is ordinary, then for every $P \in E(K) \setminus \hat{E}(K)$ and a suitable power $q$ of $p$, t...
Yukiko Katagiri, Shinichi Kobayashi· 0 citations
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