A Local-Global Criterion for Dynamics on P^1

On the minimal set for counterexamples to the local-global principle

We prove that only for powers of 2 and 3 could occur counterexamples to the local-global divisibility principle for elliptic curves defined over the rationals. For we refine our previous criterion for the validity of the principle. We also give an example that shows that the assumptions of our criterion are necessary.

A criterion for good reduction of Drinfeld modules and Anderson motives in terms of local shtukas

For an Anderson A-motive over a discretely valued field whose residue field has A-characteristic ε, we prove a criterion for good reduction in terms of its associated local shtuka at ε. This yields a criterion for good reduction of Drinfeld modules. Our criterion is the function-field analog of Grothendieck's and de Jong's criterion for good reduction of an abelian variety over a discretely valued field with residue characteristic p in terms of its associated p-divisible group.

基于局部判据的有限域上的置换多项式

Permutation polynomials over finite fields by the local criterion

本文进一步研究了局部判据，提出了一类在 $\mathbb{F}_{q^2}$ 上的置换多项式及其复合逆，并证明了线性化多项式与所有置换多项式在特定条件下均为局部置换多项式。

Irreducibility of Polynomials over Global Fields is Diophantine

Given a global field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have any root in $K$. This strengthens the known result that the set of non-$n$-th-powers in $K$ is diophantine when $K$ is a number field. We also deduce a diophantine criterion for a polynomial over $K$ of given degree in a given number of variables to be irreducible. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order definitions of $\mathbb{Z}$ in $\mathbb{Q}$.

A valuation criterion for normal basis generators in local fields of characteristic $p$

Let $K$ be a complete local field of characteristic $p$ with perfect residue field. Let $L/K$ be a finite, fully ramified, Galois $p$-extension. If $π_L\in L$ is a prime element, and $p'(x)$ is the derivative of $π_L$'s minimal polynomial over $K$, then the relative different $\euD_{L/K}$ is generated by $p'(π_L)\in L$. Let $v_L$ be the normalized valuation normalized with $v_L(L)=\mathbb{Z}$. We show that any element $ρ\in L$ with $v_L(ρ)\equiv -v_L(p'(π_L))-1\bmod[L:K]$ generates a normal basis, $K[{Gal}(L/K)]\cdotρ=L$. This criterion is tight: Given any integer $i$ such that $i\not\equiv -v…

Criterion for surjectivity of localization in Galois cohomology of a reductive group over a number field

Let $G$ be a connected reductive group over a number field $F$, and let $S$ be a set (finite or infinite) of places of $F$. We give a necessary and sufficient condition for the surjectivity of the localization map from $H^1(F,G)$ to the "direct sum" of the sets $H^1(F_v,G)$ where $v$ runs over $S$. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.