Wen-Chen Chi, Anly Li
In this paper a Kummer theory of division points over rank one Drinfeld A=Fq[T]-modules defined over global function fields was given. The results are in complete analogy with the classical Kummer theory of division points over the multiplicative algebraic group Gm defined over number fields.
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Let ϕ be a Drinfeld A-module of characteristic p0 over a finitely generated field K. Previous articles determined the image of the absolute Galois group of K up to commensurability in its action on all prime-to-p0 torsion points of ϕ, or equivalently, on the prime-to-p0 adelic Tate module of ϕ. In this article we consider in addition a finitely generated torsion free A-submodule M of K for the action of A through ϕ. We determine the image of the absolute Galois group of K up to commensurability in its action on the prime-to-p0 division hull of M, or equivalently, on the extended prime-to-p0 ad…
Local Kummer theory for Drinfeld modules
本文研究了局部域上Drinfeld模的惯性群在修改后的adelic Tate模上的作用，通过典型局部Kummer配对描述惯性群作用像，给出了其有效公式及为开集的充要条件。
Constants for Artin like problems in Kummer and division fields
本文利用 Lenstra–Moree–Stevenhagen 特征和方法，通过含 Galois 群的一般乘积公式，显式计算 Kummer 域与 Serre 曲线分裂域中 Titchmarsh 除数问题的非密度常数。
In this work we generalize the concept of injective module and develop a theory of divisibility for modules over a general ring, which provides a general and unified framework to study Kummer-like field extensions arising from commutative algebraic groups. With these tools we provide an effective bound for the degree of the field extensions arising from division points of elliptic curves, extending previous results of Javan Peykar for CM curves and of Lombardo and the author for the non-CM case.
The goal of this work is to study some aspects of the geometry of the first cover $Σ^1$ in the Drinfeld tower over $\mathbb{H}^d_K$ the Drinfeld symmetric space over $K$ a finite extension of $\mathbb{Q}_p$. It is a cyclic étale cover of order prime to $p$ and even of Kummer type from the vanishing of the Picard group of $\mathbb{H}^d_K$ shown in a previous work of the author. It is then completely described by a certain class of invertible functions on $\mathbb{H}^d_K$ via the Kummer exact sequence and the main result of this article gives an explicit description of this class thus providing …
On successive minimal bases of division points of Drinfeld modules
该研究定义了局部域上Drinfeld模的uⁿ-分点空间的逐次极小基，探究其与对应格的逐次极小基的关系，并借此研究野分歧子群作用，证明了受限情形下秩2 Drinfeld模的Szpiro猜想的函数域类比。
Kummer-faithfulness for function fields
本文研究Kummer-忠实域的函数域类比，引入Drinfeld-Kummer-忠实（DKF）域概念，通过分歧理论给出其充分条件，并构造了相关实例。
The norm residue symbol for formal Drinfeld modules
研究具有稳定约化高度为一的形式 Drinfeld 模的库默尔配对，通过形式 Drinfeld 模的对数等元素给出明确描述，推广了相关结果并扩展了公式。
In this paper, we prove explicit reciprocity laws for a class of formal Drinfeld modules having stable reduction of height one, in the spirit of those existing in characteristic zero (cf. the work of Wiles). We begin by defining the Kummer pairing in the language of formal Drinfeld modules defined over local fields of positive characteristic. We then prove explicit formulas for this pairing in terms of the logarithm of the formal Drinfeld module, a certain Coleman power series, torsion points and the trace. Our results extend the explicit formulas already proved by Anglès for Carlitz modules, …
Distributive properties of division points and discriminants of Drinfeld modules
该研究针对带指定位的整体函数域提出Drinfeld模的秩r分布新概念，引入三类基本分布推导分点与判别式的乘积公式，涵盖已知结果并得到典范判别式定义等新结论，还给出经典情形下的显式值。
Two abelian varieties $A_1, A_2$ over a number field $K$ are called strongly iso-Kummerian if the torsion fields $K(A_1[d])$ and $K(A_2[d])$ coincide for all $d \geq 1$. For all $g \geq 4$ we construct geometrically simple, strongly iso-Kummerian $g$-dimensional abelian varieties over number fields that are not geometrically isogenous. We also discuss related examples and put significant constraints on any further iso-Kummerian pair.
We examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group's action on the division points of an appropriate Drinfeld module.
Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\, Y/ \text{Br}_1\, Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron-Severi lattices. Over a field of characteristic $0$, we prove that the existe…
Ramification of Tate modules for rank $2$ Drinfeld modules
本文研究秩2 Drinfeld模分点生成的函数域扩张的分歧，定义其导子作为椭圆曲线导子的类似物，通过导子计算在特定受限情形下证明了Szpiro猜想的类似结论。
We give a global description of the Frobenius elements in the division fields of Drinfeld modules of rank $2$. We apply this description to derive a criterion for the splitting modulo primes of a class of non-solvable polynomials, and to study the frequency with which the reductions of Drinfeld modules have small endomorphism rings. We also generalize some of these results to higher rank Drinfeld modules and prove CM-lifting theorems for Drinfeld modules.
We prove an analogue of the Sato-Tate conjecture for Drinfeld modules. Using ideas of Drinfeld, J.-K. Yu showed that Drinfeld modules satisfy some Sato-Tate law, but did not describe the actual law. More precisely, for a Drinfeld module ϕdefined over a field L, he constructs a continuous representation ρ_\infty : W_L \to D^* of the Weil group of L into a certain division algebra, which encodes the Sato-Tate law. When the Drinfeld module has generic characteristic and L is finitely generated, we shall describe the image of this representation up to commensurability. As an application, we give i…
Anderson generating functions are generating series for division values of points on Drinfeld modules, and they serve as important tools for capturing periods, quasi-periods, and logarithms. They have been fundamental in recent work on special values of positive characteristic L-series and in transcendence and algebraic independence problems. In the present paper we investigate techniques for expressing Anderson generating functions in terms of the defining polynomial of the Drinfeld module and determine new formulas for periods and quasi-periods.
Let $\mathbb{F}_q[T]$ be the polynomial ring over a finite field $\mathbb{F}_q$. We study the endomorphism rings of Drinfeld $\mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm.
Let $E$ be an elliptic curve defined over a number field $K$, let $α\in E(K)$ be a point of infinite order, and let $N^{-1}α$ be the set of $N$-division points of $α$ in $E(\bar{K})$. We prove strong effective and uniform results for the degrees of the Kummer extensions $[K(E[N],N^{-1}α) : K(E[N])]$. When $K=\mathbb{Q}$, and under a minimal assumption on $α$, we show that the inequality $[\mathbb{Q}(E[N],N^{-1}α) : \mathbb{Q}(E[N])] \geq cN^2$ holds with a constant $c$ independent of both $E$ and $α$.
Let $k$ be a global field, let $A$ be a Dedekind domain with $\text{Quot}(A) = k$, and let $K$ be a finitely generated field. Using a unified approach for both elliptic curves and Drinfeld modules $M$ defined over $K$ and having a trivial endomorphism ring, with $k= \mathbb{Q}$, $A = \mathbb{Z}$ in the former case and $k$ a global function field, $A$ its ring of functions regular away from a fixed prime in the latter case, for any nonzero ideal $\mathfrak{a} \lhd A$ we prove best possible estimates in the norm $|\mathfrak{a}|$ for the degrees over $K$ of the subfields of the $\mathfrak{a}$-div…
Let $Φ$ be a Drinfeld $\mathbf{F}_{q}[T]$-module of rank 2, over a finite field $L$, a finite extension of $n$ degrees of a finite field with $q$ elements $\mathbf{F}_{q}$. Let $m$ be the extension degrees of $ L$ over the field $\mathbf{F}_{q}[T]/P$, $P$ is the $\mathbf{F}%_{q}[T]$-characteristic of $L$, and $d$ the degree of the polynomial $P$. We will discuss about a many analogies points with elliptic curves. We start by the endomorphism ring of a Drinfeld $\mathbf{F}_{q}[T]$-module of rank 2, End$_{L}Φ$, and we specify the maximality conditions and non maximality conditions as a $\mathbf{…
Let $A=\mathbb{F}_q[T]$ be the polynomial ring over $\mathbb{F}_q$, and $F$ be the field of fractions of $A$. Let $ϕ$ be a Drinfeld $A$-module of rank $r\geq 2$ over $F$. For all but finitely many primes $\mathfrak{p}\lhd A$, one can reduce $ϕ$ modulo $\mathfrak{p}$ to obtain a Drinfeld $A$-module $ϕ\otimes\mathbb{F}_\mathfrak{p}$ of rank $r$ over $\mathbb{F}_\mathfrak{p}=A/\mathfrak{p}$. The endomorphism ring $\mathcal{E}_\mathfrak{p}=\mathrm{End}_{\mathbb{F}_\mathfrak{p}}(ϕ\otimes\mathbb{F}_\mathfrak{p})$ is an order in an imaginary field extension $K$ of $F$ of degree $r$. Let $\mathcal{O}_…
Stabilization of isogeny spaces between supersingular Drinfeld modules
本文通过Brandt矩阵理论和自动形式的L函数性质，证明了奇点Drinfeld模的形态空间维度稳定公式，并由此构造了半田秩度码。
A New Characteristic-Uniform Model for Elliptic Curves -- Theory, Arithmetic, and Applications
本文提出椭圆曲线的新特征一致模型\boldsymbol{\textit{C}}_d曲线，推导其各类算术公式与相关理论，特例化显示其兼容Curve25519，是椭圆曲线理论与应用的高效模型。
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