Drinfeld-Stuhler modules and the Hasse principle

Drinfeld-Stuhler modules

We study $\mathscr{D}$-elliptic sheaves in terms of their associated modules, which we call Drinfeld-Stuhler modules. We prove some basic results about Drinfeld-Stuhler modules and their endomorphism rings, and then examine the existence and properties of Drinfeld-Stuhler modules with large endomorphism algebras, which are analogous to CM and supersingular Drinfeld modules. Finally, we examine the fields of moduli of Drinfeld-Stuhler modules.

Twists of Drinfeld-Stuhler modular varieties

Let $\cal A$ be a maximal (or more generally a hereditary) order in a central simple algebra over a global field $F$ of positive characteristic. We study the reduction of the modular scheme of $\cal A$-elliptic sheaves at all places of $F$. We show that some of these varieties - for different $\cal A$ - are twists of each other. This result can be viewed as a global analogue of the Cherednik-Drinfeld theorem for these varieties.

On class number relations, intersections, and GL(2)-tale over the function field side

The aim of this paper is to study class number relations over function fields and the intersections of Hirzebruch-Zagier type divisors on the Drinfeld-Stuhler modular surfaces. The main bridge is a particular "harmonic" theta series with nebentypus. Using the strong approximation theorem, the Fourier coefficients of this series are expressed in two ways; one comes from modified Hurwitz class numbers and another gives the intersection numbers in question. An elaboration of this approach enables us to interpret these class numbers as a "mass sum" over the CM points on the Drinfeld-Stuhler modula…

Drinfeld Modules with Complex Multiplication, Hasse Invariants and Factoring Polynomials over Finite Fields

We present a novel randomized algorithm to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial $f \in \F_q[x]$ to be factored) with respect to a random Drinfeld module $ϕ$ with complex multiplication. Factors of $f$ supported on prime ideals with supersingular reduction at $ϕ$ have vanishing Hasse invariant and can be separated from the rest. Incorporating a Drinfeld module analogue of Deligne's congruence, we devise an algorithm to compu…

$\mathscr{D}$-椭圆层与Hasse原理

$\mathscr{D}$-elliptic sheaves and the Hasse principle

本文研究Drinfeld--Stuhler簇 $X^D$ 的算术性质，建立 $\mathscr{D}$-椭圆层的显式判别准则以判定其在 $F$ 的次数为 $d$ 的有限扩张上无有理点，并在 $d=2$ 时构造违反Hasse原理的二次扩张无限族。

Factoring Polynomials over Finite Fields using Drinfeld Modules with Complex Multiplication

We present novel algorithms to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial $f(x) \in \F_q[x]$ to be factored) with respect to a Drinfeld module $ϕ$ with complex multiplication. Factors of $f(x)$ supported on prime ideals with supersingular reduction at $ϕ$ have vanishing Hasse invariant and can be separated from the rest. A Drinfeld module analogue of Deligne's congruence plays a key role in computing the Hasse invariant lift. We …