Let $p$ and $q$ be multiplicatively independent natural numbers, and $K$ the field $ℂ\left(x^{1∕s}\mid s=1,2,3\dots \right)$. Let $p$ and $q$ act on $K$ as the Mahler operators $x\mapsto x^p$ and $x\mapsto x^q$. Schäfke and Singer (2019) showed that a finite-dimensional vector space over $K$, carrying commuting structures of a $p$-Mahler module and a $q$-Mahler module, is obtained via base change from a similar object over $ℂ$. As a corollary, they gave a new proof of a conjecture of Loxton and van der Poorten, which had been proved before by Adamczewski and Bell (2017). When $K=ℂ\left(x\right)$, and $p$ and $q$ are complex numbers of absolute value greater than 1, acting on $K$ via dilations $x\mapsto px$ and $x\mapsto qx$, a similar theorem has been obtained by Bézivin and Boutabaa (1992). Underlying these two examples are the algebraic groups ${\mathbb{𝔾}}_m$ and ${\mathbb{𝔾}}_a$, respectively, with $K$ the function field of their universal covering, and $p$, $q$ acting as endomorphisms.

Replacing the multiplicative or additive group by the elliptic curve $ℂ∕\Lambda$, and $K$ by the maximal unramified extension of the field of $\Lambda$-elliptic functions, we study similar objects, which we call elliptic $\left(p,q\right)$-difference modules. Here $p$ and $q$ act on $K$ via isogenies. When $p$ and $q$ are relatively prime, we give a structure theorem for elliptic $\left(p,q\right)$-difference modules. The proof is based on a periodicity theorem, which we prove in somewhat greater generality. A new feature of the elliptic modules is that their classification turns out to be fibered over Atiyah’s classification of vector bundles on elliptic curves (1957).

Only the modules whose associated vector bundle is trivial admit a $ℂ$-structure as in thc case of ${\mathbb{𝔾}}_m$ or ${\mathbb{𝔾}}_a$, but all of them can be described explicitly with the aid of (logarithmic derivatives of) theta functions. We conclude with a proof of an elliptic analogue of the conjecture of Loxton and van der Poorten.

让 $磷$ 和 $q$ 是乘法独立的自然数，并且 $钾$ 场 $ℂ\left(X^{1∕秒}\mid 秒=1,2,3\dots \right)$. 让 $磷$ 和 $q$ 采取行动 $钾$ 作为马勒算子 $X\mapsto X^{磷}$ 和 $X\mapsto X^q$. Schäfke 和 Singer (2019) 表明，有限维向量空间在$钾$, 承载一个通勤结构 $磷$-马勒模块和一个 $q$-Mahler 模块，通过从相似对象的基础更改获得 $ℂ$. 作为推论，他们对 Loxton 和 van der Poorten 的猜想给出了新的证明，该猜想之前已被 Adamczewski 和 Bell (2017) 证明过。什么时候$钾=ℂ\left(X\right)$， 和 $磷$ 和 $q$ 是绝对值大于 1 的复数，作用于 $钾$ 通过扩张 $X\mapsto 磷X$ 和 $X\mapsto qX$，Bézivin 和 Boutabaa (1992) 已经获得了类似的定理。这两个例子的基础是代数群${\mathbb{𝔾}}_{米}$ 和 ${\mathbb{𝔾}}_{\mathrm{一种}}$，分别与 $钾$ 它们的通用覆盖的功能场，和 $磷$, $q$ 充当内同态。

用椭圆曲线替换乘法或加法组 $ℂ∕\Lambda$， 和 $钾$ 通过领域的最大无分支扩展 $\Lambda$-椭圆函数，我们研究相似的对象，我们称之为椭圆函数 $\left(磷,q\right)$- 差异 模块。这里$磷$ 和 $q$ 采取行动 $钾$通过同种异体。什么时候$磷$ 和 $q$是素数，我们给出椭圆的结构定理$\left(磷,q\right)$- 差异模块。该证明基于周期性定理，我们在某种程度上证明了它的普遍性。椭圆模的一个新特征是它们的分类结果是在 Atiyah 对椭圆曲线上的向量丛的分类（1957）上进行了纤维化。

只有关联向量丛是平凡的模块才承认 $ℂ$-结构如在 thc 的情况下 ${\mathbb{𝔾}}_{米}$ 或者 ${\mathbb{𝔾}}_{\mathrm{一种}}$，但所有这些都可以在 theta 函数（的对数导数）的帮助下明确描述。我们以 Loxton 和 van der Poorten 猜想的椭圆类似物的证明作为结论。