Abstract
Let and be multiplicatively independent natural numbers, and the field . Let and act on as the Mahler operators and . Schäfke and Singer (2019) showed that a finite-dimensional vector space over , carrying commuting structures of a -Mahler module and a -Mahler module, is obtained via base change from a similar object over . 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 , and and are complex numbers of absolute value greater than 1, acting on via dilations and , a similar theorem has been obtained by Bézivin and Boutabaa (1992). Underlying these two examples are the algebraic groups and , respectively, with the function field of their universal covering, and , acting as endomorphisms.
Replacing the multiplicative or additive group by the elliptic curve , and by the maximal unramified extension of the field of -elliptic functions, we study similar objects, which we call elliptic -difference modules. Here and act on via isogenies. When and are relatively prime, we give a structure theorem for elliptic -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 -structure as in thc case of or , 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.
Citation
Ehud de Shalit. "Elliptic -difference modules." Algebra Number Theory 15 (5) 1303 - 1342, 2021. https://doi.org/10.2140/ant.2021.15.1303
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