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-differencemodules. 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.
