TY - JOUR

T1 - Elliptic (p,q)-difference modules

AU - De Shalit, Ehud

N1 - Funding Information: I would like to thank David Kazhdan and Kiran Kedlaya for helpful discussions related to this work. I would also like to thank the referee for making useful suggestions on the exposition. The author was supported by ISF grant 276/17. Publisher Copyright: © 2021 Mathematical Sciences Publishers.

PY - 2021

Y1 - 2021

N2 - Let p and q be multiplicatively independent natural numbers, and K the field ¢(x 1/s | s = 1, 2, 3…). Let p and q act on K as the Mahler operators x ›→ x p and x ›→ xq. 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 ¢. 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 = ¢(x), and p and q are complex numbers of absolute value greater than 1, acting on K via dilations x ›→ px and x ›→ qx, a similar theorem has been obtained by Bézivin and Boutabaa (1992). Underlying these two examples are the algebraic groups Gm and Ga, 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 ¢/Λ, and K by the maximal unramified extension of the field of Λ-elliptic functions, we study similar objects, which we call elliptic (p, q)-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 (p, q)-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 Gm or Ga, 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.

AB - Let p and q be multiplicatively independent natural numbers, and K the field ¢(x 1/s | s = 1, 2, 3…). Let p and q act on K as the Mahler operators x ›→ x p and x ›→ xq. 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 ¢. 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 = ¢(x), and p and q are complex numbers of absolute value greater than 1, acting on K via dilations x ›→ px and x ›→ qx, a similar theorem has been obtained by Bézivin and Boutabaa (1992). Underlying these two examples are the algebraic groups Gm and Ga, 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 ¢/Λ, and K by the maximal unramified extension of the field of Λ-elliptic functions, we study similar objects, which we call elliptic (p, q)-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 (p, q)-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 Gm or Ga, 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.

KW - Difference equations

KW - Elliptic functions

UR - http://www.scopus.com/inward/record.url?scp=85128957165&partnerID=8YFLogxK

U2 - https://doi.org/10.2140/ant.2021.15.1303

DO - https://doi.org/10.2140/ant.2021.15.1303

M3 - Article

SN - 1937-0652

VL - 15

SP - 1303

EP - 1342

JO - Algebra and Number Theory

JF - Algebra and Number Theory

IS - 5

ER -