椭圆函数域上的超超越性与$q$-差分方程
Hypertranscendence and $q$-difference equations over elliptic functionfields
AI总结：
该文研究椭圆函数域上满足$q$-差分方程的幂级数的微分性质，证明其满足系数在椭圆函数域中的多项式微分方程当且仅当属于特定生成环，推广了射影直线上的相关结果并得到椭圆曲线差分-微分系统的可积性结论。
摘要：
射影直线上线性差分方程解的微分性质近期已得到阐明。相比之下，人们对椭圆曲线上线性差分方程解的微分性质知之甚少。本文研究复系数幂级数$f(z)$，其满足椭圆函数域$K$上的线性差分方程，对应的差分算子为$ϕf(z)=f(qz)$（其中$2\b q\bm{\b\b Z}$，该算子源于椭圆曲线的自同态）。我们的主要定理指出：此类$f$还满足系数取自$K$的多项式微分方程，当且仅当它属于由$z$、$z^{-1}$和魏尔斯特拉斯$ζ$-函数在$K$上生成的环$S=K[z,z^{-1},ζ(z,Λ)]$。这是Adamczewski、Dreyfus与Hardouin近期相关定理的首个椭圆情形推广——原定理针对系数在$\b C(z)$中的差分方程解的微分超越性，涵盖了移位算子、$q$-差分算子、Mahler算子等多种差分算子。尽管二者均采用参数化Picard-Vessiot理论的通用方法，但椭圆情形存在诸多独有特征，尤其是单值性考量的出现与环$S$的引入，带来了非平凡的困难。我们特别指出，在中间结果中，我们证明了椭圆曲线上差分-微分系统的一个可积性结果，它是Schäfke与Singer在射影直线上得到的可积性结果的亏格1类比。
英文摘要：
The differential nature of solutions of linear difference equations over the projective line was recently elucidated. In contrast, little is known about the differential nature of solutions of linear difference equations over elliptic curves. In the present paper, we study power series $f(z)$ with complex coefficients satisfying a linear difference equation over a field of elliptic functions $K$,with respect to the difference operator $ϕf(z)=f(qz)$, $2\le q\in\mathbb{Z}$,arising from an endomorphism of the elliptic curve. Our main theoremsays that such an $f$ satisfies, in addition, a polynomial differentialequation with coefficients from $K,$ if and only if it belongs tothe ring $S=K[z,z^{-1},ζ(z,Λ)]$ generated over $K$ by$z,z^{-1}$ and the Weierstrass $ζ$-function. This is the first elliptic extension of recent theorems of Adamczewski, Dreyfus and Hardouin concerning the differential transcendence of solutions of difference equations with coefficients in $\mathbb{C}(z),$ in which various difference operators were considered (shifts, $q$-differenceoperators or Mahler operators). While the general approach, of usingparametrized Picard-Vessiot theory, is similar, many features, andin particular the emergence of monodromy considerations and the ring$S$, are unique to the elliptic case and are responsible for non-trivial difficulties. We emphasize that, among the intermediate results, we prove an integrability result for difference-differential systems over ellipticcurves which is a genus one analogue of the integrability results obtained by Sch\''afke and Singer over the projective line.
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