本文针对 Abel 的代数解判定问题，证明方程存在非平凡代数解当且仅当相关 Puiseux 系数对几乎所有素数满足 Gauss 同余，并将判据用于超几何级数等问题。
摘要：
Abel 提出的一个经典问题是：当 η 是给定的 C(x) 上的代数函数时，判定微分方程 y'=ηy 是否具有一个在 C(x) 上代数的非平凡解 y。Risch 设计了一种算法，可在给定 η 时判定是否存在代数解。本文采用不同视角：当 η 在 C∪{∞} 中某点处具有有理系数 Puiseux 展开时，不失一般性可将该点设为 0。我们证明如下算术刻画：y'=ηy 存在非平凡代数解，当且仅当 xη(x) 在 0 处 Puiseux 展开的系数对几乎所有素数都满足 Gauss 同余。随后，我们将该判据应用于超几何级数：当 xη(x) 是具有有理参数的代数超几何级数时，我们完全确定了具有代数解的方程 y'=ηy，这使我们能够证明 Golyshev 利用 motive 理论作出的一项预测。我们还给出另外三个应用，尤其是对有理函数对角线和二维有向游走的应用。
英文摘要：
A classical problem due to Abel is to determine if a differential equation $y'=ηy$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $η$ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $η$, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $η$ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty\}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y'=ηy$ if and only if the coefficients of the Puiseux expansion of $xη(x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y'=ηy$ with an algebraic solution when $xη(x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present three other applications, in particular to diagonals of rational fractions and to directed two-dimensional walks.
Genus fields of abelian extensions of congruence rational function fields
在这篇论文的公开版本[Finite Fields and Their Applications {\bf 20}（2013）40--54]中，在该论文定理4.2的证明中存在错误。本文对定理4.2，4.5和5.2给出了同余函数域的亏格域的构造。首先，我们考虑割圆函数域的情况下的想法Leopoldt，然后一般情况下。作为应用，我们给出了Kummer，Artin-Schreier和循环$p$-扩张的亏格域. Kummer扩张是由G. Peng和Artin-Schreier扩张由S. Hu和Y.李
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Originally conjectured unpublished by Grothendieck, then formulated precisely by Katz, the $p$-curvature conjecture is a local-global principle for algebraic differential equations. It is at present open, though various cases are known. Katz subsequently proved this conjecture in a wide range of cases, for differential equations corresponding to the Gauss-Manin connection on algebraic de Rham cohomology. This dissertation addresses the non-abelian analogue of Katz' theorem, in the sense of Simpson's non-abelian Hodge theory, surveyed by Simpson and later developed in characteristic $p$ by Ogus…
