A refined version of the Siegel-Shidlovskii theorem

Exceptional values of E-functions at algebraic points

E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with rational functions coefficients. They were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function, and studied further by Shidlovskii in 1956. The celebrated Siegel-Shidlovskii Theorem deals with the algebraic (in)dependence of values at algebraic points of E-functions solutions of a differential system. However, somewhat paradoxically, this deep result may fail to decide whether a…

An analogue of Siegel's determinant

Siegel-Shidlovskii theory of $E$-functions involves a non-vanishing proof for the determinants attached to the linear forms $D^kR(t)$, derivatives of an auxiliary function $R(t)$. Let a non-zero function $F(t)$ satisfy $m$th order linear differential equation which we shall write using the differential operator $Δ=tD$ and let $L(t)$ be any non-zero linear form of the derivatives $Δ^i F(t)$ $(i=0,...,m-1; m\ge 2)$. The determinants $\det\mathcal A_k$ attached to the linear forms $Δ^kL(t)$ have certain simple properties that allow us to give a short proof for the non-vanishing of $\det\mathcal A…

Le théorème d'André-Chudnovsky-Katz « au sens large »

Siegel's $E$- and $G$-functions were defined in two conjecturally equivalent senses, strict and broad. By taking up and completing a sketch of André, we state and prove the analogue in the broad sense of the André-Chudnovsky-Katz theorem, which is a structure theorem on the $G$-operators in the broad sense (they are differential operators cancelling the $G$-functions in the strict sense). We deduce from that a structure theorem on the $E$-operators in the broad sense, which are differential operators cancelling the $E$-functions in the broad sense. As an application of this last theorem, we gi…

E-函数值的有效代数无关性

Effective algebraic independence of values of E-functions

本文证明了存在一种算法，能够计算 E-函数之间及其在代数点处取值的多项式关系理想生成系，并确定使取值代数相关的代数点集合，核心依赖于 Hrushovski-Feng 算法变体与 Beukers 提升定理。

二次西格尔尖型形式的傅里叶-雅可比周期的显式改进Gan--Gross--Prasad恒等式

An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms

该研究针对Sp₄的傅里叶-雅可比周期，在新分歧情形下计算相关局部积分，构建了关联二次西格尔尖型形式与半整数权形式Petersson范数的显式猜想恒等式，并指出其在范数增长等方面的推论。

关于 $E$-函数的 Lindemann-Weierstrass 定理

A Lindemann-Weierstrass theorem for $E$-functions

本文利用 André 的 $E$-算子理论与 Beukers 的结果，证明了 $E$-函数在线性独立性表述下的 Lindemann-Weierstrass 定理，并得出整超几何函数在代数论证处的所有超越值在 $\overline{\mathbb{Q}}$ 上线性独立的推论。