arXiv subjects

Gabriel Lepetit

Publications and source records attributed to Gabriel Lepetit.

4 recordsLinked to original sources

arXiv2021

Le théorème d'André-Chudnovsky-Katz

The subject of this Master 2 thesis is the study of the André-Chudnovsky-Katz theorem on the structure of the solution of the nonzero differential equation of minimal order with coefficients in $\overline{\mathbb{Q}}(z)$ satisfied by a $G$-function. We begin by presenting the theory of globally nilpotent differential operators, of which the main result is the Katz theorem, which states that they are fuchsian with rational exponents. We then give a full proof of the Chudnovsky theorem implying that the minimal nonzero differential operator with coefficients in $\overline{\mathbb{Q}}(z)$ of a $G$-function satisfies a moderate growth condition on some denominators called the Galochkin condition. We finally outline the proof of the André-Bombieri theorem establishing the equivalence between the Galochkin condition and the Bombieri condition, which implies the global nilpotence. This allows us to prove the main point of the André-Chudnovsky-Katz theorem.

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arXiv2021

Quantitative problems on the size of $G$-operators

$G$-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel's $G$-functions, satisfy a condition of moderate growth called Galochkin condition, encoded by a $p$-adic quantity, the size. Previous works of Chudnovsky, André and Dwork have provided inequalities between the size of a $G$ -operator and certain computable constants depending among others on its solutions. First, we recall André's idea to attach a notion of size to differential modules and detail his results on the behavior of the size relatively to the standard algebraic operations on the modules. This is the corner stone to prove a quantitative version of André's generalization of Chudnovsky's Theorem: for $f(z)=\sum_{α, k,\ell} c_{α, k,\ell} z^α \log(z)^k f_{α, k,\ell}(z)$, where $f_{α, k,\ell}(z)$ are $G$-functions, we can determine an upper bound on the size of the minimal operator $L$ over $\overline{\mathbb{Q}}(z)$ of $f(z)$ in terms of quantities depending on the $f_{α, k,\ell}(z)$, the rationals $α$ and the integers $k$. We give two applications of this result: we estimate the size of a product of two $G$-operators in function of the size of each operator; we also compute a constant appearing in a Diophantine problem encountered by the author.

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arXiv2021

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 give a new proof of a generalization by André of the Siegel-Shidlovskii theorem on the algebraic independence of the values of $E$-functions in the broad sense.

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arXiv2021

On the linear independence of values of $G$-functions

We consider a $G$-function $F(z)=\sum_{k=0}^{\infty} A_k z^k \in \mathbb{K}[[z]]$, where $\mathbb{K}$ is a number field, of radius of convergence $R$ and annihilated by the $G$-operator $L \in \mathbb{K}(z)[\mathrm{d}/\mathrm{d}z]$, and a parameter $β\in \mathbb{Q} \setminus \mathbb{Z}_{\leqslant 0}$. We define a family of $G$-functions $F_{β,n}^{[s]}(z)=\sum_{k=0}^{\infty} \frac{A_k}{(k+β+n)^s} z^{k+n}$ indexed by the integers $s$ and $n$. Fix $α\in \mathbb{K}^* \cap D(0,R)$. Let $Φ_{α,β,S}$ be the $\mathbb{K}$-vector space generated by the values $F_{β,n}^{[s]}(α)$, $n \in \mathbb{N}$, $0 \leqslant s \leqslant S$. We show that there exist some positive constants $u_{\mathbb{K},F,β}$ and $v_{F,β}$ such that $u_{\mathbb{K},F,β} \log(S) \leqslant \dim_{\mathbb{K}} Φ_{α,β,S} \leqslant v_{F,β} S$. This generalizes a previous theorem of Fischler and Rivoal (2017), which is the case $β=0$. Our proof is an adaptation of their article "Linear independence of values of $G$-functions'' ([FR]), making use of the André-Chudnovsky-Katz Theorem on the structure of the $G$-operators and of the saddle point method.

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