Arithmetic properties of formal power series modulo p and the Artin-Hasse exponential

The thesis investigates the arithmetic properties of formal power series over fields of positive characteristic, with a particular focus on the transcendence properties of the Artin-Hasse exponential modulo $p$. Let $k$ be a field of characteristic $p$ and let $t_1,\ldots ,t_n$ be indeterminates over $k$. We begin by presenting a criterion, due to T. Harase, for determining whether an element of $k(\!(t_1,\ldots,t_n)\!)$ is algebraic over $k(t_1,\ldots,t_n)$. This result generalizes a celebrated theorem by G. Christol, which characterizes the algebraic power series in $\mathbb{F}_p[\![t]\!]$ as those whose coefficient sequences are generated by a $p$-automation. We apply this result to generalize classical transcendence criteria and to provide new proofs of known results in this field. In addition, we present some concrete examples of applications, touching upon aspects of Carlitz module theory. In parallel, we examine the diagonals of formal power series. The second part of the thesis focuses on the Artin-Hasse exponential $E_p=E_p(t)$: a formal power series whose coefficients are shown to be $p$-adic integers. We briefly describe its analytic properties as a $p$-adic function and then, following a 2024 preprint by J. Kramer-Miller, we discuss the transcendence properties of its reduction modulo $p$, which we denote by $\overline{E}_p$. As a first result, we show that $\overline{E}_p$ is transcendental over $\mathbb{F}_p(t)$, affirmatively answering a question posed by D. Thakur in the early 2000s. For $p\ge5$, an elementary proof is obtained using the transcendence criterion mentioned above. In the general case, the proof relies on iterative Frobenius equations and iterative derivations on $\mathbb{F}_p(\!(t)\!)$. The final part is devoted to two algebraic independence results by J. Kramer-Miller. The first states that for $f_1,\ldots,f_r\in t\,\mathbb{F}_p[t]$ satisfying certain linear independence conditions, the power series $\overline{E}_p(f_1),\ldots,\overline{E}_p(f_r)$ are algebraically independent. The second concerns algebraic relations among the series $\overline{E}_p(ct)$, where $c\in\mathbb{F}_p^\times$. Both proofs require the use of iterative differential Galois theory and the Tannakian interpretation of the iterative differential Galois group. We introduce the theory of iterative differential modules and their Galois groups, then describe the Tannakian interpretation of the iterative differential Galois group, providing a concrete, self-contained proof that avoids the full formalism of Tannakian categories. Using this framework, the proofs reduce to demonstrating the transcendence of the product of certain evaluations of $\overline{E}_p$, which can be achieved using the same techniques employed to prove the transcendence of $\overline{E}_p$.