arXiv subjects
Publications and source records attributed to Daniel Vargas-Montoya.
We compute the Galois groups of the reductions modulo the prime numbers $p$ of the generating series of Apéry numbers, Domb numbers and Almkvist--Zudilin numbers. We observe in particular that their behavior is governed by congruence conditions on p.
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $f_1(z),\ldots, f_m(z) \in K[[z]]$ such that, for every $1 \leq i \leq m$, $f_i(z)$ is a solution of a differential operator $\mathcal{L}_i \in E_p[d/dz]$, where $E_p$ is the field of analytic elements. Suppose that $K$ is totally ramified over $\mathbb{Q}_p$, and that for every $1 \leq i \leq m$, the operator $\mathcal{L}_i$ has a strong Frobenius structure and satisfies the maximal order multiplicity (MOM) condition at zero. Then, we show that $f_1(z),\ldots, f_m(z)$ are algebraically dependent over $E_p$ if and only if there exist integers $a_1,\ldots, a_m$, not all zero, such that $f_1(z)^{a_1} \cdots f_m(z)^{a_m}\in E_p$. The main consequence of this result is that it provides a tool to study the algebraic independence of a broad class of $G$-functions and certain $E$-functions over the field of analytic elements.
Let $K$ be a finite extension of $\mathbb{Q}_p$ that is totally ramified over $\mathbb{Q}_p$. The set $\mathcal{M}\mathcal{F}(K)$ consists of power series in $1+zK[[z]]$ that are solutions of differential operators in $K(z)[d/dz]$ equipped with strong Frobenius structure and satisfying maximal order multiplicty (MOM) condition at zero. It turns out that this set contains an interesting class of $E$- and $G$-functions. In this work, we provide a criterion for determining the algebraic independence, over the field of analytic elements, of elements belonging to $\mathcal{M}\mathcal{F}(K)$. As an illustration of this criterion, we show the algebraic independence of some $E$- and $G$-functions over the field of analytic elements.
The aim of this paper is to investigate the algebraicity behavior of reductions of $D$-finite power series modulo prime numbers. For many classes of D-finite functions, such as diagonals of multivariate algebraic series or hypergeometric functions, it is known that their reductions modulo prime numbers, when defined, are algebraic. We formulate a conjecture that uniformizes the Galois groups of these reductions across different prime numbers. We then focus on hypergeometric functions, which serves as a test case for our conjecture. Refining the construction of an annihilating polynomial for the reduction of a hypergeometric function modulo a prime number p, we extract information on the respective Galois groups and show that they behave nicely as p varies.
Let $L$ be a differential operator with coefficients in $\mathbb{Q}(z)$ of order $n\geq2$ with maximal unipotent monodromy at zero. In this paper we are interested in determining when the canonical coordinate of $L$ belongs to $\mathbb{Z}_p[[z]]$. For this purpose, motivated by a recent conjecture due to P. Candelas, X. de la Ossa and D. van Straten~\cite{CD}, we study the situation when $L$ has a strong Frobenius structure $Φ=(ϕ_{i,j})_{1\leq i,j\leq n}\in M_n(\mathbb{Z}_p[[z]])$ such that $ϕ_{1,1}(0)=1$. We then give a necessary and sufficient condition for the canonical coordinate of $L$ to belong to $\mathbb{Z}_p[[z]]$ when $L$ has such a strong Frobenius structure.
The notion of strong Frobenius structure is classically studied in the theory of $p$-adic differential operators. In the present work, we introduce a new definition of the notion of strong Frobenius structure for $q$-difference operators. The relevance of this definition is supported by two main results. The first one deals with \emph{confluence}. We show that if the $q$-difference operator $L_q$ has a strong Frobenius structure for a prime $p$ with period $h$ and if $L$ is the $p$-adic differential operator obtained from $L_q$ by letting $q$ tend to 1, then $L$ has a strong Frobenius structure for $p$ with period $h$. The second one deals with congruence modulo cyclotomic polynomials. We show that if $f(q,z)\in\mathbb{Z}[q][[z]]$ is a solution of a $q$-difference operator having strong Frobenius structure for $p$ then $f(q,z)$ satisfies some congruences modulo the $p$-th cyclotomic polynomial. Another definition of strong Frobenius structures associated with $q$-difference operators has been introduced by André and Di Vizio and we also point out why their definition is not suitable for our applications: confluence and congruence modulo cyclotomic polynomials. Finally, we show that some $q$-hypergeometric operators of order 1 have a strong Frobenius strong for infinitely many primes numbers.