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Galois groups of reductions modulo p of D-finite series

Xavier Caruso, Florian Fürnsinn, Daniel Vargas-Montoya

TL;DR

The paper investigates reductions modulo primes of D-finite power series, focusing on when reductions are algebraic and how their Galois groups vary with p. It proposes a framework predicting uniform Galois groups across primes via a finite family of subgroups in GL_n(K) and refines this with Galois-equivariant and prime-partition concepts. Hypergeometric functions are used as test cases, with explicit annihilating polynomials Z_p and a rich interplay between Dwork maps, relation graphs, Katz's functor, and Frobenius structures to describe the reductions and their Galois representations. The results yield both upper and lower bounds on Galois groups, and in the Gaussian 2F1 case, provide concrete uniformization results for many primes, illustrating how differential and p-adic techniques illuminate reductions of D-finite series and their arithmetic structure.

Abstract

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.

Galois groups of reductions modulo p of D-finite series

TL;DR

The paper investigates reductions modulo primes of D-finite power series, focusing on when reductions are algebraic and how their Galois groups vary with p. It proposes a framework predicting uniform Galois groups across primes via a finite family of subgroups in GL_n(K) and refines this with Galois-equivariant and prime-partition concepts. Hypergeometric functions are used as test cases, with explicit annihilating polynomials Z_p and a rich interplay between Dwork maps, relation graphs, Katz's functor, and Frobenius structures to describe the reductions and their Galois representations. The results yield both upper and lower bounds on Galois groups, and in the Gaussian 2F1 case, provide concrete uniformization results for many primes, illustrating how differential and p-adic techniques illuminate reductions of D-finite series and their arithmetic structure.

Abstract

The aim of this paper is to investigate the algebraicity behavior of reductions of -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.

Paper Structure

This paper contains 46 sections, 38 theorems, 203 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation.
  3. Acknowledgements.
  4. A general picture of what we expect
  5. Getting intuition by examples
  6. $f(x) = (1-x)^{a/d}$
  7. $f(x) = \exp(\arctan x)$
  8. $f(x) = \sum_{n=0}^\infty \binom{2n}{n}^{\!2} \cdot x^n$
  9. $f(x) = \sum_{n=0}^\infty \binom{2n}{n}^{\!3} \cdot x^n$
  10. $f(x) = \sum_{n=0}^\infty \sum_{k=0}^n \binom{n+k}{k}^{\!2} \binom{n}{k}^{\!2} \cdot x^n$ and related functions
  11. $f(x) = {}_{3}F_{2}\left((\frac{1}{9}, \frac{4}{9}, \frac{5}{9}), (\frac{1}{3}, 1); x\right)$
  12. The conjectures: weak and strong forms
  13. Discarding too naive expectations
  14. First formulation of the conjecture
  15. Back to the examples.
  16. ...and 31 more sections

Key Result

Theorem 1.1

Let $n$ be a positive integer. Let $\boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_n)$ and $\boldsymbol{\beta} = (\beta_1, \ldots, \beta_{n-1})$ be tuples of rational numbers with $\alpha_i, \beta_j \not\in -\mathbb{N}$ for all $i,j$. Let $d$ be the smallest common denominator of the $\alpha_i$ and Let $p>2d{\cdot}\max\{|\alpha_i|+1, |\beta_j|+1\}$ be a prime number such that ${}_{n}F_{n-1}\left(

Figures (7)

  • Figure 1: From differential equations in characteristic $0$ to Galois groups in characteristic $p$
  • Figure 2: The function $\mathcal{M}(\cdot, 1)$ for the function ${}_{3}F_{2}\left((\frac{1}{6}, \frac{2}{3}, \frac{4}{3}), (\frac{1}{3}, \frac{1}{2}); x\right)$ in the interval $[0, 3]$
  • Figure 3: Christol's interlacing condition depicted on the unit circle for the function ${}_{3}F_{2}\left((\frac{1}{9}, \frac{4}{9}, \frac{5}{9}), (\frac{1}{3}, 1); x\right)$
  • Figure 4: Christol's interlacing condition illustrated on the unit circle for the function $f(x)={}_{2}F_{1}\left((\frac{1}{2} , \frac{2}{3}), (\frac{1}{3}); x\right)$ (on the left) and the function $g(x)={}_{2}F_{1}\left((\frac{1}{4}, \frac{1}{2}), (\frac{15}{4}); x\right)$ (on the right)
  • Figure 5: The sets $\{\exp(2\mathbf i\pi \lambda \alpha_i)\}$ and $\{\exp(2\mathbf i\pi \lambda \beta_j)\}$ for ${}_{3}F_{2}\left((\frac{1}{8}, \frac{3}{8}, \frac{1}{2}), (\frac{1}{4}, \frac{5}{8}); x\right)$
  • ...and 2 more figures

Theorems & Definitions (113)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.0.1
  • Proposition 2.1.1
  • proof
  • Remark 2.1.2
  • Remark 2.1.3
  • Lemma 2.1.4
  • proof
  • Corollary 2.1.5
  • ...and 103 more