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Hermite's approach to Abelian integrals revisited

Makoto Kawashima

Abstract

In this article, we establish a new linear independence criterion for the values of certain {\it Lauricella hypergeometric series} $F_D$ with rational parameters, in both the complex and $p$-adic settings, over an algebraic number field. This result generalizes a theorem of C.~Hermite \cite{Hermite} on the linear independence of certain Abelian integrals. Our proof relies on explicit Padé type approximations to solutions of a reducible Jordan-Pochhammer differential equation, which extends the Padé approximations for certain Abelian integrals in \cite{Hermite}. The main novelty of our approach lies in the proof of the non-vanishing of the determinants associated with these Padé type approximants.

Hermite's approach to Abelian integrals revisited

Abstract

In this article, we establish a new linear independence criterion for the values of certain {\it Lauricella hypergeometric series} with rational parameters, in both the complex and -adic settings, over an algebraic number field. This result generalizes a theorem of C.~Hermite \cite{Hermite} on the linear independence of certain Abelian integrals. Our proof relies on explicit Padé type approximations to solutions of a reducible Jordan-Pochhammer differential equation, which extends the Padé approximations for certain Abelian integrals in \cite{Hermite}. The main novelty of our approach lies in the proof of the non-vanishing of the determinants associated with these Padé type approximants.

Paper Structure

This paper contains 17 sections, 147 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Padé-type approximants of Laurent series
  4. Formal $f$-integration map
  5. Holonomic series and kernel of $f$-integration map
  6. Order $1$ differential operator and Padé-type approximants
  7. Linear independence of Padé-type approximants
  8. Study of kernels of $\varphi_j$
  9. Proof of Theorem $\ref{['nonvanish Delta']}$
  10. Estimates
  11. Absolute values of the Padé approximants
  12. Proof of Proposition \ref{['estimate PQ']} $(i)$
  13. Poincaré-Perron type recurrence
  14. Proof of Proposition $\ref{['estimate PQ']}$ $(ii)$
  15. Absolute values of the Padé approximations
  16. ...and 2 more sections

Theorems & Definitions (20)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of Theorem $\ref{['nonvanish Delta']}$
  • proof
  • proof
  • proof
  • ...and 10 more