On the arithmetic properties of complex values of Hecke-Mahler series. I. The rank one case

Federico Pellarin

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 3, page 329-374
  • ISSN: 0391-173X

Abstract

top
Here we characterise, in a complete and explicit way, the relations of algebraic dependence over ℚ of complex values of Hecke-Mahler series taken at algebraic points u ̲ 1 , ... , u ̲ m of the multiplicative group 𝔾 m 2 ( ℂ ) , under a technical hypothesis that a certain sub-module of 𝔾 m 2 ( ℂ ) generated by the u ̲ i ’s has rank one (rank one hypothesis). This is the first part of a work, announced in [Pel1], whose main objective is completely to solve a general problem on the algebraic independence of values of these series.

How to cite

top

Pellarin, Federico. "On the arithmetic properties of complex values of Hecke-Mahler series. I. The rank one case." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.3 (2006): 329-374. <http://eudml.org/doc/242951>.

@article{Pellarin2006,
abstract = {Here we characterise, in a complete and explicit way, the relations of algebraic dependence over $\{\mathbb \{Q\}\}$ of complex values of Hecke-Mahler series taken at algebraic points $\underline\{u\}_1,\ldots ,\underline\{u\}_m$ of the multiplicative group $\{\mathbb \{G\}\}_\{\{\rm m\}\}^2(\{\mathbb \{C\}\})$, under a technical hypothesis that a certain sub-module of $\{\mathbb \{G\}\}_\{\{\rm m\}\}^2(\{\mathbb \{C\}\})$ generated by the $\underline\{u\}_i$’s has rank one (rank one hypothesis). This is the first part of a work, announced in [Pel1], whose main objective is completely to solve a general problem on the algebraic independence of values of these series.},
author = {Pellarin, Federico},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {algebraic independence; values of Hecke-Mahler series at algebraic points; formal double Laurent series},
language = {eng},
number = {3},
pages = {329-374},
publisher = {Scuola Normale Superiore, Pisa},
title = {On the arithmetic properties of complex values of Hecke-Mahler series. I. The rank one case},
url = {http://eudml.org/doc/242951},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Pellarin, Federico
TI - On the arithmetic properties of complex values of Hecke-Mahler series. I. The rank one case
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 3
SP - 329
EP - 374
AB - Here we characterise, in a complete and explicit way, the relations of algebraic dependence over ${\mathbb {Q}}$ of complex values of Hecke-Mahler series taken at algebraic points $\underline{u}_1,\ldots ,\underline{u}_m$ of the multiplicative group ${\mathbb {G}}_{{\rm m}}^2({\mathbb {C}})$, under a technical hypothesis that a certain sub-module of ${\mathbb {G}}_{{\rm m}}^2({\mathbb {C}})$ generated by the $\underline{u}_i$’s has rank one (rank one hypothesis). This is the first part of a work, announced in [Pel1], whose main objective is completely to solve a general problem on the algebraic independence of values of these series.
LA - eng
KW - algebraic independence; values of Hecke-Mahler series at algebraic points; formal double Laurent series
UR - http://eudml.org/doc/242951
ER -

References

top
  1. [BS] Z. Borevich and I. Shafarevich, “Number theory”, Academic Press, New York, London, 1966. Zbl0145.04902MR195803
  2. [Ku] K. K. Kubota, On the Algebraic Independence of Holomorphic Solutions of Certain Functional Equations and their Values, Math. Ann. 227 (1977), 9–50. Zbl0359.10030MR498423
  3. [Lo-Po2] J. H. Loxton and A. J. van der Poorten, Arithmetic properties of certain functions in several variables II, J. Australian Math. Soc. A24 (1977), 393–408. Zbl0339.10027MR506055
  4. [Lo-Po3] J. H. Loxton and A. J. van der Poorten, Algebraic independence properties of the Fredholm series, J. Australian Math. Soc. A29 (1978), 31–45. Zbl0392.10034MR510583
  5. [Mah] K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann. 101 (1929), 342–366. Zbl55.0115.01MR1512537JFM55.0115.01
  6. [Mas1] D. W. Masser, A Vanishing theorem for Power Series, Invent. Math. 67 (1982), 275–296. Zbl0481.10034MR665158
  7. [Mas2] D. W. MasserAlgebraic independence properties of the Hecke-Mahler series, Quart. J. Math. Oxford 50 (1999), 207–230. Zbl0929.11021MR1697155
  8. [Ni] K. Nishioka, “Mahler functions and transcendence”, Springer Lecture Notes in Mathematics, Vol. 1631, 1996. Zbl0876.11034MR1439966
  9. [Pel1] F. Pellarin, Propriétés d’indépendance algébrique de valeurs de séries de Hecke-Mahler, C. R. Acad. Sci. Paris, Ser. I. 340 (2005), 861–866. Zbl1098.11040MR2151774
  10. [Pel2] F. Pellarin, On the arithmetic properties of complex values of Hecke-Mahler series II, preprint. Zbl1116.11057
  11. [Per] O. Perron, “Die Lehre von den Kettenbrüchen”, Chelsea Publishing Company, New York, 1957. Zbl0041.18206MR37384

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.