# 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 (2006)

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

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topPellarin, 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

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- [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
- [Pel2] F. Pellarin, On the arithmetic properties of complex values of Hecke-Mahler series II, preprint. Zbl1116.11057
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