modular embedding — 1 articles found.

Transcendence of Zeros of Automorphic Forms for Cuspidal Triangle Groups

C. R. Math. Rep. Acad. Sci. Canada Vol. 41 (1) 2019, pp. 1-6
Vol.41 (1) 2019
Paula Tretkoff Details
(Received: 2018-05-21 , Revised: 2018-09-18 )
(Received: 2018-05-21 , Revised: 2018-09-18 )

Paula Tretkoff,Department of Mathematics, Texas A & M University, College Station, TX 77843-3368 USA; CNRS, UMR 8524, Universite de Lille 1, Cite Scientifique, 59655 Villeneuve d'Ascq, France; e-mail: paulatretkoff@tamu.edu

Abstract/Résumé:

We extend some results of on elliptic modular forms. We take any Fuchsian triangle group with a cusp and look at power series expansions in a natural parameter around that cusp. Consider the automorphic forms for such a triangle group whose power series expansions in the natural parameter have algebraic coefficients. We show that the zeros of such forms are either transcendental, or are “CM.” By “CM,” we mean they correspond to abelian varieties with complex multiplication. This result is the first of its kind in the case of non-arithmetic groups.

Nous étendons certains résultats de sur les formes modulaires elliptiques. Nous prenons un groupe fuchsien triangulaire quelconque avec une pointe et examinons les développements en série de puissance dans un paramètre naturel autour de cette pointe. Considérons les formes automorphes pour un tel groupe triangulaire dont les développements en série de puissance dans le paramètre naturel ont des coefficients algébriques. Nous montrons que les zéros de telles formes sont soient transcendants soient “CM”. Par “CM,” nous voulons dire qu’ils correspondent à des variétés abéliennes à multiplication complexe. Ce résultat est le premier du genre au cas des groupes non-arithmétiques.

Keywords: Transcendence, automorphic forms, modular embedding, non-arithmetic group

AMS Subject Classification: Transcendence (general theory), Transcendence theory of other special functions, Fuchsian groups and their generalizations 11J81, 11J91, 20H10

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