A critical value on an abelian group $G$ of odd order $d$ is a value $\lambda $ such that the functional equation $f\star f (2\,t) =\lambda f(t)^2$ on $G$ has a nonzero solution $f$. We construct many critical values by using abelian varieties with complex multiplication.
Une valeur $\lambda $ est dite critique sur un groupe abélien d’ordre impair $G$ si l’équation fonctionnelle $f\star f (2\,t) =\lambda f(t)^2$ sur $G$ a une solution $f\ne 0$. Nous construisons de nombreuses valeurs critiques à l’aide des variétés abéliennes à multiplication complexe.
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Keywords: Functional Equation, Convolution, Abelian variety, Torsion groups, Complex multiplication, Theta functions, Weil number, Modular variety.
Yves Benoist 1
CC-BY-ND 4.0
Yves Benoist. Convolution and square in abelian groups II. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 355-409. doi: 10.5802/jtnb.1365
@article{JTNB_2026__38_2_355_0,
author = {Yves Benoist},
title = {Convolution and square in abelian groups {II}},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {355--409},
year = {2026},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {38},
number = {2},
doi = {10.5802/jtnb.1365},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1365/}
}
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