Kronecker’s solution of Pell’s equation for CM fields
Annales de l'Institut Fourier, Volume 63 (2013) no. 6, p. 2287-2306

We generalize Kronecker’s solution of Pell’s equation to CM fields K whose Galois group over is an elementary abelian 2-group. This is an identity which relates CM values of a certain Hilbert modular function to products of logarithms of fundamental units. When K is imaginary quadratic, these CM values are algebraic numbers related to elliptic units in the Hilbert class field of K. Assuming Schanuel’s conjecture, we show that when K has degree greater than 2 over these CM values are transcendental.

Nous généralisons la solution de Kronecker des équations Pell aux corps K CM dont le groupe de Galois sur est un 2-groupe abélien élémentaire. Il s’agit d’une formule qui relie les valeurs CM d’une certaine fonction modulaire de Hilbert aux produits de logarithmes des unités fondamentales. Lorsque K est quadratique imaginaire, ces valeurs CM sont des nombres algébriques reliés aux unités elliptiques des corps de classes de Hilbert de K. Sous l’hypothèse que la conjecture de Schanuel soit vraie, nous montrons que, lorsque K et de degré plus grand que 2 sur , ces valeurs CM sont transcendantes.

DOI : https://doi.org/10.5802/aif.2830
Classification:  11F41
Keywords: CM point, Hilbert modular function, Pell’s equation
@article{AIF_2013__63_6_2287_0,
     author = {Masri, Riad},
     title = {Kronecker's solution of Pell's equation for CM fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {6},
     year = {2013},
     pages = {2287-2306},
     doi = {10.5802/aif.2830},
     mrnumber = {3237448},
     zbl = {1295.11044},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2013__63_6_2287_0}
}
Masri, Riad. Kronecker’s solution of Pell’s equation for CM fields. Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2287-2306. doi : 10.5802/aif.2830. http://www.numdam.org/item/AIF_2013__63_6_2287_0/

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