Kronecker’s solution of Pell’s equation for CM fields

Masri, Riad

Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2287-2306.

Nous généralisons la solution de Kronecker des équations Pell aux corps $K$ CM dont le groupe de Galois sur $\mathbb{Q}$ 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 $\mathbb{Q}$, ces valeurs CM sont transcendantes.

We generalize Kronecker’s solution of Pell’s equation to CM fields $K$ whose Galois group over $\mathbb{Q}$ 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 $\mathbb{Q}$ these CM values are transcendental.

Classification : 11F41
Keywords: CM point, Hilbert modular function, Pell’s equation
Mot clés : point CM, fonction modulaire de Hilbert, équation Pell

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