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Kronecker’s solution of Pell’s equation for CM fields. (La solution de Kronecker des équations Pell pour les corps CM.) (English. French summary) Zbl 1295.11044
Summary: 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.
##### MSC:
 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11R42 Zeta functions and $$L$$-functions of number fields
##### Keywords:
CM point; Hilbert modular function; Pell’s equation
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##### References:
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