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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
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References:
[1] Asai, Tetsuya, On a certain function analogous to \({\rm log}_{η }\,(z),\) Nagoya Math. J., 40, 193-211, (1970) · Zbl 0213.05701
[2] Bruinier, Jan Hendrik; Yang, Tonghai, CM-values of Hilbert modular functions, Invent. Math., 163, 2, 229-288, (2006) · Zbl 1093.11041
[3] Buell, D. A.; Williams, H. C.; Williams, K. S., On the imaginary bicyclic biquadratic fields with class-number \(2,\) Math. Comp., 31, 140, 1034-1042, (1977) · Zbl 0379.12002
[4] Howard, Benjamin; Yang, Tonghai, Intersections of Hirzebruch-Zagier divisors and CM cycles, 2041, viii+140 pp., (2012), Springer, Heidelberg · Zbl 1238.11069
[5] Konno, Shuji, On kronecker’s limit formula in a totally imaginary quadratic field over a totally real algebraic number field, J. Math. Soc. Japan, 17, 411-424, (1965) · Zbl 0147.03603
[6] Murty, M. Ram; Murty, V. Kumar, Transcendental values of class group \(L\)-functions, Math. Ann., 351, 4, 835-855, (2011) · Zbl 1281.11071
[7] Murty, M. Ram; Murty, V. Kumar, Transcendental values of class group \(L\)-functions, II, Proc. Amer. Math. Soc., 140, 9, 3041-3047, (2012) · Zbl 1282.11082
[8] Siegel, Carl Ludwig, Lectures on advanced analytic number theory, iii+331+iii pp., (1965), Tata Institute of Fundamental Research, Bombay · Zbl 0278.10001
[9] Waldschmidt, Michel, Diophantine approximation on linear algebraic groups, 326, xxiv+633 pp., (2000), Springer-Verlag, Berlin · Zbl 0944.11024
[10] Washington, Lawrence C., Introduction to cyclotomic fields, 83, xiv+487 pp., (1997), Springer-Verlag, New York · Zbl 0484.12001
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