no. 6
Kronecker’s solution of Pell’s equation for CM fields
[La solution de Kronecker des équations Pell pour les corps CM]
Masri, Riad1
1 Texas A&M University Department of Mathematics College Station, TX 77843 (USA)
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 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 ℚ, ces valeurs CM sont transcendantes.

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

DOI : 10.5802/aif.2830
Classification : 11F41
Keywords: CM point, Hilbert modular function, Pell’s equation
Mot clés : point CM, fonction modulaire de Hilbert, équation Pell

Masri, Riad 1

1 Texas A&M University Department of Mathematics College Station, TX 77843 (USA) 
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Masri, Riad. Kronecker’s solution of Pell’s equation for CM fields. Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2287-2306. doi : 10.5802/aif.2830. https://aif.centre-mersenne.org/articles/10.5802/aif.2830/

[1] Asai, Tetsuya On a certain function analogous to log η (z), Nagoya Math. J., Volume 40 (1970), pp. 193-211 | MR | Zbl

[2] Bruinier, Jan Hendrik; Yang, Tonghai CM-values of Hilbert modular functions, Invent. Math., Volume 163 (2006) no. 2, pp. 229-288 | DOI | MR | Zbl

[3] Buell, D. A.; Williams, H. C.; Williams, K. S. On the imaginary bicyclic biquadratic fields with class-number 2, Math. Comp., Volume 31 (1977) no. 140, pp. 1034-1042 | MR | Zbl

[4] Howard, Benjamin; Yang, Tonghai Intersections of Hirzebruch-Zagier divisors and CM cycles, Lecture Notes in Mathematics, 2041, Springer, Heidelberg, 2012, pp. viii+140 | MR | Zbl

[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, Volume 17 (1965), pp. 411-424 | DOI | MR | Zbl

[6] Murty, M. Ram; Murty, V. Kumar Transcendental values of class group L-functions, Math. Ann., Volume 351 (2011) no. 4, pp. 835-855 | DOI | MR | Zbl

[7] Murty, M. Ram; Murty, V. Kumar Transcendental values of class group L-functions, II, Proc. Amer. Math. Soc., Volume 140 (2012) no. 9, pp. 3041-3047 | DOI | MR | Zbl

[8] Siegel, Carl Ludwig Lectures on advanced analytic number theory, Notes by S. Raghavan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 23, Tata Institute of Fundamental Research, Bombay, 1965, pp. iii+331+iii | MR | Zbl

[9] Waldschmidt, Michel Diophantine approximation on linear algebraic groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 326, Springer-Verlag, Berlin, 2000, pp. xxiv+633 (Transcendence properties of the exponential function in several variables) | MR | Zbl

[10] Washington, Lawrence C. Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997, pp. xiv+487 | MR | Zbl

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