The Hasse Norm Principle For Biquadratic Extensions
Nick Rome
Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 947-964.

Nous donnons une formule asymptotique pour le nombre d’extensions biquadratiques du corps des rationnels de discriminant borné qui contredisent le principe de norme de Hasse.

We give an asymptotic formula for the number of biquadratic extensions of the rationals of bounded discriminant that fail the Hasse norm principle.

Reçu le : 2017-06-29
Révisé le : 2018-03-25
Accepté le : 2018-06-05
Publié le : 2019-03-28
DOI : https://doi.org/10.5802/jtnb.1058
Classification : 11N25,  11R16
Mots clés : Hasse norm theorem, Biquadratic extensions, character sums 
@article{JTNB_2018__30_3_947_0,
     author = {Nick Rome},
     title = {The Hasse Norm Principle For Biquadratic Extensions},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {947--964},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1058},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2018__30_3_947_0/}
}
Nick Rome. The Hasse Norm Principle For Biquadratic Extensions. Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 947-964. doi : 10.5802/jtnb.1058. https://jtnb.centre-mersenne.org/item/JTNB_2018__30_3_947_0/

[1] Andrew Baily On the density of discriminants of quartic fields, J. Reine Angew. Math., Volume 315 (1980), pp. 190-210 | Zbl 0421.12007

[2] Algebraic Number Theory (J. W. S. Cassels; Albrecht Fröhlich, eds.), Academic Press Inc., 1967, xviii+366 pages | Zbl 0153.07403

[3] Henri Cohen; Francisco Diaz y Diaz; Michel Olivier A survey of discriminant counting, Algorithmic number theory (Lecture Notes in Computer Science) Volume 2369, Springer, 2002, pp. 80-94 | Zbl 1058.11076

[4] Christopher Frei; Daniel Loughran; Rachel Newton The Hasse norm principle for abelian extensions, Am. J. Math., Volume 140 (2018) no. 6, pp. 1639-1685 | Zbl 07018404

[5] John Friedlander; Henryk Iwaniec Opera de Cribro, Colloquium Publications, Volume 57, American Mathematical Society, 2004, xx+527 pages | Zbl 1226.11099

[6] John Friedlander; Henryk Iwaniec Ternary quadratic forms with rational zeros, J. Théor. Nombres Bordx, Volume 22 (2010) no. 1, pp. 97-113 | Zbl 129.11060

[7] Albrecht Fröhlich; Martin J. Taylor Algebraic Number Theory, Cambridge Studies in Advanced Mathematics, Volume 27, Cambridge University Press, 1991, xiv+355 pages | Zbl 0744.11001

[8] Henryk Iwaniec; Emmanuel Kowalski Analytic Number Theory, Colloquium Publications, Volume 53, American Mathematical Society, 2004, xi+615 pages | Zbl 1059.11001

[9] Gérald Tenenbaum Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, Volume 46, Cambridge University Press, 1995, xiv+448 pages | Zbl 0831.11001