On sums of three squares
James W. Cogdell
Journal de Théorie des Nombres de Bordeaux, Volume 15 (2003) no. 1, p. 33-44

We address the question of when an integer in a totally real number field can be written as the sum of three squared integers from the field and more generally whether it can be represented by a positive definite integral ternary quadratic form over the field. In recent work with Piatetski-Shapiro and Sarnak we have shown that every sufficiently large totally positive square free integer is globally integrally represented if and only if it is so locally at all places, thus essentially resolving the remaining open case of Hilbert's eleventh problem. In this paper we give an exposition of the ideas in the proof of this result.

Nous nous intéressons à la question de savoir quand un entier d'un corps de nombres totalement réel est somme de trois carrés d'entiers du corps, et plus généralement, s'il peut être représenté par une forme quadratique ternaire définie positive et entière sur le corps. Dans un travail récent avec Piatetski-Shapiro et Sarnak, nous avons montré que tout entier totalement positif et sans facteur carré assez grand possède une représentation intégrale globale si et seulement s'il en est de même localement partout, résolvant ainsi pour l'essentiel le dernier cas ouvert du onzième problème de Hilbert. Dans cet article, nous exposons les idées de la démonstration de ce résultat.

@article{JTNB_2003__15_1_33_0,
     author = {Cogdell, James W.},
     title = {On sums of three squares},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {1},
     year = {2003},
     pages = {33-44},
     zbl = {1050.11043},
     mrnumber = {2018999},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_33_0}
}
Cogdell, James W. On sums of three squares. Journal de Théorie des Nombres de Bordeaux, Volume 15 (2003) no. 1, pp. 33-44. https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_33_0/

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