On sums of three squares
Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 33-44.

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.

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.

@article{JTNB_2003__15_1_33_0,
     author = {James W. Cogdell},
     title = {On sums of three squares},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {33--44},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {1},
     year = {2003},
     zbl = {1050.11043},
     mrnumber = {2018999},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_33_0/}
}
TY  - JOUR
AU  - James W. Cogdell
TI  - On sums of three squares
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2003
SP  - 33
EP  - 44
VL  - 15
IS  - 1
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_33_0/
LA  - en
ID  - JTNB_2003__15_1_33_0
ER  - 
%0 Journal Article
%A James W. Cogdell
%T On sums of three squares
%J Journal de théorie des nombres de Bordeaux
%D 2003
%P 33-44
%V 15
%N 1
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_33_0/
%G en
%F JTNB_2003__15_1_33_0
James W. Cogdell. On sums of three squares. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 33-44. https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_33_0/

[1] M. Baruch, Z. Mao, Central value of automorphic L-functions, preprint, 2000.

[2] J.-L. Brylinkski, J.-P. Labesse, Cohomologie d'intersection et fonctions L de certaines variétés de Shimura. Ann. Sci. École Norm. Sup.(4) 17 (1984), 361-412. | Numdam | MR | Zbl

[3] J.W. Cogdell, I.I. Piatetski-Shapiro, P. Sarnak, Estimates on the critical line for Hilbert modular L-functions and applications, in preparation.

[4] E.N. Donkar, On sums of three integral squares in algebraic number fields. Amer. J. Math., 99 (1977), 1297-1328. | MR | Zbl

[5] W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math., 92 (1988), 73-90. | MR | Zbl

[6] W. Duke, R. Schulze-Pillot, Representations of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 89 (1990), 49-57. | MR | Zbl

[7] J.S. Hsia, Representations by spinor genera. Pacific J. Math. 63 (1976), 147-152. | MR | Zbl

[8] J.S. Hsia, Y. Kitaoka, M. Kneser, Representations of positive definite quadratic forms. J. Reine Angew. Math. 301 (1978), 132-141. | MR | Zbl

[9] H. Iwaniec, Fourier coefficients of modular forms of half integral weight. Invent. Math. 87 (1987), 385-401. | MR | Zbl

[10] H. Iwaniec, P. Sarnak, Perspectives on the analytic theory of L-functions. Geom. Funct. Anal. (GAFA), Special Volume, Part II (2000), 705-741. | MR | Zbl

[11] H. Kim, F. Shahidi, Cuspidality of symmetric powers of GL(2) with applications. Duke Math. J. 112 (2002), 177-197. | MR | Zbl

[12] M. Kneser, Darstellungsmasse indefiniter quadratischer Formen. Math. Zeit. 77 (1961), 188-194. | MR | Zbl

[13] P. Michel, Familles de fonctions L de formes automorphes et applications. J. Théor. Nombres Bordeaux 15 (2003), 275-307. | Numdam | MR | Zbl

[14] Y. Petridis, P. Sarnak, Quantum unique ergodicity for SL2(O)\H3 and estimates for L-functions. J. Evol. Equ. 1 (2001), 277-290. | MR | Zbl

[15] D. Rohrlich, Non-vanishing of L-functions for GL(2). Invent. Math. 97 (1989), 381-403. | Zbl

[16] P. Sarnak, Integrals of products of eigenfunctions. Internat. Math. Res. Notices (1994) 251-260. | MR | Zbl

[17] P. Sarnak, Estimates for Rankin-Selberg L-functions and quantum unique ergodicity. J. Funct. Anal 184 (2001), 419-453. | MR | Zbl

[18] R. Schulze-Pillot, Thetareihen positiv definiter quadratischer Formen. Invent. Math. 75 (1984), 283-299. | MR | Zbl

[19] R. Schulze-Pillot, Darstellungsmasse von Spinorgeschlechtern ternärer quadratischer Formen. J. Reine Angew. Math. 352 (1984), 114-132. | MR | Zbl

[20] R. Schulze-Pillot, Ternary quadratic forms and Brandt matrices. Nagoya Math. J. 102 (1986), 117-126. | MR | Zbl

[21] A. Selberg, On the estimation of Fourier coefficients of modular forms. Proc. Sympos. Pure Math., Vol. VIII, pp. 1-15 Amer. Math. Soc., Providence, R.I., 1965. | MR | Zbl

[22] G. Shimura, On the Fourier coefficients of Hilbert modular forms of half-integral weight. Duke Math. J. 72 (1993), 501-557. | MR | Zbl

[23] C.L. Siegel, Über die analytische Theorie der quadratischer Formen I. Ann. of Math. 36 (1935), 527-606; II, 37 (1936), 230-263; III, 38 (1937) 212-291. | MR

[24] C.L. Siegel, Indefinite quadratische Formen und Funktionentheorie I. Math. Ann. 124 (1951) 17-54; II, 366-387. | MR | Zbl

[25] J.-L. Waldspurger, Correspondance de Shimura. J. Math. Pures Appl. 59 (1980), 1-113. | MR | Zbl

[26] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures Appl. 60 (1981), 375-484. | MR | Zbl

[27] L. Walling, A remark on differences of theta series. J. Number Theory 48 (1994), 243-251. | MR | Zbl