On sums of three squares

James W. Cogdell

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 1, page 33-44
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Cogdell, James W.. "On sums of three squares." Journal de théorie des nombres de Bordeaux 15.1 (2003): 33-44. <http://eudml.org/doc/249109>.

@article{Cogdell2003,
abstract = {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.},
author = {Cogdell, James W.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {sums of squares; integral quadratic forms; theta series; Waldspurger's theorem; subconvexity estimates},
language = {eng},
number = {1},
pages = {33-44},
publisher = {Université Bordeaux I},
title = {On sums of three squares},
url = {http://eudml.org/doc/249109},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Cogdell, James W.
TI - On sums of three squares
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 1
SP - 33
EP - 44
AB - 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.
LA - eng
KW - sums of squares; integral quadratic forms; theta series; Waldspurger's theorem; subconvexity estimates
UR - http://eudml.org/doc/249109
ER -

References

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