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Quadratic Diophantine equations, the class number, and the mass formula


Author: Goro Shimura
Journal: Bull. Amer. Math. Soc. 43 (2006), 285-304
MSC (2000): Primary 11E12; Secondary 11D09, 11E41
DOI: https://doi.org/10.1090/S0273-0979-06-01107-4
Published electronically: April 14, 2006
MathSciNet review: 2223009
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  • 1. P. Bachmann, Die Arithmetik der Quadratischen Formen, Leipzig, Teubner, 1898.
  • 2. A. Borel, Some finiteness properties of adele groups over number fields, Publ. Math. IHES, No. 16, 1963. MR 0202718 (34:2578)
  • 3. Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York and London, 1966. MR 0195803 (33:4001)
  • 4. L. Dirichlet, Vorlesungen über Zahlentheorie, supplement by R. Dedekind, Braunschweig, 1894.
  • 5. E. N. Donkar, On sums of three integral squares in algebraic number fields, Amer. J. Math. 99 (1977), 1297-1328. MR 0460286 (57:280)
  • 6. M. Eichler, Quadratische Formen und orthogonale Gruppen, Springer, Berlin, 1952; 2nd ed., 1974. MR 0351996 (50:4484)
  • 7. C. F. Gauss, Disquisitiones Arithmeticae; 1801; English translation by A. A. Clarke, Yale Univ. Press, 1966. MR 0197380 (33:5545)
  • 8. G. H. Hardy, On the representation of a number as the sum of any number of squares, and in particular of five, Trans. Amer. Math. Soc. 21 (1920), 255-284. MR 1501144
  • 9. Hardy, Errata, Trans. Amer. Math. Soc. 29 (1927), 845-847. MR 1500501
  • 10. E. Hecke, Vorlesungen über die Theorie der Algebraischen Zahlen, Leipzig, 1923; Chelsea, New York, 1948. MR 0352036 (50:4524)
  • 11. S. Kudla and S. Rallis, On the Weil-Siegel formula, J. Reine Angew. Math. 387 (1988), 1-68. MR 0946349 (90e:11059)
  • 12. Lagrange, Recherches d'Arithmétique, 1773-75, Oeuvres III, 695-758.
  • 13. Legendre, Recherches d'Analyses Indéterminée, 1785.
  • 14. S. Rallis, $ L$-functions and the oscillator representation, Lecture Notes in Math., No. 1245, Springer, 1987. MR 0887329 (89b:11046)
  • 15. G. Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973), 440-481 (= Collected Papers II, 532-573). MR 0332663 (48:10989)
  • 16. G. Shimura, On the Eisenstein series of Hilbert modular groups, Revista Matemática Iberoamericana 1 (1985), 1-42 (= Collected Papers III, 644-685). MR 0836282 (87h:11038)
  • 17. G. Shimura, Euler Products and Eisenstein Series, CBMS Regional Conference Series in Mathematics, No. 93, Amer. Math. Soc., 1997. MR 1450866 (98h:11057)
  • 18. G. Shimura, An exact mass formula for orthogonal groups, Duke Math. J. 97 (1999), 1-66 (= Collected Papers IV, 509-574). MR 1681092 (2000a:11073)
  • 19. G. Shimura, The number of representations of an integer by a quadratic form, Duke Math. J. 100 (1999), 59-92 (= Collected Papers IV, 575-608). MR 1714755 (2001f:11057)
  • 20. G. Shimura, Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, vol. 82, Amer. Math. Soc., 2000. MR 1780262 (2001k:11086)
  • 21. G. Shimura, The representation of integers as sums of squares, Amer. J. Math. 124 (2002), 1059-1081. MR 1925343 (2003i:11049)
  • 22. G. Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups, Mathematical Surveys and Monographs, vol. 109, Amer. Math. Soc., 2004. MR 2027702 (2004m:11068)
  • 23. G. Shimura, Inhomogeneous quadratic forms and triangular numbers, Amer. J. Math. 126 (2004), 191-214. MR 2033567 (2005a:11047)
  • 24. G. Shimura, Integer-valued quadratic forms and quadratic Diophantine equations, preprint 2005.
  • 25. C. L. Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math. 36 (1935), 527-606 (= Gesammelte Abhandlungen I, 326-405). MR 1503238
  • 26. C. L. Siegel, Über die analytische Theorie der quadratischen Formen II, Ann. of Math. 37 (1936), 230-263 (= Gesammelte Abhandlungen I, 410-443). MR 1503276
  • 27. C. L. Siegel, Über die Zetafunktionen indefiniter quadratischer Formen II, Math. Zeitschr. 44 (1939), 398-426 (= Gesammelte Abhandlungen II, 68-96). MR 1545778
  • 28. C. L. Siegel, Zur Theorie der quadratischen Formen, Nachr. Akad. Wiss. Göttingen, Math.-phys. Klasse, 1972, Nr.3, 21-46 (= Gesammelte Abhandlungen IV, 224-249). MR 0311578 (47:140)
  • 29. G. K. Stanley, On the representation of a number as the sum of seven squares, J. London Math. Soc. 2 (1927), 91-96.
  • 30. A. Weil, Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1-87 (= Collected Papers III, 71-157). MR 0223373 (36:6421)
  • 31. A. Weil, Number Theory, Birkhäuser, Boston, Basel, Stuttgart, 1984. MR 0734177 (85c:01004)

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Additional Information

Goro Shimura
Affiliation: Department of Mathematics, Princeton University, Princeton, NJ 08544-0001

DOI: https://doi.org/10.1090/S0273-0979-06-01107-4
Received by editor(s): October 26, 2005
Published electronically: April 14, 2006
Article copyright: © Copyright 2006 Goro Shimura

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