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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
Posted:
April 14, 2006
MathSciNet review:
2223009
Full-text PDF
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Similar Articles |
Additional Information
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- 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.
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- 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,
-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.
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- 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.
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- 19.
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- 20.
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- 21.
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- 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.
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- 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)
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Additional Information
Goro Shimura
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544-0001
DOI:
http://dx.doi.org/10.1090/S0273-0979-06-01107-4
PII:
S 0273-0979(06)01107-4
Received by editor(s):
October 26, 2005
Posted:
April 14, 2006
Article copyright:
© Copyright 2006 Goro Shimura
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