|
Continued fractions and modular functions
Author(s):
W.
Duke
Journal:
Bull. Amer. Math. Soc.
42
(2005),
137-162.
MSC (2000):
Primary 11Fxx, 11Gxx
Posted:
January 25, 2005
Retrieve article in:
PDF DVI PostScript
References |
Similar articles |
Additional information
References:
-
- [An1]
- G. E. Andrews, An introduction to Ramanujan's ``lost" notebook. Amer. Math. Monthly 86 (1979), 89-108. MR 0520571 (80e:01018)
- [An2]
- G. E. Andrews, Ramanujan's ``lost" notebook. III. The Rogers-Ramanujan continued fraction. Adv. in Math. 41 (1981), no. 2, 186-208. MR 0625893 (83m:10034c)
- [Art]
- E. Artin, Galois theory. Edited and with a supplemental chapter by Arthur N. Milgram. Reprint of the 1944 second edition, Dover, Mineola, NY, 1998. MR 1616156 (98k:12001)
- [Ask]
- R. Askey, Orthogonal polynomials and theta functions. Theta functions--Bowdoin 1987, Part 2 (Brunswick, ME, 1987), 299-321, Proc. Sympos. Pure Math., 49, Part 2, Amer. Math. Soc., Providence, RI, 1989. MR 1013179 (90h:33006)
- [BC]
- S. Barnard and J.M. Child, Advanced Algebra. Macmillan, London, 1939. MR 0001185 (1:193a)
- [Be]
- B. C. Berndt, Ramanujan's notebooks. Parts I-V. Springer-Verlag, New York, 1985-1998. MR 0781125 (86c:01062), MR 1486573 (99f:11024)
- [BCZ]
- B. C. Berndt, H. H. Chan and L-C. Zhang, Explicit evaluations of the Rogers-Ramanujan continued fraction. J. für die reine und angew. Math. 480 (1996), 141-159. MR 1420561 (98c:11007)
- [Ber]
- W. E. H. Berwick, Modular invariants expressible in terms of quadratic and cubic irrationalities. Proc. London Math. Soc. (2) 28, 53-69 (1928).
- [Bir]
- B. J. Birch, Weber's class invariants, Mathematika 16, 283-294 (1969). MR 0262206 (41:6816)
- [Bor]
- R. E. Borcherds, What is Moonshine? Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. 1998, Extra Vol. I, 607-615. MR 1660657 (99j:17001)
- [BCHIS]
- A. Borel, S. Chowla, C. S. Herz, K. Iwasawa and J-P. Serre, Seminar on complex multiplication. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. Lecture Notes in Mathematics, No. 21, Springer-Verlag, Berlin-New York, 1966. MR 0201394 (34:1278)
- [Bra]
- R. Brauer, Galois Theory, 1957-1958, Harvard Lecture Notes.
- [Cox]
- D. A. Cox, Primes of the form
. Fermat, class field theory and complex multiplication. John Wiley and Sons, Inc., New York, 1989. MR 1028322 (90m:11016) - [Dar]
- H. B. C. Darling, Proofs of certain identities and congruences enunciated by S. Ramanujan. Proc. Lond. M. S. (2) 19, 350-372 (1921).
- [Deu]
- M. Deuring, Die Klassenkörper der komplexen Multiplikation, Enz. Math. Wiss., Band I-2, Heft 10, Teil II. B. G. Teubner Verlagsgesellschaft, Stuttgart, 1958. MR 0167481 (29:4754)
- [Dic]
- L. E. Dickson, Modern algebraic theories. Sanborn, New York, 1926.
- [Eis]
- G. Eisenstein, Theorema. J. für die reine und angew. Math. 29, 96-97 (1845) [ #26 in Mathematische Werke. Band I. 289-290, Chelsea Publishing Co., New York, 1975].
- [Elk]
- N. D. Elkies, The Klein quartic in number theory. The eightfold way, 51-101, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ. Press, Cambridge, 1999. MR 1722413 (2001a:11103)
- [FK]
- H. Farkas and I. Kra, Theta constants, Riemann surfaces and the modular group. American Mathematical Society, Providence, RI, 2001. MR 1850752 (2003a:11045)
- [Fol]
- A. Folsom, Modular forms and Eisenstein's continued fractions. Preprint, 2004.
- [Fri]
- R. Fricke, Lehrbuch der Algebra, Band 2, Vieweg, Braunschweig, 1926.
- [Göl]
- H. Göllnitz, Partitionen mit Differenzenbedingungen. J. für die reine und angew. Math. 225 (1967), 154-190. MR 0211973 (35:2848)
- [Gor]
- B. Gordon, Some continued fractions of the Rogers-Ramanujan type. Duke Math. J. 32 (1965), 741-748. MR 0184001 (32:1477)
- [Har]
- G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work. Chelsea Publishing Company, New York, 1959. MR 0106147 (21:4881)
- [Hei]
- E. Heine, Verwandlung von Reihen in Kettenbrüche. J. für die reine und angew. Math. 32, 205-209 (1846).
- [Hir]
- M. D. Hirschhorn, On the expansion of Ramanujan's continued fraction. Ramanujan J. 2 (1998), no. 4, 521-527. MR 1665326 (99i:11006)
- [Jac]
- C. G. J. Jacobi, Allgemeine Theorie der kettenbruchaehnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird, J. für die reine und angew. Math. 69, 29-64 (1869) [in Mathematische Werke, VI, pp. 385-426, Chelsea, New York, 1969].
- [Kl1]
- F. Klein, Weitere Untersuchungen über das Ikosaeder. Math. Ann. 12, 509-561 (1877) [in Gesammelte Math. Abhandlungen II , 321-384, Springer, 1922].
- [Kl2]
- F. Klein, Über die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades. Math. Ann. 14, 111-172 (1878) [in Gesammelte Math. Abhandlungen III, 13-75, Springer, 1922].
- [Kl3]
- F. Klein, Über die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann. 14, 428-471 (1879) [in Gesammelte Math. Abhandlungen III, 90-136, Springer, 1922].
- [Kl4]
- F. Klein, Lectures on the icosahedron and the solution of equations of the fifth degree. Translated into English by George Gavin Morrice. Second and revised edition, Dover Publications, Inc., New York, NY, 1956. MR 0080930 (18:329c)
- [Kno]
- M. I. Knopp, Modular functions in analytic number theory. Markham Publishing Co., Chicago, 1970. MR 0265287 (42:198)
- [Mag]
- W. Magnus, Vignette of a cultural episode. Studies in numerical analysis. 7-13, Academic Press, London, 1974. MR 0347521 (50:24)
- [MM]
- H. McKean and V. Moll, Elliptic curves. Function theory, geometry, arithmetic. Cambridge University Press, Cambridge, 1997. MR 1471703 (98g:14032)
- [Mu1]
- T. Muir, New general formula for the transformation of infinite series into continued fractions. Transactions of the Royal Society of Edinburgh, 27 (1876), 467-471.
- [Mu2]
- T. Muir, On Eisenstein's continued fractions, Transactions of the Royal Society of Edinburgh, 28 (1876-1878), 135-144.
- [New]
- M. Newman, Classification of normal subgroups of the modular group. Trans. Amer. Math. Soc. 126 (1967), 267-277. MR 0204375 (34:4217)
- [Ogg]
- A. P. Ogg, Rational points of finite order on elliptic curves. Invent. Math. 12 (1971), 105-111. MR 0291084 (45:178)
- [Per]
- O. Perron, Die Lehre von den Kettenbrüchen. Chelsea, New York, 1950. MR 0037384 (12:254b)
- [Ra1]
- K. G. Ramanathan, On Ramanujan's continued fraction. Acta Arith. 43 (1984), no. 3, 209-226. MR 0738134 (85d:11012)
- [Ra2]
- K. G. Ramanathan, On some theorems stated by Ramanujan. Number theory and related topics (Bombay, 1988), 151-160, Tata Inst. Fund. Res. Stud. Math., 12, Tata Inst. Fund. Res., Bombay, 1989. MR 1441329 (98g:33027)
- [Ram1]
- S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
- [Ram2]
- S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904 (20:6340)
- [Ram3]
- S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, and Springer-Verlag, Berlin, 1988. MR 0947735 (89j:01078)
- [Ro1]
- L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
- [Ro2]
- L. J. Rogers, On a type of modular relation, Proc. London Math. Soc. (2) 19, 387-397 (1921).
- [S]
- Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale, Math. Ann. 113, 1-13 (1937).
- [Sc]
- B. Schoeneberg, Elliptic modular functions: an introduction. Translated by J. R. Smart and E. A. Schwandt. Springer-Verlag, New York-Heidelberg, 1974. MR 0412107 (54:236)
- [Sch]
- I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, Berl. Ber., 302-321 (1917) [#28 in Gesammelte Abhandlungen, Band II, 117-136, Springer-Verlag, 1973].
- [Se1]
- A. Selberg, Über einige arithmetische Identitäten. Avh. Norske Vidensk.- Akad. Oslo I, 1936, Nr. 8, 23 S. (1936) [#1 in Collected papers, Vol. I. With a foreword by K. Chandrasekharan. Springer-Verlag, Berlin, 1989].
- [Se2]
- A. Selberg, Reflections around the Ramanujan centenary, Ramanujan: essays and surveys, 203-213, Hist. Math., 22, Amer. Math. Soc., Providence, RI, 2001 [in Collected papers, Vol. II, Springer-Verlag, Berlin, 1991]. MR 1862753, MR 1001304 (90g:01055)
- [Ser1]
- J-P. Serre, Extensions icosaédriques. Séminaire de Théorie des Nombres des Bordeaux, 1979-1980, Exp. No. 19 [#123 in Collected papers, Volume III, Springer-Verlag, New York, 1985]. MR 0604216 (83d:12003)
- [Ser2]
- J-P. Serre, L'invariant de Witt de la forme
. Comment. Math. Helv. 59 (1984), no. 4, 651-676 [#131 in Collected papers, Volume III, Springer-Verlag, New York, 1985]. MR 0780081 (86k:11067) - [Ser3]
- J-P. Serre, Cohomological invariants, Witt invariants, and trace forms (notes by Skip Garibaldi). Univ. Lecture Ser., 28, Cohomological invariants in Galois cohomology, 1-100, Amer. Math. Soc., Providence, RI, 2003. MR 1999384
- [Shi]
- G. Shimura, Introduction to the arithmetic theory of automorphic functions. Princeton University Press, Princeton, NJ, 1994. MR 1291394 (95e:11048)
- [Sil]
- J. H. Silverman, Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994. MR 1312368 (96b:11074)
- [Sta]
- H. M. Stark, On complex quadratic fields wth class-number two. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 289-302. MR 0369313 (51:5548)
- [Syl]
- J. J. Sylvester, On a remarkable modification of Sturm's theorem. Philosophical Magazine 4, 446-457 (1853) [#61 in Collected Math. Papers, Vol. I, 609-619, Chelsea, New York, 1973].
- [Tot]
- G. Toth, Finite Möbius groups, minimal immersions of spheres, and moduli. Universitext, Springer-Verlag, New York, 2002. MR 1863996 (2002i:53082)
- [Wa1]
- G. N. Watson, Theorems stated by Ramanujan (VII): Theorems on continued fractions, J. London Math. Soc. 4 (1929), 39-48.
- [Wa2]
- G. N. Watson, Theorems stated by Ramanujan (IX): Two continued fractions, J. London Math. Soc. 4 (1929), 231-237.
- [Web]
- H. Weber, Lehrbuch der Algebra, III. Braunschweig, 1908 [reprinted by Chelsea, New York, 1961].
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(2000):
11Fxx, 11Gxx
Retrieve articles in all Journals with MSC
(2000):
11Fxx, 11Gxx
Additional Information:
W.
Duke
Affiliation:
Department of Mathematics, University of California, Box 951555, Los Angeles, California 90095-1555
Email:
wdduke@ucla.edu
DOI:
10.1090/S0273-0979-05-01047-5
PII:
S 0273-0979(05)01047-5
Received by editor(s):
November 4, 2003
Posted:
January 25, 2005
Additional Notes:
Research supported in part by NSF Grant DMS-0355564.
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|