Ramanujan graphs and Hecke operators
HTML articles powered by AMS MathViewer
- by Arnold K. Pizer PDF
- Bull. Amer. Math. Soc. 23 (1990), 127-137
References
- Frédéric Bien, Constructions of telephone networks by group representations, Notices Amer. Math. Soc. 36 (1989), no. 1, 5–22. MR 972207
- Béla Bollobás, Graph theory, Graduate Texts in Mathematics, vol. 63, Springer-Verlag, New York-Berlin, 1979. An introductory course. MR 536131, DOI 10.1007/978-1-4612-9967-7
- J. Brzezinski, On automorphisms of quaternion orders, J. Reine Angew. Math. 403 (1990), 166–186. MR 1030414, DOI 10.1515/crll.1990.403.166
- F. R. K. Chung, Diameters and eigenvalues, J. Amer. Math. Soc. 2 (1989), no. 2, 187–196. MR 965008, DOI 10.1090/S0894-0347-1989-0965008-X
- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
- M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 320, Springer, Berlin, 1973, pp. 75–151. MR 0485698
- Hiroaki Hijikata, Explicit formula of the traces of Hecke operators for $\Gamma _{0}(N)$, J. Math. Soc. Japan 26 (1974), 56–82. MR 337783, DOI 10.2969/jmsj/02610056 [H2] H. Hijikata, Private communication, 1988.
- Hiroaki Hijikata and Hiroshi Saito, On the representability of modular forms by theta series, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 13–21. MR 0357332
- H. Hijikata, A. Pizer, and T. Shemanske, Orders in quaternion algebras, J. Reine Angew. Math. 394 (1989), 59–106. MR 977435
- Hiroaki Hijikata, Arnold K. Pizer, and Thomas R. Shemanske, The basis problem for modular forms on $\Gamma _0(N)$, Mem. Amer. Math. Soc. 82 (1989), no. 418, vi+159. MR 960090, DOI 10.1090/memo/0418
- Yasutaka Ihara, Discrete subgroups of $\textrm {PL}(2,\,k_{\wp })$, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 272–278. MR 0205952 [J] S. Jun, Private communication, 1988.
- Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR 766911, DOI 10.1007/978-1-4684-0255-1 [L] W. Li, Abelian Ramanujan Graphs, 1989, preprint. [LPS1] A. Lubotzky, R. Phillips, and P. Sarnak, Explicit expanders and the Ramanujan conjectures, Proc. of the Eighteenth Annual ACM Sympos. on Theory of Computing 18 (1986), 240-246.
- A. Lubotzky, R. Phillips, and P. Sarnak, Hecke operators and distributing points on the sphere. I, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S149–S186. Frontiers of the mathematical sciences: 1985 (New York, 1985). MR 861487, DOI 10.1002/cpa.3160390710
- A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261–277. MR 963118, DOI 10.1007/BF02126799 [M] G. Margulis, Manuscript in Russian on graphs with large girth, 1987.
- Arnold Pizer, On the arithmetic of quaternion algebras. II, J. Math. Soc. Japan 28 (1976), no. 4, 676–688. MR 432600, DOI 10.2969/jmsj/02840676
- Arnold Pizer, The representability of modular forms by theta series, J. Math. Soc. Japan 28 (1976), no. 4, 689–698. MR 422162, DOI 10.2969/jmsj/02840689
- Arnold Pizer, Theta series and modular forms of level $p^{2}M$, Compositio Math. 40 (1980), no. 2, 177–241. MR 563541
- Arnold Pizer, An algorithm for computing modular forms on $\Gamma _{0}(N)$, J. Algebra 64 (1980), no. 2, 340–390. MR 579066, DOI 10.1016/0021-8693(80)90151-9
- Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
Additional Information
- Journal: Bull. Amer. Math. Soc. 23 (1990), 127-137
- MSC (1985): Primary 05C35; Secondary 11F25
- DOI: https://doi.org/10.1090/S0273-0979-1990-15918-X
- MathSciNet review: 1027904