A Tower of Ramanujan Graphs and a Reciprocity Law of Graph Zeta Functions

[ 1 ] N. Alon, Eigenvalues and expanders, Combinatrica 6, No. 2: 83–96, 1986.

[ 2 ] N. Alon and V. Milman, λ1, isoperimetric inequalities for graphs, and superconcentrations, J. Combin. Theory, Ser B 38: 73–88, 1985.

[ 3 ] A.O.L. Atkin and J. Lehner, Hecke operators on Г0(m), Math. Ann., 185: 134–160, 1970.

[ 4 ] H. Bass, The Ihara-Selberg zeta functions of a tree lattice, Intern. J. Math., 3: 717–797, 1992.

[ 5 ] R.F. Coleman and B. Edixhoven, On the semi-simplicity of the Up -operator on modular forms, Math. Ann., 310 no.1: 119–127, 1998.

[ 6 ] H. Darmon, F. Diamond and R. Taylor, Fermat’s Last Theorem, Currents Developments in Mathematics, 1995, Intenational Press: 1–154, 1994.

[ 7 ] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular Function of One Variables II, Springer Lecture Notes 349: 143–316, 1979.

[ 8 ] B. H. Gross, Heights and the special values of L-series, C.M.S. Conf. Proc. 7: 115–187, 1987.

[ 9 ] K. Hashimoto, Zeta functions of finite graphs and representations of p-adic group, Adv. Stud. in Pure Math. 15: 211–280, 1989.

[ 10 ] H. Hida, A p-adic measure attached the zeta functions associated with two elliptic modular forms I, Inventiones Math. 79: 159–195, 1985.

[ 11 ] J. W. Hoffman, Remarks on the zeta function of a graph, Proc. Fourth Inter. Conf. Dynam. Sys. and Diff. Eq. 413–422, 2003.

[ 12 ] Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan, 18: 219–235, 1966.

[ 13 ] N. Katz and B. Mazur, Arithmetic Moduli of Elliptic curves, Ann. of Math. Stud., Princeton Univ. Press, 1985.

[ 14 ] W. C. Li, New forms and functional equations, Math. Ann. 212: 285–315, 1975.

[ 15 ] W. W. Li, Character sums and abelian Ramanujan graphs, J. Number Theory 41: 199–217, 1992.

[ 16 ] W. W. Li, Zeta and L-functions in Number Theory and Combinatorics, CBMS Regional Conference series in Math. 129, AMS 2019.

[ 17 ] A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8: 261–277, 1988.

[ 18 ] J.-M. Mestre, La méthod des graphes. Examples et applications, Proc. Int. Conf. on class numbers and fundamental units of algebraic number fields, Katata, Japan, 217–242, 1986.

[ 19 ] M.R. Murty, Ramanujan graphs, J. Ramanujan Math. Soc.1, 1–20, 2001.

[ 20 ] A.K. Pizer, Ramanujan graphs, Computational perspectives on number theory (Chicago, H. 1995), AMS/IP Stud. Adv. Math. 7, Amer. Math. Soc. Providence, RI: 159–178, 1998,

[ 21 ] K.A. Ribet, On modular representation of Gal(Q/Q) arising from modular forms, Inventiones Math., 100: 431–476, 1990.

[ 22 ] J.H. Silverman, The Arithmetic of Elliptic Curves, 2nd edition GTM 106, Springer, 2008, ISBN 9780387094936.

[ 23 ] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, Princeton Univ. Press, 1971.

[ 24 ] H.M. Stark and A.A. Terras, Zeta functions of finite graphs and coverings, Advanced in Math., 121: 124–165, 1996.

[ 25 ] K. Sugiyama, Zeta functions of Ramanujan graphs and modular forms, Comment. Math. Univ. Sanct. Pauli 66, 1–2: 29–43, 2017.

[ 26 ] R.M. Tanner, Explicit construction from a generalized N -gons, SIAM. J. Alg. Discr. Math. 5: 287–294, Press 1984.

[ 27 ] A. Terras, Fourier analysis on finite groups, London Math. Soc., Cambridge Univ. Press, 1999.

[ 28 ] A. Valette, Graphs de Ramanujan et application, Sém. Bourbaki 1996-97, n◦ 829, Astérisque: 247–276, 1997.